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σeð ¤0
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cf0pQp¥ ¬e6§ñ60¦(^c¤ ^d~
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u1 + u 2 + u 3 + · · · + u n + · · · =
∞ X
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n
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,e^cC
^¤¡p¯ª0¢¼ 6 cI e¥ cVe¥ p6=^d9§¦ n→∞
2n
S2n = (u1 − u2 ) + (u3 − u4 ) + · · · + (u2n−1 − u2n ).
?7ªe¥ cw^c¤ ^d9 e¥ ¥
,ª6p.cô=e^¤,pC ce6¶e^f cf0?¦! ^c=¤ «,p¥ ¬ ¹cpµ ñ6cd#ª S c=C¤,=e6p=6y ¤ ~c=C¤,=e6p=¢M7^d n ,¹^¤ ^^¤ª0 ,¤ª^d S 2n
2n
S2n = u1 − (u2 − u3 ) − (u4 − u5 ) − · · · − (u2n−2 − u2n−1 ) − u2n .
c¾§0¤,
p CdTcdT;,c¶=e6! e^ ^± ,,p»=e7e6¤,ª0pd9CdP ce6S¶d9e^cf cc=f0c? ¦w cw ^c=c=C¤ ¤, =«,e6pp=¥6 ?¬ cw c?^ ¤,== S ^≤,hue^^b#¤=¦0f ª #d c=cd#ª¶ d9^6V ¤ ¥
¥ lim S = S. ° ^0
c ,c S = S + u cñ6cd#ª 2n
1
2n
n→∞
2n+1
2n
2n
2n+1
lim S2n+1 = lim (S2n + u2n+1 ) =
n→∞
n→∞
= lim S2n + lim u2n+1 = S + 0 = S.
b# d9=f ^6 (d c¤ ¤,pCcdT,=e6c= ¡[,§p V¤ e6¥ª0
d9¥ dP(¤ 0c
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c^4¤ ¡ pe¥ c¯p7e¥ 0=¦7¥= ^6d9§ c¦; e¥ c óP ¥n
c→p∞¥ ¬ c ¤0
weA¦ cC
eð b6c¤ ^dP7
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ò
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] >?RpuN u´`=@N pD,:=_h: g¯Y6F,a?D, N N,:?G&Z[> YD¥Y(>ô`=>?@]6Y^ZR=>?:=] Z[>Q>Q _!x =,(>x Y^]^=,Y6R=x Z pD,:=_]^R=>?Y¥>ô`Y¸@R=>W¥>ô\dY6D,:ji N@Nh@ :R=Y6DtDW ºlk~Nt`> (>Q
n→∞
1
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×ß6Ø=ß7Ê?Ë2 Ö ÎÐ× Ñ¥ØQÔòÍPÙÚpÛ0Ô
rn = (−1)n un+1 − (−1)n un+2 + (−1)n un+3 − · · · = = (−1)n [(un+1 − un+2 ) + (un+3 − un+4 ) + . . . ].
cyyf ,?[
ñ6¤,pc ~§^¦we6e^¬f0c^,f0=f~?¦w ^¤ cp ccQ^¡Vc0 ¥ ^,¬ c c=e6w=pC,f0=f r c ¤ ¥
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. ¹¤ C,=fþáI^± «, p , 6eÅ,=e6 Td´e¥ ,ª0,=^d´ ¤ C,=f0l¯ ¤ 0¦ =§^e¥ ♦ cp cQ¡V¬ v = u p w = (−1) Iª0 ^e6¬ =c nµ¸,=e6 Tòe6ª0d9d9t¤0
P (−1) c^¤,= ^ lim v = lim u = 0 <~ ' ( P >x =,x(#0
n
n
n
n
n
n+1
n+2
n+3
n+1
∞
n
n
n
n=1
n→∞
n
n→∞
n
∞ X (−1)n+1
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ñ6e^ce¥ ^c¥
¤c0
p[¬he6ª0,d9ôd9eðc¦±~cC
^ ^cd9c e6^¤ ¬ #§°¦~« ^6 T¤ ¬w¥c=¦h·7e¥ =f6=ª6 dP
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@9 x0=e^e^dPp¤ =^d9T±¤0
[eA¦ cC
eð. c=e^f0cp ¬fªIªC
cp 6c¤ 6(e^^dwªe¥ cpµ d^c¤ ^d9 áI^± «,, ce¥ ¥
cp¥ ¬ ce6¬ 1/(2n) d9c c=c c[ª0T=6V 2
1 = 0. n→∞ (2n)2 lim
¹ ,c cn ¤ =¥
4¥ ^ ¢¯e6ª0d9d#ª¶¤0
d9cQ¡V c7Q= eCp¬0
0
|r4 | <
r4 > 0
cp ¬=·7( e6 c±he(c=·7 f c± :î ùï
? =
|r4 | <
÷| 6¤ªC
cVQ=d96¬ c
,¾*,=e6 ce¥
1 1 1 1 − 2 + 2 − 2 + r4 . 2 2 4 6 8 |r4 | < |u5 |
S = S 4 + r4 =
ó§cA =e^ c[^c¤ ^d9IáI6± «,
S = S n + rn
1 (2 · 5)2
1 = 0,01 100
ø¥óP ¥
cp¥ ¬ c ,=±0
^ ,p~e6ª0d9d&[¤0
S≡
115 ≈ 11,93 9,64
|rn | < 0,01
ìYînmtó<;tó>5ä]äÜuÒ×uÖVàYÐá?8¸ÙuÐäãdÖ]×@Ø@ÓÖRäá?8¸âYçd× ∞ X sin nϕ
n=1
n
.
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R7 L A ì$ ìYü ù ì@óo7"×ÐÙrÙ@ÖRÓäÜ@ådæÐVÒÛrârØrÝ]ÙuÐÏqp±Òß0VÙrØruÐÙ@ÒÛrârØ@ÓÒÙrØ@ÓÚÛ@Ö4G]áWÖRÓå àWÖRäÛ@ÖÜ8RÝ]åYÒ]Óä
ç ÛrârØrÝ]ÙuÐÏdÖRÓf.±ØrârØZãdÜuÒRêsÖÜuÖbÑØ@Ó v = 1/n Ú w = sin nϕ ØådæráWÒ]ÓÚræráWÖ T
n
n
1 = 0, n→∞ n lim
ÐàWä]ÒæuÐVäáRØrærÙrÞÒ äåZÓÓÞ"àWäÛuÖRÓÖ43ÐáWÒÜ8RÙuÖ43OÖ âdçd×Ð ∞ X
7 ê à
sin nϕ
n=1
4Ö 3]âuÐÙrØræ@ÒÙrÞ±êDBáWÖ ÖVÝ]ÙuÐ]æuÐVÒáVÚ@æráWÖØ@ä
ãYÖO×@ÙrÞßÍâdçd×ä
ãYÖO×@ØráWä
çê t 3]âuÐÙrØræ@ÒÙrÙ@ÖRä]á?8 æuÐVäáRØrærÙrÞã&äåZÓÓ¥âYçd×ÐF 7 ê à ÙuÐØ0RÖÜuÒ]ÒÛrâ@ÖRäáWÖÍ×uÖVÏrÐÝÞàYÐVÒáWäçgä Ø@ä Û@ÖÜ8RÝOÖVàYÐÙrØ@Ò]Ó ÏdÖRÓÛZÜuÒÏZäÙrÞãærØ@ä]ÒÜ êl. Òß@äáRàRØráWÒÜ8RÙ@ÖrÚÏdÖRÓÛZÜuÒÏZäÙ@ÖRÒ ærØ@äÜuÖ z = cos ϕ +/ ÛrârØÍàWÖVÝ]àWÒ×uÒÙrØrØà¸äáWÒÛ@ÒÙ 8×ÐVÒá z k = cos kϕ + i sin kϕ ê 2 4Ö 3×Ð i sin ϕ m X
m
zk =
k=0
1 − z m+1 X = (cos kϕ + i sin kϕ). 1−z k=0
áWävu×ÐÛ@ÖÜ@ådæuÐVÒ]ÓÔäÜuÒ×@å!uÊÕ å!uÉÖ4r@ÒÙrÏråDw t
√ X m+1 | m 2 2 2 (cos kϕ + i sin kϕ) = |1 − z ≤ = =√ . |1 − z| |1 − z| |1 − cos ϕ − i sin ϕ| 1 − cos ϕ
7 ê0
k=0
2 ÐÏrØ@Ó&Ö?0VâuÐÝOÖRÓÚYÜuÒàYÐç æuÐVäá?8x 7 ê 0 ÖRäáVÐVÒáWäçÖ43]âuÐÙrØræ@ÒÙrÙ@ÖVߣÛrârØ ÚdÒäÜ@Ø ϕ 6= ±2πn Ú →∞ Ê ê 5 ݦÖ4@ r ÒÙrÏrØ& 7 ê 0 äÜuÒ×@å!u á Ö4r@ÒÙrÏrØ ÚYÖVæ@ÒÙ8 æuÐVäáWÖ±Ø@äÛ@ÖÜ8Rm ] Ý W å ] Ò Ó Þ ¦ Ò ÛrârظÛrârØ@ÓÒÙ@ÒÙrØrØ n Ûrâr=ØrÝ]0,ÙuÐ∞ÏdÖVà E R0 ÒÜ@çØy± . ØrârØZãdÜuÒÊ×rÜ@çàWÒÕÒ]äáRàWÒÙrÙ@Þãâdçd×uÖVà √ X ∞ 2 cos kϕ ≤ √ , 1 − cos ϕ k=0
√ X ∞ 2 sin kϕ ≤ √ . 1 − cos ϕ
7 ê 7
k=0
Wá ÓÒáRØ@ÓÚZæráWÖÖ4r@ÒÙrÏråWÚrÐÙuÐbÜuÖ43]ØrærÙrå!uz 7 ê 7 ÚZÓÖOÑÙ@ÖÛ@ÖÜ@ådærØrá?8@ÚdØÙ@ÒÊÖ?0VâuÐÕÐçräv8Ï£ÏdÖRÓ ÛZÜuÒÏdäÙrÞÓ ærØ@äÜÐVÓê4. Òß@äáRàRØráWÒÜ8RÙuÖrÚRàRÞærØ@äÜ@Ø@ÓòäåZÓÓå S ä-ådæ@ÒáWÖRÓ â@ÒÝ]åbÜ8áVÐáWÖVà±Ûrâ@Ø@ÓÒâuÐ / m M êà w t
m X
Sm =
m
sin kϕ =
k=1
k=1
=
t
1 2 sin(ϕ/2)
m h X k=1
áWävu×Ð×rÜ@çàWä]Òã
X 1 ϕ 2 sin sin kϕ = 2 sin(ϕ/2) 2
h 1 i 1 ϕ 1 1 i ϕ − cos k + ϕ = cos − cos m + ϕ . cos k − 2 2 2 sin(ϕ/2) 2 2
Ú
ÚZØ@ÓÒ]Ò]Ó
x 6= 2mπ m = 0, ∞ m √ X 2 2 sin kϕ ≤ |Sm | = =√ . 2| sin(ϕ/2)| 1 − cos ϕ
e¥ =(l¥= ,^hd9§C,¦ =f c ^¤ ^d9^ c^c¶¤ 0
w÷ n0úq?øÉe^ce6p= d²¤0
=e^cp ¢M §¦!¥ !6^c X ÷ n0 jø |u |. |u | + |u | + |u | + · · · + |u | + · · · = e¥ ¤ 0
h÷ n0 jøeA¦ cC
eð c¤0
w÷ n0úq?ø,pCT=¢ÉI=e^cp ¢M c 0 ¥^ªe¥ c c / eA,¦pcQC
,T7=¢M d9[eðª e¥ / ce¥ tc ,¡0 y~¤ 0C
=÷ n0e^cpúq? øI¢MeA¦ cCc
eA¦eÅcC
,Ph7¤ d9
eÅ ÷ n0 jø(¤,=eA¦cQ
eð #c!¤0
÷ n úq?ø l¯¥ ^ eA¦cC
,70¦ eðô¤ 0
c7,7=e^cp ¢M cVªe¥ c ceA¦ cC
,7 ^eðôc ^ ¬e6ª 7¥µ e6eA¦cQ
,^ 7c ^°eðte^ ¤c0
Te^cAc
± he6f0=¶ff0ªce¥ ^c §cw¦eA¦e6cQª0
,d9d7 ^^eð¤ t^ c^eðf c=eðc¤ Tc?d9 t¬f0côV,ñ6¶0=¦»e^e^cp c¢M± e6 c (cp ?
=¢MQ k=1
∞
1
2
3
n
n
n=1
× ß6Ø=ß7Ê?Ë2 Ö ÎÐ× Ñ¥ØQÔòÍPÙÚpÛ0Ô <~ ' (P xEH x ?e^e¥ ¥
cp¬V,[=e^cp ¢Mª0¢ ôªe¥ cª0¢ eð¦cC
d9ce6¬V¤
1
BRN
∞
X 1 1 1 1 1 (−1)n−1 ; a) 1 − + − + · · · + (−1)n−1 + · · · = 2 3 4 n n n=1
ø ø
∞ X 1 1 1 1 1 1 1 1 − − + + − − + ··· = (−1)n(n−1)/2 ; 2 3 4 5 6 7 n n=1 r r √ 4 3 1 5 2 1 6 5 1 7 3 3 1 + + − − 2− − 4 3 3 4 2 5 5 6 6 r ∞ n/2 X 1 8 7 n(n−1)/2 1 n + 1 + ··· = . − (−1) 7 7 n n n=1
d9cC @
,9ªQ ^± # ?x =Pø¦¥?=e^¤ e¥d9 c¥
, ª ^d9^e^Tf ±~ ±þ¤0¤
~0
c7¤, =¤ eA¦cQC
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c«,7peA¥¦ cC¬
c Peð ¤,0
*cC
,øI=f c7p ,¤ 06
ôeð ªe¥ c ceA¦ cC
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,± ª0¢M «, d9eÅ ^ ¾§¤ cd9e^ ^cp ¬=dT^ ªp^=d9f¶eð¶f= f¶¤ ¤C0,
ô=f0 cpd» , lI6 e𤠶0C¦ , ?=µ cp cQ¡V vn =
k^d ¤ ~ñ6cdT,c
1 , n
wn = (−1)n(n−1)/2 . 1 = 0, n→∞ n
,=e6 T¯e6ª0d9d9 e^ cd9c¥p¥ ¬ c^c¤ 0
lim vn = lim
n→∞
∞ X
c^¤,= ^
n=1
wn =
∞ X n=1
S1 S2 S3 S4
(−1)n(n−1)/2 = 1 − 1 − 1 + 1 + 1 − 1 − 1 + . . .
= w1 = 1, = w1 + w2 = 1 − 1 = 0, = w1 + w2 + w3 = 1 − 1 − 1 = −1, = w1 + w2 + w3 + w4 = 1 − 1 − 1 + 1 = 0.
4° e^¢§
e¥ ¥
,ª6p c[¤0
~ øPeA¦ cC
eð ,cC
,=f0c¯ªe¥ c c ce^f cp ¬fªVe^cc=6e6=ª0¢Éµ 7 ±~¤0
w¯=e^cp ¢M §¦~¥ P (1/n) ,f0=f~~[e¥ ,ª0,=¯ø¥ ¤,=eA¦cQ
eð C,=f cø#dþ¾½4ñ6c¥d© , e¥ ,l(ª0, ,=ôMñ6 ¤ c^c[C,Q==f áI·76± ^d e^«, c=d9=cf¥¡pÉ¥ ^¬ ¤ T ±wd9¤^ 0
dT0¾§ce^ cp ¬=^ª^d9eðV ¤ 0µ ∞
n=1
∞ X
1 (−1)n(n−1)/2 , n n=1
e^c,?
=¢M7 ±~e(¤0
cdþ ø¥ ~ ce¥ ¥
cp¥ ¬ ce6¬ vn =
n + 1 n/2 n
.
?Ë ØÑ¥Õ Î]¥Ø=ßIßW«QÎÅÑ¥×U ¥Ò ÑMÖÝÎÐ× Ñ¥ØQÒ Ñ(Î¯Ñ Û0Ú+V4ÖZÎ ÚIÙÚpÛÑ¥Ø b#d9=cf c=f0=cfy ce^ ±~côd9cc¥^p¤,=¥ ¬ ^T ±¶ c¤±
y p , 6eðyeð¦cC
,7 d9eÅ I ce¥ ¥
cp¥ ¬ ce6¬ { }|
lim
n + 1 n/2
= lim
1+
BZM vn
4
1 n/2 √ = e, n
c¯^ ¤ 0T
¶±h ø#I¯d9e^ccCc=
,ªQ 6^±~e6e¥= =¶¥=eM^d9 §¤ ¦hCñ6,=cf ^ccdt¤ ½40
¥ ,Q= p ,e^ T6=eðy6eAeŦhcQ
,[70
d9 eð #0
e^ce6==pµ n→∞
n
n→∞
∞ X 1 n/2 1 . 1+ n n n=1
ó§¤,= p¯ñ6c=¤0
[ c4 ¤ ¥
¥ ¬ cd#ª¯ ¤ C,=fª¯e^¤,= ^ 7e§¤,=eA¦cC
, 7 d9eð¤0
cd P (1/n) ∞
n=1
lim
h1
1+
1 n/2 . 1 i 1 n/2 √ = e, = lim 1 + n→∞ n n n
Qb#==f0f ¢4dº,c=^¤,dTp;Cc=dÉcô ¤¤00
w
; e^nø¥c e6f0p==f~p ~^ ¤ 0T
±ø§~d9cQ
,ø¥ªC eð¦^±cC
e¥ =eÅ¥~=^ªd9e¥ §c¦ ¤c 0
V ø¥;¤,=eA¦cQ
eð n→∞
~±
) z +- !§! *] + $ § * _| * ´
?e^e¥ ¥
c= ¯cQ
c^cV¯p¡V ^±·70¦hc ¤ ce^c[õc¤,p6 !heA¦ cC
e67^e¥µ f « c ^, ? §¬ ¦c¤(0¤,
pc6( [ f cI ^d9 6 ¡[§
,¦ªôe6=ª0d9e^d!c? ¢M, Q cV ¶^d©ªe¥eò c^c ¤ c^eAd9¦cQ
,ªe67= =d9,= peð h ¤0=
¢M=7d9C ±V ¤ 0µ %& ' Q >x =,xIHap]^> ºTZD>y] >Q< ¹N,F] ¶@ ,<¶R=]^Y ÅCB D,>V`p@Y6
x;¹ªe¥¬V¤ 0
t÷ n0úq?ø§eA¦ cC
eð!=e^cp ¢M c ? = ∞ X
|un | = T.
°c=C,Q dº c c¤0
fªô cp cQ¡V¥ ¬ TIe¥ =¥=^d9TIñ6c^cV¤0
n=1
v1 , v 2 , . . . , v n , . . . ,
c=¤ «,p¥ ¬ É e/ ¥ S mµ ô,pe6 ,phe6ª0−w d9dP[, −w ¤ 0
~, .÷ .n0.ú,q?−w ø¥c, .^.(. d9c?¡V c ¤ ¥
e6=p¬V[0
X X X ÷ júq?ø w , v − u = S = pA
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þe6cQc=
C^¤,¤=¡Ve6p= f ©c ^ e¥ > c ôm > e¥ ce¥ e¥ l=6 =¥kµ d9§¦cC
c^c~C,=f0.c~e^=ª =« »e6ª 7^e6^ chª0 ¤ c==6eÅø¥¾ e^0 ,ª=e^cp ¢M c± 1
2
n
m
n=1
n
n
n
m
k
l
m
n=1
n=1
× ß6Ø=ß7Ê?Ë2 Ö ÎÐ× Ñ¥ØQÔòÍPÙÚpÛ0Ô eAe6¦ª0cQd9
d d9ce¥V¤ 0
[÷ n0úq?ød9c c=c c(c=C¤,=e6p=¢M7 M ce¥ ¥
cp¥ ¬ ce6V,=e6 ò¦ X Xw v c^¤,= ^ e^^¤¦0ªô e¥ cd T C cc=C,Q,=6?cc¤0
X Xw v eA¦cQ
,eðh ,e¥ ¥
cp¥ ¬= c X X X ÷ jùgø w . v − u = b#=f dºc¤,pCc=dÉ ^¤ pô,=e6¬^c¤ ^d9
cf0pQ=, ¤ 06 ¦cCªQ
^ddfô^ª ^¤ ^¬»¤¡[e^
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¼¤ ÷0n0
úq? ø7eð¦cC
eÅþªe¥ c c §¾´ñ6cdKe¥ ,ª0,=hd9 X Xw v
cp ,¡V £ò¬º¤,=eA¦ cC
,7 d9 eð Tl¯^± e66 ¬ c Écpµ¸ ^¤ §¦;É^e¥ £ñ6 ¤0
eA¦cQ
0 e6¬ c¤0
»÷ n0úq?ø9§ ~²=e^cp ¢M ceA¦ cC
,7 d9eð c ¤ c= c¤ ^ ªe¥ cpµ ¢¯?¾§cpµ¸=c¤ §¦;?^e¥ 7cp ¬f c4cC
¯¤0
?,= ¤ d9^¤ P v Q¤,=eA¦ cC
0 eð =ò
¤ª0^c± eA¦cQ
0 eð cw¤0
÷ n0úq?ø¥.f=f©d9Àª00
d ¡=§ À¤,=eA¦cQ
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¥ = ¤ t ^c^¤,= ^ cd c=C¤,=e¥== t e^¥ l k e6ª0d9dP P v ^c^¤,= ^0µ cyc=C¤,=e6p=6? Vc¶¤ ^d#wf=fwe6ª0d9dP P w e6¤ ^d9eÅfwf c ^ cd#ªw ¤ ¥
¥ ,ª.¾
¤ 6^ªCeð ¬Q&=pc©[,e^ =e6c ,¤ pc=e6 ª0d9c¤ dP^ ÷ j©úq?ªøÉe¥ ^cc^ ¤,¢¯=§ ° e6^p = 6côeÅþc=C¤,¤ =6e6p =±*6~=w¤ ¤ ,0=
þQ÷&n0f úq?cøÉA
¤,=eðc¦cp µ ¤0
b#=¤,f = eðd¦cCc
,¤,peÅC c dT¦c=ªQe¥h 0c¦ô ¤,c[pCeA¦ cCc
,e6¬7d9 ± c?eð¡h6¤0
hò ¤ ¬ f c« ^ , ?c ±»¬ ÷|e^cdT ,¥ ¤ ¬= 6d9~^¤,¤!=g0e^e^úq?dPø¥p ¤ 0µ p¬hf0=f©e6ª0d9d#ª!
=ª¦eð¦cC
,70¦ eðt¤0
c cC
©Vf c=c¤ §¦»e^ce6c!V cp cQ¡V0µ ¥ ¬? (§¦~^ce¥¤ ^=d96=6dPj&úq(¦ô
ò= ^ f0c=^6c[y¤ c0
^0,
7
c¤ ª0^c±¶öIc=¤ «,p¥ ¬ ò¦; 1
BRS
l
k
n
n
n=1
n=1
∞
∞
n
n
n=1
n=1
∞
∞
∞
n
n
n
n=1
n=1
n=1
∞
∞
n
n
n=1
n=1
∞
n
n=1
l
n
n=1
k
n
n=1
+e¥ = ¥="9^d9 § ¦~ ¤0Q
x>h=,x>÷=,n0x úq?/ ø¥e¥ ? =eð¤¦0cC
÷ n0eÅ *jø¥¤0c
PeA¦e^cQc
e6==peð ~^C , =Tf0±*c 6!¤ ^=d9^e^ cp, T¢M±w ¤§0¦
÷ n0¥ú q? ø¥ l¯¤ª0^ d9he¥ c=d9 ,=e^cp ¢M ceð¦cC
,7 ± eÅh¤0
weA¦cQ
eð =e^e^d9c=¤ d^ ^¤ ¬ .f 6d9ªw ¤ cC
,ô ^¤ ^e6== c=f ©eÅ =6=^d9§¦þ÷ý? = Cd9^ ¥µ (0¦ô c¤0
fø§^e^f c ^ c^cV¤0
%&DY! D, :¥ @ 'rÇ :dQ ºTEx D Zx¯Y2´¥>t:=ap]]^> >Q ?D]6>hZ] N >QN < D¹[Y
Y ((] Y6!Du@ ºT,Z
?Ë ØÑ¥Õ Î]¥Ø=ßIßW«QÎÅÑ¥×U ¥Ò ÑMÖÝÎÐ× Ñ¥ØQÒ Ñ(Î¯Ñ Û0Ú+V4ÖZÎ ÚIÙÚpÛÑ¥Ø BQà Ue^,Q, ? 4 +-c= e^cpx ¢Ml( , c*
eðc¦fcCp
,Qp7¥ ± ¬eÅe6¼¤ 0
^÷ ¤ n0úcq?øô± ,d9=^e66²¼cp^ c¬¤ f ^c*d9 cp ¤ c?¥¡V
cp¥ c?¬¡V òd e¥ =¥=^d9T=P°c=C,? d ^¤ 6 S mµ¸,=e6 ª0¢õe6ª0d9d#ª» eð¦cC
c^c¤0
#h ^¤ 6 ¸ µ , = 6 e
0 ª ¢ 6 e 0 ª 9 d # d ª ¤ 0
p c , 0 ª
^
c ^ ¯ c M
^
¤ = c , ? , ? ¬
c ^ ( c
¤ c C p c ¬
c ±
¥ µ T k ¤ ^e6p= ccf c,±=e6^^ cô ,e¥ p=ô6e6=ª0^d9d9dP§¦;Ai e^e¥cC
c ^¤k¡?e^ ^=A
e^¬ô d9cQ¡V0 ?c¶ = TT¤, pcp ¬ô0, =ce6e^¬[cp ¬^f ¤,c~===^cp e6¬=·7=c m 0ó»
¤ª0^c±Ve6c¤ c 0S c( e¥ ,ª k d9cQ¡V c(TT¤,p¬I,=e6cp ¬f c¯cp ¬=·7cÉ e¥µ S ≤ T c l ?c,btcA,
= e6 Tòe6ª0d9d9 T e^cC
^¤ ¡[? e^¬4 S Q? =pT cp 0 ce^¬ ^¤,==^ e6=c T ≤S ÷ j ø T ≤ S < S, S ≤ T < S. ?y÷ j ø§~e6ª 7^e6c= h ¤ ¥
¥ e¥ ¥
,ª6ye6ª 7^e6c= ¯ ¤ ¥
¥ lim S = S ? = ¤0
e7 ¤ cCcp ¬= c¶Cd9^ ^ Td c¤0
f cd²e6ª0d9d9 ¤ c= wcQ¡[ eAlim ¦cQ
Teð=h~S ^^ce6ª0d9dP[¤,=,e6ª0d9d9I ^¤ c,Q,? ¬ c^c¤0
= ^ e ^ e 9 d = c
¤
´ d ^
^
¤ t ¬ c = 7
± ¥ e , 0 ª , = ± § f c ð
© = ^ e p c M ¢
t c A e ¦ Q c ,
7
± ð e * ¤ 0
÷ n ú ? q ø e^¤ cC^
d9^¤¡Vjúq ~^^f0c7=fd9cQ ¡Vcp c?cI¡V ¤ ¥¥
e6¬= =T= ¬p=¯f0
c=¤ ÷ j ù«,gø¥p0¹¥ ^¬¤ ^Ée6p= e¥ c=¥f =^e¥d9 T==¥=^ó§d9c§A ¦ =e^ cô=ªC
^6cp µ e
^cTc=±ô¤60
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d9ce6» d9^ 6h0¦e6ª0d9dTóP ¥
cp¥ ¬ c ^¤ 6ve6== cwpµ f0~e¥ =¥=^d9§¦ u ¤ 0
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,7 ± e𶤠0
¶¯c= c=·7^ ¶ ^¤ ^d9^e6¥ ¬ c^c e^c± ¹e6^¤ ^[±0
¥^
d6Vf che^^=Q~c¤ c±©cp , =(e6f0©=f~f ^cc¤ ^^d9 ,p~¹e6cpª0 d9cQdP¡V ¥ ¬ Td e¥ cd cc=^,? d e6ª0d9d#ªf»f c=c¤ c±»eV cd9c=7¬¢ö ^f c=c¤ c±» ^¤ ^e6== cf te¥ =¥=^d9§¦»A¤ 0
~d9 ¦cpµ d e^^e6© eA¦cQ
ª0¢ëe6ª0d9d#ª
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e¥ d9c cc? e6 c~c?h¡VeA¦¤ cC0
,
¥ ~7¬÷ ^n0T^úcq?ø§eÅ©0 d9¤^0^ ^
d ¤÷ 0n0
úq?ø¥ . ¹ª−we6¬ f0=c=f¤ 0 µ { }|
m
k
m
m
k
k
k
k
l
l
k
l
m
m→∞
k→∞
k
m
k
n
n
n
n
n
n
n
n
lim |un | = 0
e¥ ¥
cp¥ ¬ c lim v = lim w = 0. m¤ cd9c^c f0=fôe¥ ¥
,ª6y(^c¤ ^d9júq,¤0
n→∞
n→∞
n
n→∞
∞ X
÷ j 3 ø
n
÷ j0ùnø
vn
n=1
÷ j0 jø p , ¢Með[¤,=eA¦cC
,7 d9 eÅ pl7? ^T ^¤ 6 σ δ cc=C,Q dc=¤ 6Cf ¤0
c[÷ j0ùnø ©÷ j¾§ jTø¥ ^e^¤ c^c=d 6^¤ e6c,^ Q,,c ? ¬ T±[c=¤ 6Ccf C ceA
¥ p¬©e60 ,ª¤,=eA¦ cC
d9ce6¤0
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d9pcQ=¡[f µ
cd9c²e6¤,pC¤ 0c
e6¬ ÷ jσ jø¥,l¯−cδ= e67dþpQ?p ^ ^^dd9¤ ^¬p c=¬=p·7 ¤ cp «,AcQp¡V 6 ¬¥c pT¬= ±fT¢y¡±c=¢ô¤ c=c=6CC¤ d9c6c?f C¡Vcf c¯σe^p0= ,pf ªy=f ¤, =eðc¦cpcp µµ σ −δ +σ >A δ cp ¬=·7c^cV e¥ h p = h f
? ^ = ; ¹ c ¥ e (
c 6 e = ? c
ô c σ − δ + σ − δ < A ·=^cf0c? ^= þ,=e6 §¦e6ª0d9dÀcf cp c e¥ A =ªC
,ª »e^c^¤·p¬eðe^f0cp ¬ ª0^cC
cdP? cd ^¤ ? =,¹ce^f cp ¬fªy7e^0 ,ª÷ j 3 ø v w e6¤ ^d#eÅôf¶ªQ ¢ ¤ ∞ X
wn
n=1
nm
mn
n1 0
n1 0
m1 0
n1 0
m1 0
n1 n2
n1 0
n1 0
m1 0
m1 0
n1 n2
n1 n2
m1 m2
m1 m2
n
n
× ß6Ø=ß7Ê?Ë2 Ö ÎÐ× Ñ¥ØQÔòÍPÙÚpÛ0Ô e6^¤ c^^d9¤,= ¬ eÅw^ f~ cdªQ ¢¯c=C¤,=ce6=ñ6= cy ~ c=0C,
Q^f ,e^=c6?n cVme6?ª0d9cdP7¤, pcCe6dP?¤ ¦7c^f cp c^^cV= p =f ±7 dcQ¡cò¤,=pªCC
c6d ¤0
¤,=, A cô¤ ^c? ce^¬
cf0pQp¬ <~ ' (Q x>=,x4l( ,ôªe¥ c ceA¦ cC
,7^^ceðh¤ 0
X ÷ j0; :ø 1 (−1) n ¯,=ª0±¢¼©e6ª0 d9^d#¤ ^ªôe6=¤=0
c=[fVª;g7¤,pø4Qª0 ¥ =¢M¯ª0¢öe6ª0d9d9ª¤
ôqùn¶¤,pQ. ø4ª0d9^ ¬=·=¢Éµ @9 x§kPe¥ c,pºeA¦ cC
d9ce6¬»¤0
þ÷ j; :ø7c ^0
,9eC=dÀ¤0
¼÷ j; :ø¥9e^cA =e^ c ¤ « ªtáI^± «,#eð¦cC
eÅ &!¥=¤ d9c ^e^f ±þ¤ 0
P n ¤,=eA¦cC
eÅ §°cpµ C,Q dþ ^¤ 6 S e6ª0d9d#ªô¤0
h÷ j; :ø§ôQ= ·76d^^c¤,pC^¤ ª cd0
÷ j oø 1 1 1 1 1 1 1 S = 1 − + − + − + − + ... k67d9( cQcQ¡V
ôd´eð¦÷ jcC
,oø(7, ©± eÅWq heg0¤0¾
¤ 62^ªQ ¬Q3=p¶4 we^50 ,ª©6eð¦cC
7 d9c8e6 ÷ j oø(d9cQ¡^d Q= eCp¬ B0
1
∞
n+1
n=1
∞
−1
n=1
1 1 1 1 1 1 1 1 1 S= − + − + − + − + ... 2 2 4 6 8 10 12 14 16
Ç = ·7^dþ cp ,ª0 ^ T±h¤ 0
h cQ
h eð¦cC
Td¼÷ j oø§e¥ ¥
,ª0¢M dc¤,pCcdT S =1−
1 S= 2
1 1 1 1 1 1 1 1 1 1 1 + − + − + − + − + − + ... 2 3 4 5 6 7 8 9 10 11 12 1 1 1 1 1 1 − + − + − + ... 2 4 6 8 10 12
óP7 cQ ¡V
dh¤ª0I^ ^cC¤
~¬
c¤ª0M^eAc¦ dTcC
,¾²7¤ 6^^eðªC[ ¬?¤p0p
MI ccp0 , ª0^ dc =eAe^¦ ccC¥
,
7 ± 7eðhd9¤ ^0e6
hòe(e¥ e6ª0=d9¥=d9^cd9±Tq=ùn e6Scpµ ÷ j ø 1 1 1 1 1 1,5S = 1 + − + + − + . . . c4¤0
h÷ j ø. cp ,ª0 0 eðI÷ j oø.Iª03fpCT2=65 e^f 7cd#ª04¢ ^¤ ^e6== cfªc=¤ «,p¥ ¬pµ cC
T Tc[e¥c= =¤ ¥= ^«,d9pT¥É ¬^ ¤ c=¥
¥=¢MeÅ[p=f cº ce¥ ò
=ª¦ cp cQ¡V¥ ¬ §¦[e¥ ¥
c? c b^ ^¤ ¬ ,= ¤ c= ^¤ ¥
^d¼c=¤ «,pÅ ¬ Tye¥ =¥=^d9T==f cÀQôcC
µ dþ cp cQ¡V¥ ¬ Tdþe¥ ¥
c? ô
c=¤ «,p¥ ¬ §¦ ,? = ÷ júqQiø 1 1 1 1 1 1 1 1 1− − + − − + − − + ... ¹¤ ^c¤,p^ª^dþñ6c=y¤0
w2e¥ ¥
,4ª0¢M73 d*6c¤,?8CcdT 5 10 12 1 1 1 1 1 1 1 1 − + − − + − − + ··· = 2 4 3 6 8 5 10 12 1 1 1 1 1 1 1 1 − + − + − − − + ··· = = 1− 2 4 3 6 8 5 10 12 1 1 1 1 1 1 − + ··· = = − + − + 2 4 6 8 10 12 1 1 1 1 1 1 1 = 1 − + − + − + . . . = S. 2 2 3 4 5 6 2 1−
b#=f dºc¤,pCc=dÉ ^¤ ^e6p= cfh÷ júqQiø&ª0d9^ ¬=·=6ye6ª0d9d#ªô¤0
[7
¤,pQ
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, 76^ceÅI¤ 0
P e^d9^¤ ^^e60p =[^¤¬ª0 p,= f cce¥ ¥
^c¤ª0p ¥ª ¬p §c¦we¥ c=¥
¤ c «,pp¥¥ ¬=¬ ò§¦ô¦h e¥cp =cQ6¡V=6dP&¥ ¦;¬ §c¦¶e6ª0e¥ d9=dP¥=(−1) ^¤d90§
¦ /n Cd9^ eðwfô¥ q ln 2 + ln(p/q) (=+ w .
c=c@ ¤ 9 p«, ¥p ¬¥ T¬Ç d9T=~ d9 w·7cp e¥^ cQd ¡V=S¥=^d9¥ T¬d9 4 T,,d9=?~e6 =e¥ e^ =cC¥
ª0=^¢*^¤d9¡e6Tª0pd9d9¯~d#ª0ª¢ =ke^c^e6¤ª0c? ¯~ª0e e¥¢ q= ¥ =c^ke¥d9 ^¥§¤
¦;ª0c ?e ¥p ¬ c=Ée¥ d9¥ µ k(p + q) { }|
∞
n
n=1
1 2
k(p+q)
Sk(p+q) =
1 1 1 1 + + + ... + − 1+2·0 1+2·1 1+2·2 1 + 2(pk − 1) 1 1 1 1 − − − ... + . − 2·1 2·2 2·3 2kq
~e^^¤ª0 ¤ c? y0¦[ cIC,pf=dTc¯If c ^ c±Ve6ª0d9d94d9cQ¡V c(eð
¥ p¬¾§¥
^d l = kq ?l¯c= d![T^de¥ =6=^d9TTeTc=¤ «,p¥ ¬ Td9 Cc ,=c=f0C,=Qd9 ^ ¾² ¤ L66ªQ= ¬Qkp−1 =p¯,=e6 ª0¢ e6ª0d9d#ªôd9cQ¡V c ¤ ¥
e6==¬y[0
1 1 1 + + ... + − 1+2·0 1+2·1 1 + 2L 1 1 1 1 1 1 1 − − − ··· − − − ··· − + + ... + = 2·1 2·2 2l 2(l + 1) 2L 2(l + 1) 2L 1 1 1 1 1 = − + − + − ... 1+2·0 2·1 1+2·1 2·2 1+2·2 1 1h 1 1i 1 . ... − + + + ... + 2L 1 + 2L 2 l + 1 L Sk(p+q) =
¹ce¥ ¥
^Ie¥ =¥=^d9c( ¤ ^c¤,p^ª^d*e¥ Å
,ª0¢M dc¤,pCcdT
1 1 1 1 1 1 1 1 + ... + = 1 + + ... + + + ... + − 1 + + ... + . l+1 L 2 l l+1 L 2 l
¾*¤ 6^ªQ ¬Qpp¯,=e6 ,phe6ª0d9dP[ ¤ d96V0
1 1 1 1 1 1 Sk(p+q) = 1 − + − + − · · · − + + 2 3 4 5 2L 1 + 2L 1 1h 1 1 1 1 i . + 1 + + ... + + ... + − 1 + + ... + 2 2 l+1 L 2 l k → ∞
e¥ t^ ^¤ ¬©! cp ,ª0 ^ cd T¤,p¡^ ^¤ ^±þf ¤ ¥
¥ ,ªt ¤
/ cp ¬=Ccp¬=eÅ~¨(c¤ d#ªC =d9»÷ÐqúqQø§»÷Ðq iø¥ cp ,ª0 d
tce¥µ
1 lim ln(kp − 1) + C + εkp − ln kq − C − εkq = 2 k→∞ 1 kp − 1 1 p = ln 2 + lim ln = ln 2 + ln , 2 k→∞ kq 2 q
S = lim Sk(p+q) = ln 2 + k→∞
cô¤ ^c=? ce^¬ cfpQp¬ e¥<~ =¥='^d9 ( §¦ôQ ªEx H e¥x ckf pcQeðp¦cC¬
,p=7f^ª0^¢ ceÅh ^¤¤ 06
e6= = cfª cp c?¡V¥ ¬ §¦ºc=¤ «,p¥ ¬ ò¦ ∞ X (−1)n+1
c ^^ce6ª0d9dP[Cd9^0 =e^¬V¤,pQ n=1
n
,
1
BRB
@9 ó&ª0d9dP eA¦cC
c^cV¤0
[C^e6, ∞ X (−1)n+1
n
×ß6Ø=ß7Ê?Ë2 Ö ÎÐ× Ñ¥ØQÔòÍPÙÚpÛ0Ô
= ln 2 = S 0 .
?h ¤ ¥
§
,ª ^^ct ¤ d9^¤,©C^e6 c òc»¤0
*d9c?¡V cCd9^ ¬»p=f &c 6^c e6ª0d9dP[e6p? [¤,= c± n=1
bcA
p r ln p/q 1 p p 00 = ln 2 log2 2 + log2 = S = ln 2 + ln = ln 2 1 + 2 q ln 2 q r r p p 0 = S log2 2 . = ln 2 log2 2 q q S 00 = n = log2 2 S0
r
p = 3. q
4° ¥ e^¬¢§
Td9 cp ,e¥ ª0= ¥ =dT^d99Td9c p/q
cp ,=¡V 16cþ e¥C ¥
ccc=pC,¬Q,cQ=
6? 9cþc=c!¤ Q w«,p·7¥^ e6¬, ?c
=«,(pÇ =¬d9¢ 6 c?dT Mc?¡V0c µ ¤ d9^¤hjúq p , 6eÅ~,=e6 Tde¥ ,ª0,=^d,=e6cQ7^^cy ¤ d9^¤,0 ^e^f c= e^^e^ d9 c=T¤ ¯ ¤d0
^ ^l(¤ ,¬ôñ6c=Ccd9^cVcQ¡V ¥c
e6^dþ¬ºe¥ ^¥¤
,^ª0 ¢M^e^7C ^¯ c e^¤ c¥
6¥p p^ 6 ¬ c^ce^c± e6º, #0
X ÷ júqq?ø ϑ , ϑ + ϑ + ··· + ϑ + ··· = ª¶f c=c¤ c^c ϑ = u + ··· + u , ϑ^^^=^^u^^^6^^^+^^·^·^·^+ ^^u6^^^,^ , ÷ júq?gø ϑ^^^= ^^u^^^6^^^^^+^^·^·^·^+6^^u^^ ,, ,pCT=6eðhe^^¤ª0 ¤ c= Td*¤0
cdº¤ 0
~÷ n0úq?ø¥ %&÷ j úqq? ø( '] >QQ@ Fw,<¶B ÷ Y¯n0úb] q?Yø¯ ] (>QYC[N] @ `,`,N=@>?Rp:=D,D,E9Fy@ ,< U -+
~c ^0
c , ce^f cp ¬fªô ¤ ¶ ¢4cd n ∞
1
2
n
n
n=1
1
1
2
m1 +1
m1
n
mn−1 +1
m2
mn
ϑ1 + ϑ 2 + · · · + ϑ n = u 1 + u 2 + · · · + u mn
lim (ϑ1 + ϑ2 + · · · + ϑn ) = lim (u1 + u2 + · · · + umn ) = S.
7!^e6^f0#b cc=eðf f ø# d© ¤ cc¶¤,CpCCcpd9c dT^¬
cI ,[0T¦t ¢4¤, =c c¤ ^0c¯T
ÉeAf0¦^cQ9¤
,ª0¹ 7cp ,^ª0^ ce^eð^c¶ e^ ¥¤
ò0 ±
0y¦==÷|e¥If =dc¥d»=^c d9 ¤,§e¥p¦ Cc4d9dc?¡V¤ ª e¥0
ºccQ== ªQf0
c¯ 6eA¢4¦ ,cCeðp
,¦0cp¬ µµ
e^c± ¬e6eðyVc f d9c 6^ ¬¯ =§ª¦~¡e6Mª0d9e6ª0dd9¤,d#=ªe^0 ¤ cc7e6V¤, =eA¦ cC6
eÅTh±¶,[¤0
e^0(¾þeA¦ñ¥cQ
,cdt7e^ d9^Teðhe¥ ¤ 0
e^c 6pp6 ¬ c óP cQ¡V ^Ice6c[
¥ ce(c¤,p Td ^¤ ¥¦cQ
cdT ?4 ¢4e6¤,=« ^±wc=C f0=¢M70¦ 6
^e^¬h¤ªC
c e6¤,=^e^±f ¤ d9òcQ¡ 6¯we^e¥f ,cª ¡Vcfô[¬wf0eAc=¦cQc
,¤ c7dº ± ¤ eðt e^cQ^
¤ª0 f~ ¤,¤ =ceA¦=cC
, T7±t^d#¤ ª0eð
h(1¤ 0−
,ª 1) + (1 − 1) + . . . 1− 1 + 1 − 1 + ... n→∞
n→∞
?Ë ØÑ¥Õ Î]¥Ø=ßIßW«QÎÅÑ¥×U ¥Ò ÑMÖÝÎÐ× Ñ¥ØQÒ Ñ(Î¯Ñ Û0Ú+V4ÖZÎ ÚIÙÚpÛÑ¥Ø BWG c(cmdT¤ 0cd9&c¯c4^c eð¦=cC^
e¥ 7d9cª0e6 ^¶e6f0¬(=f0cC±0^µ e6 ccIT cCT
dP cpe¥ ^¥dP
pc p ^¥e^ f0¬c ^cc(e6=,y?Q Q
Q=4 ª c±y^¤ ¡[ce¥
^¥
cp µ p¥ ¬ ce6! ¯e¥ ¥
,ª6ôeA¦cQ
d9ce6¬¶ ce¥ ¥
^± ;cyd9cQ¡V cVc=d96¬ cy7eð¦cpµ
eA¦d9cQc
e6 c=t^cVe^^¤¤0ª0
h ÷ n0¤ úcq?ø¥= c^c!¤0
t÷ júqq?ø¥cc=7^cc¤ Ve¥ ¥
,ª6eA¦cC
d9ce6¬ <~ ' ( Q Kx Jx(¾²¥=¤ d9c ^e^f cd¤ 0
{ }|
∞ X 1 , n n=1
Td9 ^h¤ e¥^ e6p==¥p= ,^d9ôTd9^^wc[e¥e¥ ¥=
¥c=^?d9 § ¦;,c=Cd9¤ ^ «,0p ô¥ 0¬¦V §C,¦;=f l¯¶cpf0=pf Q p¬ c*cyQ cpp ,ª0 cp^ cQ ¡VT±w¥¤ 0¬p
µ p =ªC
6Veð¦cC
,7 d9eð f0p@ ¡[9
c± , ¹x cp6 ,ªCª0
^ dºdºe6Cª0,d9=d9f c ¤ ^c¤ ¥p
,ª0¬[¢M¤0
~ ± ^eŤhª0 ¤ ,0=
d9ô c p cC
¤ 0
~0
,ª 70¦ôeÅ =6=^d9§¦¶ X X ÷ júqQø 1 (−1) A , A = . (n − 1)p + i #^0c
¤ ^÷ d9j úqQáIø7^ ± p , 6«,eð.q?eAø ¦ AcC
,>70 d95 geð ø §A c>e^f Acp ¬fªº5 cø ºlimªQ
cAp 6=0c¤ 6te^^dÀªe¥ c d ¹ªe6¬ e6ª0d9d&¤0
÷ júqQø¥ ,=e6 ,pe6ª0d9dP e¥ =¥=^d9§¦ eA¦ cC
c^c ^e^^¤ª0 A¤ c4 = c^c¤0
=p6
¥ Sd n4 , p e§ce6ppf0cd r nn = kp+r 0 ≤ r ≤ p−l c±0
,ª »¤ c c e¥ =6=^d9§¦þh¤0
÷ júqQø7 ^¤ §¦þe¥ =¥=^d9§¦ ¤,¾´w=e6e¥^ª0 d9¥e6d#
,ªª0¢Mc S7^±²^¤ª0 A k&e6ª0d9d#ªþf c=c¤ §¦cc=C,Q dK r^¤ 6 A &¹cp ,ª0 d X ÷ júVq 3 ø S − (−1) A = (−1) A . ¹^¤ ^ ±0
,cp ,ª0 ÷ jdúVq 3 øÉ^¤,f==^ e^cpe6 ¢Mc Td ¥ ,=dwª0¥ A
c¶ cp c±e6ª0d9d9 p
∞
n−1
n
n
n=1
i=1
n
n
n+1
n→∞
n
n
n
∗ k+1
k+1
k
i−1
n
i
k
∗ k+1
i=1
∗ k+1
Ak+1
= ^(§ c cf0pQ= c ,c
k X i−1 Sn − (−1) Ai ≤ Ak+1 .
¤
i=1
Ak → 0
k→∞
?y÷ júq?nø§e¥ ¥
,ª6
∞ X (−1)i−1 Ai = A, lim Sn =
÷ júq?nø
cÇ c==Cd9,6Q, =dT6V=eA¦ccQ
¤ d9¯c
e6¤¬Vª0^ ceA±¦cQ^
¤ª0 c^ c ¤ c^e^f0^¤Tª0, = ,¤ ¤ cd9^=¤ cc ^cV¤ 0cC
¤ 0
[0
,ª 70¦[e¥ =¥?µ ^d9§¦;=¤0
=ªC
6C,=f c cp cQ¡V¥ ¬ Td7^^ce6ª0d9dP[÷|^e¥ 2pc,4e6ª 7^e6=ª^ ød9cQ¡[6 ô (e^c,?
p¬e(e6ª0d9d9c±~ eð¦cC
c^cV¤0
<~ ' ( Q Ex P x ?e^e¥ ¥
cp¬V,eð¦cC
d9ce6¬V¤ 0
n→∞
1−
i=1
1 5 n(n + 1) − 1 + 1 − + ··· + 1 − + ... 2 6 n(n + 1)
ô^^ce^^¤ª0 ¤ c= ɱh¤0
1 5 n(n + 1) − 1 1− + 1− + ··· + 1 − + ... 2 6 n(n + 1)
× ß6Ø=ß7Ê?Ë2 Ö ÎÐ× Ñ¥ØQÔòÍPÙÚpÛ0Ô cp @ 9 ^ ~ x c °^ ¤,^=« 0
±~ c f ¤ª0cA e^^§¤¦hª0 e^ f c ¤,f0c?¦ô= ^ ^Éc±~d9cQ¤¡V0
~ ceA¦QcQ=
eCeðp ,¬V [ce^f 0cp
¬fªy ce¥ §µ 1
BRI
∞ X 1 5 n(n + 1) − 1 1 , 1− + 1− + ··· + 1 − + ··· = 2 6 n(n + 1) n(n + 1) n=1
eA¦cQ
d9ce¥¬!f c=c¤ c^ch¤,= ^V§ ~ ^cC
cf ¤,p chªe6p= cp ^,÷|¤,pC Td9©d96cQ
?µ d9ø¥i cÉf=eC=6eð¯ eA¦cC
c^c4¤0
^cMc(¤,=eA¦cQ
eð ? ce^f cp ¬fª4
,I ^^cÉ PT cp µ 6eð! ^cC¦cC
d9T±! ¤ C,=fweð¦cC
d9ce6 l¯^± e6¥ ¬ c ;0 ^ ¤ 0
e¯ ^ 6pµ Td9~ 0
^f eC=d9h¤,= ¥
« y÷ u = 1ø¥ ,e¥ ¥
cp¥ ¬ c 2k+1
lim u2k+1 = 1 6= 0.
Pª0d f0pc=QV°M=
,¤ c p±ôf0d9c ^ ¤~^^c6e¥¤ & ^~d9( ,¤,pe6I ª eA 7¦cC^cQ
,e6
=T^ª¤ ±~6¡¶¤
c0=
h6¤,?,p? ,pcQc[¡V ^c¬[¤ ^dPcp ^j [¡[[^e6c=f 7c^dþ¤ e¥^ ,ª0c,==I =^ c=c¤,p
,0 µ %& 'Q xKJxò]
N@,<©÷ n0úq?ø pD,:=_ >Q`>>CB N,Z[Y
Y6DG(:»Y¥>!] @ `,`,N=@>?Rp:=D,D,E9F@,< ÷ júqq?ø(] >QNV@ ,
x ¹¤ ¶ ¢4cd n ,=e6 ,phe6ª0d9dP¤0
h÷ n0úq?ø S ªQ
c? 6c¤ 6 k→∞
C
n
Sn = u 1 + u 2 + · · · + u n ≤ u 1 + u 2 + · · · + u n + · · · + u m n = ∞ X = ϑ1 + · · · + ϑ n ≤ ϑn .
#b =f d c¤,pCcdT9d9c c=c cc=C¤,=e6==¢Mpþ ce¥ ¥
cp¥ ¬ ce6¬ µ ¦,=e6 0µ ñ6§c¦¶~e6ª0c=d9C,d Q,S=6?¤ 0
ôc÷ n0¤0úq?
ø&c÷ n0^¤,úq?=ø§ eA ¦ cQ^
,[eðe^ ^¤¦0ªôe6ª0d9d9c±ôeA¦ cC
,7^^ceð~¤ 0
nô÷ júqq?ø¥ c ¾Q=f0 ¢4 ^ [¤,=e^e^d9c=¤ dc= ¤ ce7cyeA¦ cC
d9ce6¤ 0
¤ ¥
e6p=p , ¢M^^côe^cpµ c±ô ¤ cC¥
^
=ª¦~¤0
c %&D>^ G Z[ >['@ ,<Q Ex ÷P g0x jòø¶
] ]N[>Q@QD< >Q,FZ[`=] @>QGPN `==@R=N,Y6\
x ó§cA =e^ cª ^¤¡[
^ ¢¼^c¤ ^d9 d9^^d X X X ÷ júqQjø w = u · v =R=S S , A
=,e^cð =e^ cc ¤ ¥
¥ ^ ¢¯ n=1
n
∞
∞
n
n=1
∞
l
m
v
m=1
l=1
wn =
u
n X
uk vn−k+1 .
T[÷ g e6ª0ùnd9øVd9eA ¦ cCe^
cc=eð 6=e6e^=cpª0 ¢M¢M 0c ¦É¤t0
¤ c0
l( ,²° cc= C¤ ,¥Q
¥ d^ ^c¤ e66 S ¤ ¥S
cp cQS¡V ndTµ¸T,=e6cº ¤ 0
K ÷ g0 3 øTõªe¥ c c bcA
;cpµ¸ ^¤ §¦; P |v | = P < ∞ .;cpµ¸=c¤ §¦; e^d9cC
,ªQ k=1
u n
v n
w n
∞
n
n=1
?Ë ØÑ¥Õ Î]¥Ø=ßIßW«QÎÅÑ¥×U ¥Ò ÑMÖÝÎÐ× Ñ¥ØQÒ Ñ(Î¯Ñ Û0Ú+V4ÖZÎ ÚIÙÚpÛÑ¥Ø BRL µ¸,=e6 §¦e6ª0d9 d S c^¤,= ^ ö ^f c=c¤ Td e¥ cd L p=f #c |S − S | ≤ n |S |Ç + ?
|S==·7| ≤e^¬2L e¥ cd ε c ¤ ¥
¥ d cd9^¤ N p=f #c´
, m > N n > m T cp 0 e^¬ ^¤,=^ e6 X X ÷ júWq :ø ε ≤ ε , . u |v | ≤ 2P 4L ccA
7
,~e^¥¦ n ≥ N e^ ¤,=¥
c« ^ f0 { }|
u n
u n
u m
u n
n
∞
k
k
k=m
|Snu Snv
u m
k=N
X n n X m X X n vk − ul ul vm−l+1 = − Rn | = l=1
m=1 l=1
k=1
= |(u2 + · · · + un )vn + (u3 + · · · + un )vn−1 + · · · + un v2 | ≤ ≤ |u2 + · · · + un | |vn | + |u3 + · · · + un ||vn−1 | + · · · + |un ||v2 | ≤ ε ≤ 2L(|vn | + · · · + |vN +1 |) + (|v2 | + · · · + |vN |), 2P
f c=c¤,p~eª0 ¥cd ÷ júqW:ø& ¤ dP=6y0
|Snu Snv − Rn | ≤
ε ε + = ε. 2 2
cc=C,Q,=6?c lim S S = lim R = R = S S , côC ¤ p^ºc=^?c= ¤ c^dPe^¬þ
c^f¤ p^Qe6p==¬ 6 *ò¬þe^ ¤,=¥
c± M^e¥ c¤0
÷ g0 3 øyÀ÷ g0ùnø eA¦cQ
,♦eð~ªe¥ c c <~ ' ( Q Ex Q x((=±h ¤ cCÅ
^ ¯¤0
c ÷ júqQoø b b b 1 + b + + + ... + + ... 2! 3! n! ÷ júqQø a a a 1+a+ + + ··· + + ... 2! 3! n! @ 9 xáI^6f0c¶ª0¥
¬eð ,= ¤ d9^¤ . ¤ cd9c=7 ¤ C,=f0yl7? =d9^¤,;c ¤0
÷ júqQoø4÷ júqQøeA¦cQ
,eð ¤ ^d ¤ a < 0 ¤ 0
²÷ júqQø4eð¦cC
eÅt= pe^ ,c? 6¢Meðp µ eA ¦c cQ9
,b#=7f d9d eðhc¤¤,0p
CccdTdT9ó§ c¤ Ac =Ce^ ¥c
^c ¤ w¥
¤¥0
^c ¢õe6ª ÷7g0^ je6ø¥ = ªd96t^^d
, ¢4§¦ a C
n→∞
u v n n
n→∞
2
3
2
3
u
n
v
n
n
b2 b3 bn a2 a3 an + + ... + + ... 1 + a + + + ... + + ... = 2! 3! n! 2! 3! n! b2 a2 + ...+ = 1 + (b + a) + + ba + 2! 2! h bn n−1 n−2 2 2 n−2 b a b a ba ban−1 an i + + ... + + + ··· + + + n! (n − 1)! (n − 2)2! 2!(n − 2)! (n − 1)! n!
1+b+
÷ jùg=iø =e^e^d9c=¤ dc=7 ±~0 ^h¤0
¤,=c=±~,=e6t÷ jùg=iø¥ / eÅ ~Q= eCp¬V^^c[0
n(n − 1) n−2 2 n(n − 1) 2 n−2 1h n n−1 n−1 n b + nb a + b a + ... + ba + nba +a , n! 2! 2!
× ß6Ø=ß7Ê?Ë2 Ö ÎÐ× Ñ¥ØQÔòÍPÙÚpÛ0Ô ,cô 6 w6e^¤cªQ
c±© c~ ª0 0cd
¥¬ ¬¢Mccô,hT¤,0p
¡[ ^(b += a)e6cQ#7ó¼^ª0 ô6f cd¼?
ñ6¤,pc^cw c ±¤ e^cf0cCf ¥
=.^ ¤ ¥
÷ e6jpùg==ipµø d9cQ¡V c7Q= eCp¬Vf0=f 1
GVN
n
1+b+
b2 b3 bn a2 a3 an + + ... + + ... 1 + a + + + ··· + + ... = 2! 3! n! 2! 3! n! (b + a)n (b + a)2 + ... + + ... = 1 + (b + a) + 2! n!
÷ jùg0q?ø ^ ce^¤ ¥
e6^ c±þ ¤ c^¤ f0c±tª06¡[
=^d9eÅ #c¤0
¤,=c±,=e6 ÷ jùg0q?øIeð¦cpµ
eðV=e^cp ¢M c ¤ ^d© ¤ a = −1 ·7¬(cQ
c(^^c(e¥ =¥=^d9cÉc=Q cIc=(ªQ , 1 + 0 + 0 + ··· + 0 + ...
<~ ' (Q xUT;x(¹cfpQp¬ pc ¤ c=C¥
^ f ?
¤,?ªe¥ c ceA¦cQ
,7 ± eð¤ 0
∞ X
1 (−1)n+1 √ n n=1
^¤ ¥¦cC
y[¤,=eð¦cC
,7 ± eÅ @ 9 x.ó§cð =e^ cc ¤ ¥
¥ ^ ¢ ¤ cC¥
^ w¤0
ch÷ g0 jø¥ d9^^d 1 1 1 1 1 ih 1 i 1 − √ + √ − . . . + (−1)n+1 √ = 1 − √ + √ − . . . + (−1)n+1 √ n n 2 3 2 3 i h 1 1 i h 1 1 1 1 = 1 + 1 · −√ + −√ · 1 + 1 · √ + −√ −√ + √ · 1 + . . . + 2 2 3 2 2 3 n X 1 1 + (−1)k+1 √ (−1)n−k √ + ··· = n − k + 1 k k=1 n 1 X √ 2 2 n+1 p =1− 2+ + √ + . . . + (−1) + ... 2 3 k(n − k + 1) k=1 h
Ae¹¦cpcQ ,
ª0 d9^c e¥ T.±w ¤ 0
!¤,=eA¦cQ
eð ; ce^f cp ¬fª~ ¯T cp 6eð! ^cC¦cQ
d9T±w ¤ C,=f l¯^± e66 ¬ c
, n > 1 1 ≤ k ≤ n d9^6Vd9^e6cV ^¤,=^ e6c ÷ jùggø 1 1 p ≥ , n (n − k + 1)k [e^ ¤,=¥
ce6hf0c=c¤ c^cd9cQ¡V cª0=¥
¬eÅ c=C¥
,~^6cf ?
¤,p 0
1 1 ≥ 2 (n − k + 1)k n n2 − (n − k + 1)k ≥ 0. (n − k + 1)kn2
b#=f¶f=f¶C,=d9^,p¥ ¬Vñ6c±ô
¤ c~ cp cQ¡V¥ ^ ,c ^cC¦cQ
d9c[ª0Å
¬eÅ c ÷ jùg=ø n − (n − k + 1)k ≥ 0. 2
?Ë ØÑ¥Õ Î]¥Ø=ßIßW«QÎÅÑ¥×U ¥Ò ÑMÖÝÎÐ× Ñ¥ØQÒ Ñ(Î¯Ñ Û0Ú+V4ÖZÎ ÚIÙÚpÛÑ¥Ø GdM ¹¤ ô cd9c=7~ ^¤ ^^¤ª0, ¤ cf weÅ =6=^d9§¦~ ¤ 0¦cQ
dþf~c ^0
cd#ª¶ ^¤,=^ e6=ª { }|
cC
,^¤¡[
=¢M7^d#ª»µ¸±ô÷ j0ùg= ^ø&h ¤,0e^
c[c=d9cQ6¡Ve6 c^c « ^c #÷ j¬Vùge¥g ø¥¥
,ª0¢M7 d*c¤,pCcdT ¾²e^0 ,ª÷ jùggø n (n − k)2 + k(n − 1) ≥ 0,
n X
n n X 1X 1 p = ≥ 1 = 1. n n (n − k + 1)k k=1 k=1
1
?d9^ c[ cñ6cd#ªô ^cQ¦cC
d9T±h ¤ C,=f~eð¦cC
d9ce6w (T cp 6eð k=1
¾þQ=f0 ¢4 ^ (¤,=e^e^d9c=¤ dº ¤ d9^¤ ¤ cC
cp ,¡=¢M7 ±¶0 ¢4e6¤,=« ¢¼c ^¤,=« ¢
¥¤ ^d9 ^¤,~¶cQg0
;:0 c=^c=cd9¤60
dT, 7
cy¤ª0 ^¤ c~± ,
¥ ¥^
^ wª0
¢ p¡¤,¯= =^=e^cp °¢Me^ cc¶TeA¦pcQ
,e^¬[7,0[¦eŤ !6^¤ªC0
¬Qc=yp=¤ ?¥¦ µ ^ªC ¬? ¤ª0¢M ±¤0
©d9cQ¡6ôcf0pQp¬eð¤,=eA¦cQ
,7 d9eð 6 cp ^ cC
¤ c c7¤,=e^e^d9c=pµ ¤ ^ (ñ6c^cVc ¤ ceC§¦cQ
Q¤,=d9f ~,p·7^^cVfª0¤ eC <~ ' ( Q Ex X x(#p6
¥ ¬V¤0
÷ jùYg 3 ø 16 2 4 + · · · + (−1) + ... 1− + 3! 5! (2n − 1)! ,¤0
÷ jùgnø 1 1 1 + ... 1 − + + · · · + (−1) 2! 4! (2n − 2)! @ 9 x ůY¸@RpE9F»]6`>?]^>Qa=3;l( ,wªQ
c=e6¶ e^ cp ¬=Cc= c ¤ ¥
¥ ^ ÷ g0 jøɧµ ·7^dº T±~0
w ^e^f cp ¬f 0¦~ ^¤ §¦~e¥ =6p^d9§¦hcc0¦¶¤ 0
c n−1
2n−2
n−1
16 64 4 u1 = 1, u2 = − , u3 = , u4 = − , . . . , 3! 5! 7! 2n−2 2 ; . . . , un = (−1)n−1 (2n − 1)! 1 1 1 v1 = 1, v2 = − , v3 = , v4 = − , . . . , 2! 4! 6! 1 . . . , vn = (−1)n−1 , (2n − 2)!
cA
[f0cñ^¨¨( « ^ e^f0cd9c^c[¤ 0
d9c?¡V c7Q= eCp¬V[0
u1 = 1; v1 1 1 u2 − w 1 v 2 4 =− ; w2 = =− −1 − v1 3! 2! 3! 1 u3 − w 1 v 3 − w 2 v 2 16 1 1 1 − = ; = −1 − − w3 = v1 5! 4! 3! 2! 5! u4 − w 1 v 4 − w 2 v 3 − w 3 v 2 = w4 = v1 1 1 1 1 1 64 1 =− −1 − − − − − =− ; 7! 6! 3! 4! 5! 2! 7! ..........................................; un − w1 vn − w2 vn−1 − · · · − wn−1 v2 wn = = v1 1 = (−1)n−1 . (2n − 1)! w1 =
1
GVS
b#=f dºc¤,pCc=dÉ
×ß6Ø=ß7Ê?Ë2 Ö ÎÐ× Ñ¥ØQÔòÍPÙÚpÛ0Ô
4 16 22n−2 + + · · · + (−1)n−1 + ... 3! 5! (2n − 1)! = 1 1 1 n−1 1 − + + · · · + (−1) + ... 2! 4! (2n − 2)! 1 1 1 = 1 − + + · · · + (−1)n−1 + ... 3! 5! (2n − 1)!
1−
÷ jùg=jø ¤ 6^ªC°4 ¬Qd9=p6= wdT÷ j#ùg=jcwø§d9e^cQ¡V ¤ c[t ¤ ¤0c=
^h¤ t÷¬Vjù g=^j¤ ø^d9eð¦ cCcQ
,¡[^ eðº ^=de^e^c?c cp¢M6 c e6#=~ªe^¢M 7¤,=0¦h¥
¤ 0
cc e6 ¬ cT2É Z[ §>A@¦ô>Qd9F© ]6c`^>?c]^0>Q a=^3. #cp 6
¥ d²¤ 0
þ÷ jùgY3 øÉ,!÷ jùgnø¥f0=f!ñ6cV
¥ =¢Mô ¤ ~
¥ ^ 4 16 64 + − 3! 5! 7! 1 1 1 1− + − 2! 4! 6! 11 57 1 − − + 3! 5! 7! 1 1 1 − − + 3! 2!3! 4!3! 1 22 − 5! 7! 1 1 − 5! 2!5! 1 − 7! 1 − 7!
4
1−
4 4 4
+ ... + ...
1 1 1 + − + ... 2! 4! 6! 1 1 1 1 − + − + ... 3! 5! 7! 1−
+ ... + ... + ... + ... + ...
ô+?. .
. ¹¤ cC
cp ,¡V cp ,ª0 dþ¤ 0
,e6c,?
=¢M7 ±heV÷ jùg=jø¥
Ë PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔPÕÙÚpÛ ËY«¥×ß?Î]O¿[Î> ÑÐÛ0Ö ÞÑQÎ]¥Ö ÁáI½¾§½ g SSôãô ç 9Fæ ôéëêíì[îïwé
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}
#$Êed! ,'z
" ! +-,. * %& +
¹ªe6¬[
=,^e^f0c 6 ,ph ce¥ ¥
cp¥ ¬ ce6¬y¨ª0 f « ±
c ¤ ¥
¾§¥ T^¤, p ¡§^¦h , cQ
cdºôcdº¡ ^¤ ? =
u1 (x), u2 (x), . . . , uk (x), . . . ,
u1 (x) + u2 (x) + · · · + uk (x) + · · · =
∞ X
÷¯:0úq?ø
un (x)
, pCT¹¤ =6cCeð
h 0¨¦wª0 C,f ?« ^c ,?0 ¦ ¬ T¤d0
©¤0d9
cQc¡dT6ôò¬ôeð¦cC
,7 d9eÅ . ¤ w
¤ª0^0¦w ¤,=eð¦cpµ x
,7 d9eð e¥ ~eð¦cC
eÅh e¥ cc±~¤0
/ n=1
u1 (x0 ) + u2 (x0 ) + u3 (x0 ) + · · · + uk (x0 ) + . . . ,
c^cc¤?,c¤0
P u (x) eA¦cQ
eðh[c f x = x ó§ccfª0 ce¥¬¯e^¥¦[C,Q ^ ±V ^¤ ^d9^ c± x ¤ [f0c=c¤ §¦¤0
P u (x) eA¦ cpµ
eð ,pCT=6eðhcp =e6¬¢¼eA¦ cC
d9ce6~
= c=^c¨ª0 f « c,? ¬ c^cV¤0
(= ¤ d9^¤ ,¤ 0
e A¦7cQ
e^cc=eðh6[e6 =ª0¢M^¤ ? ±h ] −e¥1 1,c+1[cx±ô+=¤=0xfô
hf+ =¤ fôx¥
+¤ e6=¶·=· ?· ,¢4+,¥cxyde^+Cc,.Qc. ±¶.^ ^ e^ f0cx ^ I ñ6ccª0^cT =¢M^¯¤ ª0?¢ µ ^^cd96¤ ^eCfª0¢´ ¤ c^¤ ^e^e^ ¢¯¹¤ ñ6c=~¤0
¤,=eA¦cC
eÅ f=fw§ c¶ cf0?µ Q= c[°p =¤ e6Å
¬§¢K
,eðª0¦7cC
^d* d9¤,cpe66
¥ = |x| ≥ 1 ¨ª0 f « c,? ¬ c^c~¤0
¶d9cQ¡[6ôcf0pQp¬eð e¥ cc d9 cQ¡^e6c[eC=d9c^c[ ¤ cCcpD ¬ c^c[e6¤ c^ ôë ^¤ ? cp ,ª0 ^¤ ? c=¤ 6Ccf ¶?
ú¹7¤ =º=f ¡^c(C¦ 0cC¦ô
e^ cd9cce6fª0 ccpe6 = e6¬©eA¦cQ
d9ce6 cC
¤,p6
¥ , ¢M»,cp =e6=e^cpµ ¢M♦ c±~ôªe¥ c c±~eð¦cC
d9ce6 K ¾ c p = 6 e ð e ¦ C c
9 d c 6 e þ ¤ 0
! ^ ^ c c ^ e
c
T y ¦ = , ¤ = f ^
¤
6 e
, f Ð µ p t , = 6 e
, p n ±hce6p¨pª0c f f « r ch,?e6 ª0¬d9 dPc ^cVS¤4 0
=~ªQ
,÷¯ª :0yúq?¨øª0÷|f0 =f f~« ô d9 he¥ c=c xc^c ¾ ø ce6ª0pd9 =dPe6 ~S eA¦,cQc=
¤ d96Ccce6f hσ e6ª0d9 ndPµ¸ TS(x) ¤ ¥
e6p=p , 6e^cc± ¤ ¥
¥ º ce¥ ¥
cp¥ ¬ ce6,=e6 §¦»e6ª0d9d {S (x)} n = ? = 1, ∞ ¯÷ :0ùgø S(x) = lim S (x). C cc=C,Q,=6?c
,~ce6ppf0¤ 0
¯÷ :0 ø lim r (x) = lim [S(x) − S (x)] = 0. ¾*¤,=e^e^d9c=¤ ^ cdò·7I ¤ d9^¤ (cp =e6¬eA¦cQ
d9ce6 D ë ^¤ ? ] − 1, 1[ ∞
n
0
n=1
∞
n
n=1
2
n
3
mn
n
n
n
n→∞
n→∞
n
n→∞
n
1 1 − xn = , n→∞ 1 − x 1−x
Sn (x) = lim Sn (x) = lim n→∞
n
x∈D
G0
1
0 lim rn (x) = lim
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
1 1 − xn xn − = 0. = lim n→∞ 1 − x 1−x 1−x
c=C,?,=c¯6Q
; ,c ¢4
,~c^ c ¢4c^cy,= ^¤ T¥
! cpQ ?µµ
= ó& cª ^7c ^εe6>c0=e6 ª 77^e6 ¤ =¥ª
6¥[ cp=f0÷¯c:0±¶ùg øTctd9^÷¯¤:0 NøT(x) n > N (x) ¢Meðh ^¤,==^ e6 ¯÷ :0 3 ø |S(x) − S (x)| = |r (x)| < ε, x ∈ D. n→∞
n→∞
ε
n
ε
n
@× ârå!3OÖVߣäáWÖVâ@ÖVÙrÞ±ÚYÛuÖRäÏZÖÜ8RÏrå ÝbÐb×ÐOæuÐ Ö ä
ãYÖO×@Ø@ÓÖRäáRØâdçd×Ð G ê M G]ÏràRØràYÐbÜuÒÙráRÙuÐ ÝbÐb×ÐOæ@Ò Ö¸äãdÖ]×@Ø@ÓÖRäáRØ Û@ÖRäÜuÒ×uÖVàYÐáWÒÜ8RÙ@ÖWäáRØ Òv3OÖ æuÐVäáRØrærÙrÞã äåZÓÓ {S (x)} Ú@áWÖrÚ@Ø@äãdÖO×@çÍØrÝÖ?0VÕÒv3OÖ ÏrârØráWÒârØrçÍèÖVéØ×rÜ@çxådÙrÏrrØ@ÖVÙÐbÜ8RÙrÞãÛ@ÖRäÜuÒ×uÖVàYÐáWÒÜ8RÙ@ÖWäáWÒnß ÚuØ@ÓÒ]Ò]Ó î ù -ZìYî ù S þ í & ù ú &x ü ø ù íuü ðdúû@ü í Ríî ð w.Ü@çáW4Ö 3OÖÊæráW?Ö 0VÞâYçd×@ G ê M ä
ãYÖO×@ØZÜuä
ç à&?Ö 0ÜÐVäáRØ D ÚÙ@Ò]?Ö 0OãYÖO×@Ø@ÓÖØÔ×uÖRäáRÐáWÖVærÙ@ÖrÚ-æráW?Ö 0VÞ ×rÜ@çÜ u0R4Ö 3OÖ ε > 0 ØÜu^0RÖ43]Ö x ∈ D äådÕÒ]äáRàWÖVàYÐbÜuÖáRÐÏdÖRÒ ÚuæráWÖ¸×rÜ@çàWä]Òã m > N (x) Ú n > m ×rÜ@ç ÖVáRâ@ÒÝ]Ïrиâdçd×Ð σ (x) ε nm àRÞÛ@ÖÜ@ÙrçdÜuÖRvä 8¸Ù@ÒâuÐàWÒÙ@NäεáR(x) àWÖ G ê 7 |σnm (x)| = |Sn (x) − Sm (x)| = |um+1 + · · · + un | < ε. sÊâ@ÒØ@ÓådÕÒ]äáRàWÖ±ÏrârØráWÒârØrç¸èÖVéظä]ÖRäáWÖVØrá à±áWÖRÓÚRæráWÖ äãdÖ]×@Ø@ÓÖRä?á 8 âdçd×ÐÓÖOÑÙ@Ö ÖOãdÐâuÐÏ / áWÒârØrÝOÖVàYÐ?á 8@Úuä]ÖVàWÒâréÒÙrÙ@Ö Ù@ÒÊÝ]ÙuÐçØ£×ÐÑÒÙ@ÒÊådÛ@ÖRÓØrÙuÐçÍÖ äådÕÒ]äáRàWÖVàYÐÙrØrØ S(x) ê
m ^ e 0 f p Q =
c # d ª ò 7 · M ¥ e ¥ ,
ª 6 ¯
c =
¬ ¯ c
¥ e =
, V f = c c
¤ § V ¦ T
p c
µ N (x) ¢Meðt C =^ª!¤,=ce^^c e6^ ce6÷¯:0¬~ 3 ø¨²ª0 ÷¯f :0« ùn ø¥c,?c c¬= 7§¦^¤c0
cc¤ ô d9¤, p6¤, = e^ e^ d9Kc=
¤ , »d ¤,¶pCe¥ &¥
,¦»ª0¢Mc ^^d f x ¤,c,p∈6?
D¥¬ =c^c¹c¤f0y
¡d97cQQ¡V= d9c6 e^dT cp ¬=Cc¶c
p ,¬Vc e^¤ I¥
¤ ¥ ^C ,=©f cp cp= ,e6ª0 ^eA ¦ cCT
7 d9¤,c=e6 ^©(
¨ ,ª0h f « 0e¥µµ c§¦ô¤0
c ,¹¤ c0 ¢4e6¤ ¤ ªCdñ6c ^e^f0cp ¬f d9h ¤ d9^¤,=d9 <~ ' y T;>x =,x((=±hcp =e¥¬eð¦cC
d9ce6w¨ª0 f « c,? ¬ c^c¤0
ø X p(−1) , ø X (−1) . ε
∞
∞
n
n2n
(x − 1)n
@9 x øÉóP p6=^d9TI¤0
[c ¤ ¥
¥ ^ ¤ n−1
|un (x)| =
c , ¤ d9^ ¤ C,=fwmc=·7 , d9^^d lim
n→∞
s n
n2n (
n−1
n
(x + 2)2n
x>1
,p=f~f0=f
1 √ , n2n ( x − 1)n
1 1 1 1 √ = √ = √ lim √ . n n n→∞ n x − 1) 2 x−1 2 x−1
óP ¥
cp¥ ¬= c eA¦cQ
ò±w¤0
w=ªC
6Veð¦cC
¬eð!=e^cp ¢M c ,^e¥ ?¤0=
¤
√
x − 1 > 1/2
0
1 < 1, 2 x−1 √
x > 5/4
;¹¤
x = 5/4
∞ X (−1)n n−1
n
,
cp ,ª0,=^dC,=f0c ^¤ ¥
,ª0¢M7,± eð
Ë PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔPÕÙÚpÛ ËY«¥×ß?Î]O¿[Î> ÑÐÛ0Ö ÞÑQÎ]¥Ö GV7 f ¤0c=
c ¤ T± cpeð ,¦ª0cC
^eŤ ? c¶ ¤ C,=fªwáI^ ± «,;b#=f d²c¤,pCcdT;cp =e6¬¶eA¦ cC
d9ce6 øT4 óP =¥=^d9T¯¤0
Dc =¤ ¥[5/4,
¥ ^∞[ ¤ x 6= −2 ,p=f~f0=f }
|un (x)| =
1 , |x + 2|2n
c , ¤ d9^ ¤ C,=fôl7? =d9^¤,~÷|d9cQ¡V cmc=·7ø¥, d9^^d 1 |un+1 | |x + 2|2n = lim = . 2n+2 n→∞ |un | n→∞ |x + 2| |x + 2|2 lim
?eA¦cC
T±h¤ 0
h=ªQ
6VeA¦ cC
¬eðw=e^cp ¢M c ^e¥ 1 < 1, |x + 2|2
? =, ¤ |x + 2| > 1 0 x > −1 x < −3 ¹¤ c^¤ dþeA¦cC
d9ce6¬y eA¦cC
c^cV¤0
,f c «,?¦ô ^¤ ? c ¹¤ x = −1 cp ,ª0,=^dþ¤,=eA¦ cC
,7 ± eðw¤0
∞ ∞ X X (−1)n = (−1)n , 2n (−1) n=1 n=1
=,¤ cþ?0 ¦ªccCe¥
^ c d ,c4fhp Q¤ =f0eA¦ xcC¢4
= ^d9−3 c e6¢¯ ¬* cp ,¤ ª00,c¶
= ^eAdw¦pcQ=
c=f ¡[T¡©±&c¤eC =0¤ d9
¥
Teð¥±7¦ cC¤,6
,=*eA¦eÅcC
,^¤,=7=e^ c?^± eÅe6¢M¤ c0c¶d
, |x/ye¥d9+ 7cQ2|ª0¡ ^^>e6e61¬ S D =] − ∞, −3[ ] − 1, ∞[ <~ ' y T;Ex D x((=±hcp =e¥¬eð¦cC
d9ce6w¤0
ø X cos√nx , ø X n! , ø X(−1) n . x n n @ 9 x F ø&l( ,ôe^¥¦ x e^ ¤,=¥
c ^¤,=^ e6c ∞
∞
∞
n −x
n
n=1
n=1
n=1
cos nx 1 √ ≤ 3/2 . n n n
¹ ce^f cp ¬fªô¤0
P n eA¦cQ
eð cV,[ce^ cp h ¤ C,=f0e^¤,= ^ ~Q=f0 ¢Éµ ,=^ dTøòp¹¤ c ¤d90^
[ eA¦ cC¤
CT,±[=fôeA¦l7cQ
? =eðd9[=^¤,e^cp ,¢M=±0
c^d ,4e^^±7 e¥ cc±7ce^ D =]−∞, ∞[ ∞
−3/2
n=1
|un+1 | (n + 1)!|x|n 1 = lim = lim (n + 1) = ∞ > 1. n+1 n→∞ |un | n→∞ |x| n! |x| n→∞ lim
°4e^ ¢§øT
ó§¤,e¥= Å
, ª6? eA¦ cCc
¤T0±h
w¤¤,0=
weA¦ ecC
ªe¥ eðc~
c ,ôeA¦ cCe^
,¥¦ 7|x| d9<eðh∞ ¤ ~=e^cp ¢M cVeA¦cC
,7 d9eÅh ¤
∞ X (−1)n
¤0
cd
¤0
cd
n=1
α>1
0<α≤1
nα
∞ X 1 , nα n=1
Q=f0 ¢4,=^dTcV e^e¥ ¥
,ª^d9ɱw¤0
!eA¦cQ
eðhªe¥ c c[
, x > 0 h=e^cp ¢M c[
, ,b=f dþc=¤,pCcdT,¤0
weA¦cQ
eðh,[ ^¤ ? D =]0, ∞[ x>1
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô <~ ' yT;xEH x((=±hcp =e¥¬eð¦cC
d9ce6w¨ª0 f « c,? ¬ c^c¤0
ø X 2 , ø X sin(πpn + |x|), ø X cos r x , n §
¥ cp =e6h=e^cp ¢M c±h¶ªe¥ c c±~eA¦cQ
d9ce6 @9 xPøIó§¤,= w eA¦ cC
T±t¤0
e^c~C,=f0c cp cQ¡V¥ ¬ Td¼¤0
cd P 2 eA¦cQ
,7 d9eðw ¤ α > 0 ,d9c?¡V cQ=f0 ¢4 ¬ ccheA¦cC
eÅw ¤ sin(x ) > 0 ,? p
, 2kπ < x < (2k + 1)π k = 0, ∞ pC¤ 6·7 yñ6c ^¤,=^ e6cyc= ce^¥ ¬ c x ,=±0
^dþcp =e6¬V=e^cp ¢M c±~eA¦ cC
d9ce6h¤ 0
[0
(c=³¥
^ hd9 cQ¡^e6 1
GVB
∞
∞
∞
−n sin x2
2
n=1
n=1
2n2
n=1
∞
2
−αn
n=1
2
√
2kπ < x <
p
(2k + 1)π,
p √ − 2kπ > x > − (2k + 1)π.
¤0
ø. ¾§¤ c^e^c cp¤, p¬=^Cªc^d==fô·7 0e^
,¬4ª e^c± e6=d9¯¤ ^c cd96¤ , ^e^f,0¦7¨ª0 f « ± pc=7 ±¯0 ^ p n2 + |x| − πn + πn) = p p = sin π( n2 + |x| − n) cos πn + cos π( n2 + |x| − n) sin πn = p π|x| = (−1)n sin π( n2 + |x| − n) = (−1)n sin p . n2 + |x| + n sin(π
C
c c=Ccp , 6Q= e^p¬V eA¦ cC
T±h¤ 0
hf=f¶C,=f c ^¤ ¥
,ª¢M7 ±,eð ∞ X
∞ X p π|x| 2 , sin(π n + |x|) = (−1)n sin p n2 + |x| + n n=1 n=1
f c=c¤ T± Qe^cA =e^ cM ¤ C,=fªáI^± «,^ p , 6eðIeA¦cQ
, 7 d9eð Q ¤ ^dTQf0=f( 6¤ªQ
µ cQ=d96¬ eA¦cC
,7 dPeð~ªe¥ c c
,~e^¥¦ |x| < ∞ ø° ^0
c ;côñ6c=~¤0
¤,=eð¦cC
eÅ ¤ x = 0 .Ç =¨( f0e^ ¤ c= x cp cQ¡V0µ ¥ ¬ Td (x > 0) eª0 6cdþc^c c cos px/n > 0
, n 1 cp ^c^c cos
r
x x ∼1− , n 2n
ce^ cp ¬=^ª^d9eðh¤,?
f0? ¬ Tdþ ¤ C,=f0cdmc=·7 s n
lim
n→∞
r r 2 x 2n x 2n = lim cos cos = n→∞ n n x 2n = e−x < 1. = lim 1 − n→∞ 2n
b#=f dºc¤,pCcdT cp =e6¬ 0 < x < ∞ p , 6eðhcp =e6¬¢ =e^cp ¢M c±heA¦cQ
d9ce6 ¤0
<~ ' yT;xKJx ?e^e¥ ¥
cp¬V,eð¦cC
d9ce6¬V¨ª0 f « c,? ¬ T±h¤ 0
ø X (n + x) , ø X sin n sin n , ø X 1 . ∞
n=1
n
nn+x
∞
n=1
2
ln2 x + n
∞
n=2
n(ln n)x (ln ln n)p
Ë PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔPÕÙÚpÛ ËY«¥×ß?Î]O¿[Î> ÑÐÛ0Ö ÞÑQÎ]¥Ö GRG 0 @
9 xø&ó cd9c=7¬¢ ñ¥ ^d9^p=¤ ,§¦¶ ¤ ^c¤,pCc= ±¶¤0
ôd9cQ¡V cIQ= eCp¬ }
∞ X (n + x)n
=
,ôf0c=c¤ c^ce^ ¤,=¥
[c« ^ f0 n=1
∞ X
nn+x
∞ X
1+
n=1
x n 1 , n nx
∞
1+
X 1 x n 1 ex x , < n nx n n=1
x > 0.
?(ñ6c±hc« ^ f ~e¥ ¥
,ª¥pc¤0
weA¦cQ
eðh ¤ ôªe¥ c
xe^d9 >c=1¤ d»e^ cd9c¥p¥ ¬pµ # ø § ¾ c ^ e
p c = ¬ ^ ª ^ 9 d Å e V
¤ C , = f c d ¯ l
¤ 0 ¦ = ( l , [ 6 ñ c ^ ¯ c , ¤ = ^ e T±~¤0
n=1
∞ X
sin n sin n2
ce¥ ¥
cp¥ ¬ ce6¬ (ln x + n) ¹ce¥ ¥
cp¥ ¬ ce6¬ S kµ¸,=e¥ §¦»e6ª0d9d e^ cd9c¥p¥ ¬ c^cV¤0
7 p , 6eÅ~c^¤,= ^ c± ce^f cp ¬fª÷|e^dT, ¤ d9^¤fWfWfWfWf?ø n=1
2
k X
−1
k
k
1X 1 sin n sin n = [cos(n − 1)n − cos n(n + 1)] = [1 − cos k(k + 1)], Sk = 2 n=1 2 n=1 2
e6=? cò¬
1 |Sk | = |1 − cos k(k + 1)| ≤ 1. 2
l¯mc ¤ ± ¤ c,d90e6¦ M ¤ =^c;d#^óPc 7 ¥C
c± e¥ceð wÅp
fhc¥ ¬pªC ¥c ¢¯ ¬¤ b#0c=
he6f ª ¬ e¥dº (lncc ¤,xcpC+eAc¦dTn)cQ
,Teð
h ,cp¶ c e^p¢M¥ ¦ =eÅe6xh> c0 ªpe¥ , c6eðhô d9¤ c Cc=,=cf00 µ ø9¾§ce^ cp ¬=^ª^d9eðye^ ^« ,? ¬ Tdt ¤ C,=f cdtmc=·7w÷|e^dT^xc¤ >^d#0ª 3 ; :ø¥=l( ,ñ6cpµ ^c[¥
^dþe^ cd9c¥p¥ ¬ T±w¤0
2
∞ X
−1
2k , 2k (ln 2k )x [ln(ln 2k )]p
f c=c¤ T±hd9cQ¡V c[ ¤ ^c=¤,pCcp¬f~0
,ª k=2
∞ 1 X 1 . x x (ln 2) k=2 k [ln k + ln(ln 2)]p
ó§¤,= ( cp ,ª0 ^ T±V¤ 0
Veɤ0
cd øM ¤ d9^¤,¸fWfWfWfQ=f0 ¢4,=^dTc
, ¤0
!=e^cp ¢É cVeA¦cQ
eðh ¤ x > 1 ,7
, p > 1 4 ¤ x ≥ 1 <~ ' y T;Ex P x((=±hcp =e¥¬eð¦cC
d9ce6w~e6ª0d9d#ª¶¤ 0
∞ X cos nx n=0
n!
.
p≤1
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô ¤ c @^9c[ f0c=dP 0 r?6f0e^e^e¥ c¥^
,cVª^¤d90É
±V,¤ [0
Vd9 ^¤ ¥
c e6 p=p , 67e^cc±7
^± e6¥ ¬ª¢,=e6¬I ^f c=cpµ 1
GVI
∞ X cos nx n=0
ix
x!
= Re
∞ X cos nx + i sin nx n=0 cos x
= Re ee = Re ecos x+i sin x = e
n!
= Re
∞ X eix
n!
n=0
=
Re[cos(sin x) + i sin(sin x)] = ecos x cos(sin x).
#b =f dºc¤,pCc=dÉ ¤0
!=e^cp ¢M ceA¦ cC
eðw,e^^±~ e¥ cc±~ce^ ¹¤ y¤,=e^e^d9c=¤ ^ ,ô¨ª0 f « c,? ¬ c^c¤ 0
0eA¦ cC
,7^^ceðô7 ^f0c=c¤ cd ¤ cd9¥µ ¡ª f =ò e^ d c ¤ ce=e^cC¦¤,= ¢Með
,¨ª0 f « c,? ¬ c^ch¤0
P u (x) ÷| ¤ ¥
e6p=p , ¢M7C^ce^cc± Q^e¥ 7^cc¤ ¬4¨(c¤ dP? ¬ c u pe6ª0d9d#ª(^e^f0c ^ c^c4 e¥ ¨ª0 f « ± ¢ øòce^ c TIe^c± e6[e6ª0d9d9 f c ^ c^c e¥ [¨ª0 f « ± [ d9^ c q?øye6ª0d9dP[ ^ ¤ ^¤ T ò¦h¨ª0 f « ±h^e6¬V¨ª0 f « ~ ^ ¤ ^¤ T,p gøy ¤ §c¦~c=CcCf0
p,¡[p»
ce6±ôª0d9d9I ñ60÷
¦~ ¨¨(ª0¨( ^f ¤ « ^ ± « ¤ªCd9§¦øI¨ª0 f « ±»¤,=,~5 e6ª0d9d9V ¤ cCcC
µ øy ^^¤,? Vc=e6ª0d9d9 ÷| ^ ¤ ^¤ T §¦5 ø¨ª0 f « ±[¤,=^7e6ª0d9d9ò ^^¤,? cc=f0p¡[µ
c±~Iñ60¦~¨ª0 f « ± ¾²cp7^dþe¥ ,ª0,=Ic=6V,[ñ6~c ¤ ce^ c=¤ «,p¥ ^ e6^e6^ c[ c òpp¬eðôc ¤ ¥
¥ ¬ f0=f d©
c cp ¥ ¬ Tdªe¥ c d©
cp µ ¡^c©/ ªQ
c
p , 6 ^c^¤c¬§ ¤0
e^ ÷|0¤, =t¥
f0= f dÄ
cp^ ,¤ ¡^ ^ te¥ ^ò ,¬hT¦ t=¤,=òf·7»^¤e^eð¦ccC±
e6 d9ce6f cº ^¤ 0
§ø¥¦ e6eð~ª0d9¤dT0
P¾§¢?ò e^ ^ ~ñ60¦ªeÅ c ±þ ¤ cC
©f c ¢ p¤,= cd9^¤ ceA¦ cC
,7^^cpµ ∞
n
n=1
±
! %&
* _|u ($ÊId! ,'³Ê '
X x>=,x u[ w). º*) "# +
0 »m(÷¯=:0f»ùnø¥ª ¡[ycª0= 7c^d9dþ e¥, ,?ª0 ,c==e^¬ (9 p ,eÅ 6c eðNh¨=ª0 Nf « (x) ^±wc ¥ ¤ ¥
¥ , ε¢M7^x~ ^¤,=^ e6º÷¯:0 3 ø ®¯ª0 f « c,? ¬ T±¤ 0
w¯÷ :0úq?ø¥=,pCT=6eŤ,= cd9^¤ c(eA¦cC
,7 d9eÅcp =e6 D ,^e¥ ôe^
,¤,ô= ¥¢4
c^ccVε ^>¤,=0^e6 ª e67^=e6c =ª6 N p=f0c=c
,~e^¥¦ x ∈ D ¶
,~e^¥¦ ε
ε
n > Nε
|S(x) − Sn (x)| = |rn (x)| < ε.
e6¶l¯¤÷¯:0ª0 ^3 øPd90! e¥ ÷¯c:0ùn=ø#d9ªQ
,c
p , h6¤,=c¤ c¢Md9^eŤ ô cCcy
eA ¦ cCdt
, 7 ^e¥ ^cceðd NVc=p N=e6e^ ¤,pD^ª[¤
0 ,
yV e^^¥¤,¦ =x^0 µ cp l(=e6 ,* Dc^cc£cp ^!,=A ,0
c c? e^ ¬¤,pC «ª²d96¡[
,ª¤,= cd9^¤ c±² ^¤,= cd9^¤ c±~eA¦ cC
d9ce6¬¢¯¤,=e^e^d9c=¤ de¥ ¥
,ª0¢M ¯ ¤ d9^¤ <~ ' ( X >x =, x ?e^e¥ ¥
cp¬[¦ =¤,=f^¤heð¦cC
d9ce6w,=e6 §¦he6ª0d9d ø S (x) = x , ø S (x) = nx 1+n x 1+n x ,c=¤ 6Cf D = [0, 1] @ 9 x øòb=fôf=f ε
n
n
2 2
lim Sn (x) = lim
n→∞
n→∞
x = 0, 1 + n 2 x2
2 2
=Ë ß6ØQÒ Ñ6ÞÍ Ù0Ò ÑIÎ¯Ñ Û0Ú+V4Ö ÍÅÎ Ú&£PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔòÍ9ÙÚpÛ0Ô ceª0 6c=d*e^ccp c=·7^ 0 1 + n x (1 − nx) ≥ 0 d9^^dþc« ^ fª
GVL
¡ +¢
2
|0 − Sn (x)| =
2 2
≥ 2nx,
x x 1 2nx 1 = ≤ , = 2 2 2 2 2 2 1+n x 1+n x 2n 1 + n x 2n
x ∈ D.
°4e^¢§
ye¥ ¥
,ª6p cV
,wc^c ;c¼ ^¤,=^ e6c §µ ¬¤ I
¢4 ,≤ c1/(2n) 4« ¥ c4< ε cpe¥ , ªc µ
ªC
cpc p 06 ce^c¬¯¤cQ
¢M 7cCV¤ ^ d9^^¤, = cI^
e6 ,y=ª e^¥¦ x D
ce6bp=pf .c, |0= c−¤ = ^Sd9^(x)| d N = 2
, ε = 1/4 4 e^Ncc=>61/(2ε) ¤,pCcdT e6^ c N = 3 h?
b#=f ε d=c1/2 e¥C ,ª0,=7ø nµ¸,=e6 ,pwe6ª0d9dPVeð¦cC
eÅ!fwe6c^d#ªh ¤ ¥
¥ ,ª S(x) = 0 ¤,= cd9^¤ c e=T g0dPU¤QÅø¥µ ¹p¤ tce6¤,=^± ·7c^d9(^ ¤ e^,e¥p *¥
eAc=¦cC=
d9c e6c¬tfp¦CcT¤ c==·76Qc 0; c f c=¢4¨e6ª0c ¤ ¤ f T « ±¤ ªe76¤ SeÅc²e6(x)^c¤,d=¯¨( c e¥ ^f e^ f xÀ=e^÷|d9¤ 1/n 6=6eÅ ¢MôdP=f e^ d#ª0dT ¤,= T± S (1/n) = 1/(2n) n p ^c 0cC
c¤ ^d9^ c7ª0d9^ ¬=·p e^¬¯e^c^ddP=f e^ dP? ¬ cd»C,Q ^ 0l( , ε = 1/2 e6ª0d9 d ó²Sª0d9(x)^ ¬=·7,Q^ ,,^dpwe ne^I=^¤,2= ¨(
,f h e^¥µ¸¦ ,pxe6∈ D §¤,¦!=e^e6 ª0cpd9 dT?µ ¥=e^¢MI^eð¤,!=¨( cC
!f w ¤ , =d9e6c ± ε §=¦!1/2 ==f¡=ªC
,ª ~¤,=e^ cp =¥εp¬eð ¡7 ¤ n d9c± ;e^cc=6e6=ªµ ,¢MQ7 ^ ±,pQ!?
e =N c>d#ª 1/(2ε) .,=A ,0
ch 0 ¢4e6¤ ¤ ª=»¤,= cd9^¤ ª0¢öeA¦cQ
d9ce6¬he¤ ce6cd ε ,ª ~ f ^ e c ^ # d ¶ ª
¤ ¥
¥ N S(x) = 0 n
ε
1/2
1/4
n
n
n
ε
ε
9 e=g
ø§b#=fôf0=f c
lim Sn (x) = lim
n→∞
n→∞
nx = 0, 1 + n 2 x2
nx nx 1 n2 x 2 1 = |0 − Sn (x)| = = ≤ . 2 2 2 2 2 2 1+n x 1+n x nx 1 + n x nx
Q?
= c^c ε h¨( f e^ ¤
cce6=p p cc^ c ctx >=^0
¬ , h¢4ce6cª µ 7« °4¥^ e6e^c¢§7p
V^ eÅe¥ ²¥c
, ªªC6^
?¤,c;=p ^6 c[e¥
c ,¤ô ¢M |0¢47−CcS^c[ (x)| 1/ε ;mKñ6cd#ª;cQ
,=f c e¥ ¥µ ^¤,=^≤ e61/(nx) =ª N <> 1/(xε)
,,ª=6±0I
6
ceðºpc ¬ f c¯e^f cp ¬Iþcp ¬=·7 d©d9 É=^0 e¥ c N Icp =e6 D e^^A
CcdT#
,Q¶e^ e^¥6¦ ª0¥ ^ ô e¥ x N= 1/N ¤ 9« f ,c=? c¬¤ cc[± S^c=(1/N Cd9cQ¡V) =c7eð1/2
Å 9pb#¬ =f S d (x)c<¤,p1/2 C ,cQd9 ^^¤ ± px ¤ [ f Dc=cQc
¤ ccd¤ ^¤ d9 ^ ¥
c ^ ( c=É ¤ ò d9·7^T¤ pª ^¤,¡=P
6 ,e6 εcI=T1/2 cp ^=0c= Ccd9e^c?¬¡V» cM
ª0 ,fpQpe^¥¦¬ N e^¤,p^ª?
m(=fVV¯e¥ ,ª0,=ø¥ ¤ ce6^±·7C e^e¥ ¥
c= 4 cfpCT=6?0c x ∈ D e6d9c^d¢MI dPe¥= f e^ d#e^ª0d9dT6¤,p=6 eðTw± p cC^
,c =,f0 c ¨ª0 f « S (x)÷|¤ cpµ¸e= ;¤ g 6ú¡Va^ø¥ ^C d#ª7c=VdPc= f0f e^ d#xª0=d1/n ( e
¤ c S Cd9(1/n) ^=e^1/2 c^6cdP=f0e^ dP? ¬= c^c!C,Q ^ 9¤,= c^ctWq eg0m(n=ft0
cô¤ e=Pg0úa? n
ε
N
N
n
n
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô ¦ c=^Q ô cy cp fpC¡[f0cy
c ±y¤ ^p¤ ¡[f0=?¢M ,eÅ!=^fwcce^±ô c=C
e7¥¤ c¬e6 cce6d , c cyf ôw^cQ¤,
=,¨(V f f c¤ Spw(x)« ¥ ^e6f f0ccpd µ ÷|f0=f»,h¤ e=9g ú:Qø( y ¤ d9Tf0=6f»ñ6Ox c±tce^»,he^n^d ¤ c=?¡^ tc=¤ 6Cf0 [0, 1] b#e6ª0=d9f d9c§ c¥
^ T,=A ,0
c0 ¢4e6¤ ¤ª6I ^¤,= cd9^¤ ª0¢*eð¦cC
d9ce6¬(,=e6 c± S <~ ' (X xED x ?e^e¥ ¥
cp¬[¦ =¤,=f^¤heð¦cC
d9ce6w¤0
1
IRN
n
n
∞ X
∞ X (xn − xn−1 ) un = 1 +
[cp =e6 D = [0, 1] @ 9 x ¾²¤,pC^¤ ª c±~Q= e^h eA¦cQ
T±h¤0
w d9^6V0
n=0
n=1
=c fªQ
¯ ^^f c[ cp ,ª0,=6eð~T¤,p¡[6 4
, nµ¸,=e¥µ c±e6ª0d9d9 ¤ 0
S (x) = x n = 1, ∞ .¾§e^ ñ6I¨ª0 f « I ^ ¤ ^¤ T ?cC
,=f0c S(x) = lim x
, = lim x = 1
, x = 1 °4pµ 0 e^¢§
¯e^x¤,p6=^ªV1 e¥ ¥S(x)
,ª6[ ^c=Cd9cQ¡V ce6¬7T cp ^
^c»¤,
= ,^ e6e^¥¦ |S (x)| =òc=xC,Q<,=¢Mε <p1 cQ
^¤, =c== ¤ c^d9d9^^0¤0µµ ª0¢ eA¦cQ
d9ce6x¬∈¤ 0D
99 epPh
=6 ¤ ¥
e6p=p ¥µ cd9¶^cw¤ e^c±c^eAc¦ cC¤,
p Cd9 ccde6 ¦ .=¤,e6ª0=fd9d&^¶¤ ô¤0,
= ¤ª ·7^ ª º¡[¤,7= p µ 9 e= p , 6eðh ^ ¤ ^¤ T c±w¨ª0 f « 6± ;[S(x) ^dP6,¥eð e^f0Q f0c=d÷ S(x) = 0
, 0 < x < 1 S(x) = 1
, x = 1ø¥ ¤ Tb# =§f ¦ dô¨cª0 ¤,f p« C c± dT§Ce6 ª0hd9 dPpÉ , ^6¤,eð=* ccQd9^Q¤ p cÉ¥ e𬦠cC
,ct 7^^ ^¤ c^e𤠯T¤ 0
c±²Ce^¨ce6ª0 cQf « 7 ^^^± c4ò©P d9^cQ ¡[¤ 6¥ µ ¤ òT¬V §¤,¦tpC¨¤ Tª0 f c« ± ±º¹f0¤ =¥f
e6¤,pp=¶p c^e^ c7=¤, pc!C¤ ,Tw ¤ § ¦hd9¨^ ª0^ f « þ ±w ^e(¤,= c d9c=c=dP76¬¤ ¢ c¤0eA¦
cCc
, 7^0 ¦ ¤ e𥠵 ¤0
c ( = ¤ 0 ,
ª ^ e É c ^ e
c ^ e c = 9 d I ò
^ e
^
( ¦ = , ¤ = f ^ , ¤ M A e ¦ Q c
9 d c ¥ e 7 ¨ 0 ª
f
«
c , ? ¬
c ^ M c ¤ 0 µ
ò e^ cp ¬=^c= Td9 ¤ d9^¤,?¦;Td9c?¡V ct e^ cp ¬=Cc=p¬ºf ¤ ^¤ ± mc=·7 / ^c ¨(c¤ d#ªC ¤ cf0y cp ce6¬¢Ke^c,?
=6¶e7¨(c¤ d#ªC ¤ cf0c=±!c=7^^c¶f ¤ ^¤ ©mc=·7 ^¤ e6C,=cf ¯÷ :0ù½4nø¥9¥ , Qe~t=¥l¯
e^¤ 0~e6¦ c= ^~,xp =c∈f±*¡Dc~^;cd9? cQ=c¡V ¤ f cpc± , P6^¤ eð^c»¨( c¨¤ ª0d#e¥ ªCf c « N ¤ ^c&±c=p cp¤ ¬» ¥¬
e^f c¥c ,c= ε¢M 6N7e6C=w=ª0 N¢M^¤,.= ¹dK c¤,pCcdTP0µ ¡¶d9e^¨(c=¤ d9ªC ¤ªCdÀcQ
º~,= cp ^ô ¤ ce6§¦þº,p7~e^^^c e^ cp ¬=^ª^d9ò¦ ¤,¤ =C,=f ¨cª0 4f « ¤ c,C?, =¬fô ¾§c^^cV± ^¤¤0·¯
~¤,÷_=: e^úq?eCø¥ °M
,=f c¯e^,?,? (
?
d»c ¤ ¥
¥ ^ dPp¡[cpµ Ç;,=f c cp cQ¡V¥ ¬ T±h e¥ cc=±~¤ 0
X ÷ o0úq?ø c ,^e¥p CT=6eð©dPp¡c¤ ¤ª0¢M7 d
,¨ª0 f « c,? ¬= c^ch¤0
©¯÷ :0úq?ø4,¶c=¤ 6Cf0 [a, b] ÷ o0ùgø |u (x)| ≤ c
,ô#0e^
t¥¦ ÷ oxúq?∈ø&,[a,pCb]T k=¢M=1,==∞f¡ dPp¡c¤,=c±¶¨ª0 f « c,? ¬ c^c[¤ 0
ô¯÷ :0úq?ø&cp ?µ e6 D 1 + (x − 1) + (x2 − x) + · · · + (xn − xn−1 ) + . . . ,
n
n
n
n→∞
n
n→∞
n
n
ε
∞
n
n=1
k
k
=
=Ë ß6ØQÒ Ñ6ÞÍ Ù0Ò ÑIÎ¯Ñ Û0Ú+V4Ö ÍÅÎ Ú&£PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔòÍ9ÙÚpÛ0Ô IZM %&>Qa : ]6Z 'N X G x>=¶ ) ²v7 9 , ·px¸ò]N
n
n
n
n
∞
n
n
n
n=1
] >Q](Y¸@D>RV>Qa:]6ZN 3 D U +-
x,l¯cf0p¡[^de^,?,? 7ª ^¤ ¡[
^ 7gø¥ ¹ªe6¬ô¤0
P c p , 6eð!eA¦cQ
,7^± eðdPp¡c¤,=c±w¤0
!÷¯:0úq?ø¥cA
yye^0 ,ª ÷ oùgø§ d9^^dþ ^¤,=^ e6c u1 (x) + u2 (x) + u3 (x) + · · · + uk (x) + . . .
∞
n
n=1
÷ o ø e^d9 ^cQc¤,¡V¤ = ^d9ct¥
eA &
¥p =cpfI
¬þf0 ,=e^wf f &cpe^ e^c¥¬tA¦ ª0x=^e^cC∈
c» Dcºf ¤ dP°4? ^e^c¤ ¢§±
ò¢ä¶ m^ctc= ·7
ce^ , ¤ ²&¥
e6e6¤,ª0=d9=d9^ª0,£¢£ cô»,= e6ò^¬»^cf0± =^6,¤,¶p=e6ª ^ e6^ ¤ p¡[* ,
^÷ 6o0 eð ø ªf0e¥p Qpc¬ ^d²¤,= cd9^¤ c±weA¦ cC
d9ce6 ¤ ^d=e^cp ¢M c± ;cyh¤ ^c? ce^¬V
c?µ k4^¤¡[
^ q?ø&
cf0pCT=6eðw=,? c^ c <~ ' (X xEH x4l¯cfpQp¬ cV¤0
X sin nx ø X 1 a) , n n +x ¤,= cd9^¤ c[eð¦cC
,eðh,[e^^±~ e¥ cc±~ce^»÷ x ∈ Rø¥ @ 9 x;b#=fôf0=f¶
,~e^¥¦ x T cp ¢Meðh ^¤,=^ e6 |σnm (x)| = |um+1 (x) + · · · + un (x)| ≤ ≤ |um+1 (x)| + · · · + |un (x)| ≤ cm+1 + · · · + cn ,
∞
∞
2
2
n=1
sin x 1 2 ≤ 2, n n
2
n=1
1 1 ≤ , n2 + x 2 n2
n = 1, ∞,
? =f0p¡[
còe¥ =6=^d9cTcc0¦¨ª0 f « c,? ¬ §¦¤0
c( T ¤ ^ò·=67e^cc==6e^=ªµ ¢M7^^ce¥ =6=6dPc=^cVeA¦cQ
,7^^ceðh e¥ cc^c¤0
∞ X 1 , n2 n=1
cyVe^cc=6e6 e¯ ¤ C,=f cd²¾§^± ^¤·¯¤,=e^eCVQ=f0 ¢4,=^dT;cyñ6w¤ 0
¤,= cpµ d9^¤ c°eA ¦^cC
,0
eðhc ,,ce^¤,^=±~ cd9e¥ ^c¤ cc±~eAc¦e^cQ
,7 d9 eðh=ªQ
,ª V~¤0
♦ ∞ X sin nx
np
,
∞ X cos nx
np
,
∞ X
np
1 , + x2
^e¥ ôcp ¬f c p > 1 =e^e^d9c=¤ d» ^e^f cp ¬f0c( ¤ d9^¤ c f cð
¤ C,=fV¾§^± ^¤·¯¤,=e^eCI ^ ¤ d9^ dT n=1
n=1
n=1
IRS
<~ ' (X xKJx ?e^e¥ ¥
cp¬V,eð¦cC
d9ce6¬V¤ 0
1
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
∞ X (−1)n−1 √ . n + x2 n=1
áI @^± 9 «, #x¦eA?¦eðcC¦
cC
eźT±,w¤0
e^^ ±ºp , 6eÅ eðc»cC±,=cf0e^c #^ ¤ ¤ ¥
, ª0 ¢M^d 7,ªe¥d9 eðc c #e^ cAc e^=f e^cp c!¬f ªt¤ ¤C0,
º=fª d9cC
,ªQ ^± ∞ X
√
1 n + x2
, e^¤ ^±~C, = f!e¥ ¾§c^± c^±~¤c·¯e^ô¤, =e^peQ ,y 6 ^eÅ ~¤ ¤, =d9eA^¦ cC
,dT 7¾§ cd9e^eð cp; ¾*¬=^e^ª0^ ,d9ª¶eÅwªe¥ Cc^ e6c ±hceA±¦cCc
« ^d9 f cce6± wn¤µ¸^0c µ ce6ppf0[C,=f c ^¤ ¥
,ª0¢47^Cceðh¤ 0
n=1
rn (x) =
e^cA =e^ cf c=c¤ c±
∞ X
(−1)n+1 (−1)n (−1)k √ √ + + ..., =√ n + 1 + x2 n + 2 + x2 k + x2 k=n+1
(−1)n 1 1 √ √ |rn (x)| ≤ √ ≤ < ε. = n + 1 + x2 n + 1 + x2 n+1
¹ce¥ ¥
^ ^¤,=^ e6cc=C,Q,=6?,cf=f cc² ô§ c e^^A
,=±0
6eð pe6=f ccy e¥ c N =
,(ε»e^¥−¦ 1) ,hch
e¥ , ©cce^±»¥¦ ce^nt>cC
N c=¤ ªC^
d96^! ε e^ c >¤,9=0°4¥e^
¢§
wc!t e¥^ ¤,¥=
,ª^60 µ ¤,= c|rd9^(x)| ¤ ,pô<eA¦ εcC
d9ce6¬ªxe¥ c ceð¦cC
,7^^ceÅh¤0
[,e^^±~ e¥ cc±~ce^ <~ ' ( X Ex P x ?e^e¥ ¥
cp¬V,eð¦cC
d9ce6¬V¤ 0
X (−1) x ÷ o 3 ø . ε
−2
ε
n
∞
n 2
(1 + x2 )n
@9 x #0
weð¦cC
eÅh,e^^±~ e¥ cc±~ce^ l¯^± e66 ¬ c
,~¤0
n=0
÷ o0ùnø fô¤ ^¤,x==^ 0e6^^=cyª eA¦ cC
d9ce6¦c ^0
, V ¤ x 6= 0 ¤ C,=fhl7? =d9^¤, ¤ cC
∞ ∞ X x2 (−1)n x2 X = (1 + x2 )n (1 + x2 )n n=0 n=0
. x2 1 x2 = lim < 1, 2 n+1 2 n n→∞ 1 + x2 n→∞ (1 + x ) (1 + x ) lim
ñ6c=f Vc=¤0c
w¤ ceA^¦c~cQ
e¥ ¥
,eðwª6=pe^cp ch¢Mw
c ,,[e^e^¥^¦ ±~x 6=e¥ 0c¤ c0±~
*c÷ e^o 3 ø4eð¦cC
eÅ b#=f dc¤,pCcdT ¹cf0p¡^d©^ ^¤ ¬ c ^e6dPcp¤ ¶,¯=e^cp ¢Mª0¢ eA¦ cC
d9ce6¬¤0
¶÷ o 3 ø¦ =¤,=f0µ e6ª ^¤ 7 ^e6eð¦cC^
d9cc¤,e6ph6 ñ6 ,c=^¢Mc^eð, =9f0~c ^d9¤ ^¥
, ª0¢Mc ,^h^ceÅe^h^±t¤0
e¥ ¶cC=,c=±tf cc e^»cp ¤c?0¡V
÷ o0¥ 3 ¬=ø( eAc¦ ^cCcw
÷ o0ùeðn ø ¤,= cd9^¤ c ,[¤0
÷ oùnø 4 ^¤,= cd9^¤ c
=Ë ß6ØQÒ Ñ6ÞÍ Ù0Ò ÑIÎ¯Ñ Û0Ú+V4Ö ÍÅÎ Ú&£PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔòÍ9ÙÚpÛ0Ô IQà cf~° d9¤,^p6V d90eð
Ie^,Q,? ÉfC,=f c ^¤ ¥
,ª0¢M^d#ªeů¤0
,ª÷ o 3 ø¥¥
,If c=c¤ c^c nµ¸±ce6=?µ ¡ +¢
∞ X (−1)k x2 (−1)n+1 x2 (−1)n+2 x2 rn (x) = = + + ... (1 + x2 )k (1 + x2 )n+1 (1 + x2 )n+2 k=n+1
C c=t¤0
²d9cQ¡V c©¤,=e^e^dPp¤ p¬tf0=fe6ª0d9d#ªþ6e^f0c ^ c±*^^cd96¤ ^e^f0c±* ¤ cpµ ^¤ ^e^e^ teV ^,¤ bcTA
d 0 ^ cdT¤,= Td (−1) x /(1 + x ) C,=d9^,p¥ ^d q = n+1 2
2 n+1
2 −1
−(1 + x )
=
X ∞ (−1)k x2 (−1)n+1 x2 . 1 = 1 + |rn (x)| = = (1 + x2 )k (1 + x2 )n+1 1 + x2 k=n+1
x2 x2 x2 x2 1 < = < = < ε. 2 n 2 2 n 2 2n 2 (1 + x ) (2 + x ) (1 + x ) 1 + nx + · · · + x nx n
4° e^¢§
ôe¥ ¥
,ª6?c¶
,! ¢4c^c ε > 0 =e^^ð
ô,=±0
6eð==f c[ e¥ c N = ε cC
cc
,¤ w^d9^e^ ¥ ¦ c n¹>cNe¥ ¥
= ªC
^[6ôee6ª0 ¤,6=c¥d
c^c¶c . ^¤,c=^ e6c |r (x)|
,we^¥¦ < ε6h¤,= cd9^x¤ 6=ª0¢ 0 = c C , Q , = eA¦cQ
d9ce¥¬y¤0
~÷ o 3 ø§,[e^^±~ e¥ cc±~ce^ r (0) = 0 cf~¹ ^d9¤ ^^6±0V
^d»0
^ ^¤ ¬fC,=f c cp c?¡V¥ ¬ cd#ª¤0
,ªw÷ oùnø¥
,Vf c=c¤ c^c nµ¸±ce6=?µ ε
ε
−1
n
n
rn (x) =
∞ X
x2 x2 x2 = + + ... (1 + x2 )k (1 + x2 )n+1 (1 + x2 )n+2 k=n+1
e^f0c±* ¤ cpµ ^ ¤ ^c=e^te^ ¤V0eM
² d96cQ¤ ¡VT d»c©0¤, =^e^ e^dPcdTp ¤ ¤, =p T¬td f0x=f/(1e6ª0+d9d#xªþ) 6e^f0[c C^, = d9c^±*,p^^¥c d9^6d ¤ q = ^1/(1 +x ) bcA
C
2
2 n+1
2
∞ X
. 1 1 x2 x2 1− = rn (x) = = . 2 k 2 n+1 2 (1 + x ) (1 + x ) 1+x (1 + x2 )n k=n+1
4° e^¢§
e¥ ¥
,ª6?c ¤ x → 0 ÷| c[ (¤,= cdþªC ¢¥¤ùø lim r (x) = 0 r (0) = 0 e¥ ¥
cp¥ ¬ c ,ce6ppcf~¤0
d9c?¡6ò¬[d9^ ¬=·7( ¤ cCcp ¬ c^c cpµ c=ycQ¡V¤0
¥w ÷¬o c3 ^ø¥ceAε¦
cC
,eðe^¥¦ ,x cCe^
^ ±wcr ¤ (x) ^e¥d9 ^c c±wc cC e^hc( c=^C¤,,=Q, =c6d9?^¤ c(c .¤°40
w ^÷ o0ùn
ø¥ pc c=cQC
, = f c c©¤0
¼÷ o0ùnø7=ªQ
6»eA¦cQ
¬eð¤,= cd9^¤ c,! ¢4cd c=¤ 6Cf =& ~e^cQ
^¤¡[p7^d c ¾fª Qx=f0= ¢40 6 T e¥ d e6ª0d9d9 ¤0
c÷ o 3 ø4º÷ oùnøMe^cc=6e6^ ,c~ e^¤,= dþ0¦ô c¥
^ I[cf ¤ ^e6 ce6~Sc Sf x = 0 ° ^0
c 9c=¾§c ò!ñ6¤0c
±7!c! f c e6f0ª0 d9xdP4=C,0=f eAc¦ cQ
,^¤ ^d9eðþ^ f ce6^ª0c(d9¤ dP0=
dT9÷ o¤,=3 ø. fT=d f7e6ª0ªQd9 dP¢¯ S ^e^(0)f c =^ S c(0)±h^=^c0d96¤ ^eCf c±h ¤ c^¤ ^e^e^ w¤,=, x→0
n
n
1
1
2
2
S1 (x) = x2
.
1+
1 x2 (1 + x2 ) , = 1 + x2 2 + x2
7C,=f c cp cQ¡V¥ ¬ c^cV¤0
~÷ oùnø§e^cc=6e6^ ,c S2 (x) = x
2
.
1 = 1 + x2 . 1− 2 1+x
n
I0
1
ó§¤,= [ ¤ ¥
¥
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
x2 (1 + x2 ) = 0, x→0 x→0 2 + x2 lim S2 (x) = lim (1 + x2 ) = 1 lim S1 (x) = lim
e^c¶C,Q ^ d9 S (0) = S (0) = 0 ;Q=f0 ¢4,=^dT.c~e6ª0d9dPô¤,= cd9^¤ côeA¦cQ
,7¥µ ^ceð»¤0
t÷ o 3 ø¥e^ce6cQ7^Cc!y ^ ¤ ^¤ T ò¦t¨ª0 f « ± p=f¡ p , 6eÅ» ^ ¤ ¥µ ¤ IT ^ c±»¤ ^¨¤ Tª0 f « § ¦h^± ¨9ª0h f e6« ª0 d9± dP,w p ,^ ¤,6=eð ~cd9¤,^p¤ C¤ Tc!eð ¦ccC±h
,¨7ª0^ ^f c« eð ^¤ ± 0
tce^÷ f ocpùn ø¥¬fe^ª ce6cQ7^^c x→±0
1
x→±0
2
lim S2 (x) = 1 6= S2 (0) = 0.
°4d96 dTMcQ
,=f c Écþe6ª0d9dP ¤ 0
÷ oùnøy=ªQ
6 ^ ¤ ^¤ T c±¨ª0 f « ^± , ¢4cdc=¤ 6Cf =0 Ée^cC
^¤¡p7^dSc fª x = 0 ce^f cp ¬fª,p=f cd©d9 c?¡^e6ɤ 0
=ªC
6Veð¦cC
¬eðw¤,= cd9^¤ c ¹
¤
¥
^ d ^ e c
± 6 e 7 , ¤ =
c 9 d ^
¤
c A e ¦ C c ,
7 0 ¦ Å e ô ¤ 0
c
0 0 f = c c
¤ § ¶ ¦ 6 e p =
6 [
^ e
c ± « ^ ce6¬V c~¤,= cd9^¤ c±~eA¦cQ
d9ce6h¤0
x→±0
2
X xED x
' C- "¦-;Qþ ;;"# ¹ªe6¬V¤0
O
∞ X
÷ o jø
un (x)
p ,l( 6 ,ôeð~¤,=¤,= c d9cd9^¤ ^ ¤ c ceA¦ eAcC¦
,cC
,77^^ cd9eðheðh [cp¤ 0=
e6[ e^ D¤,= ¥
eÅ ¥
,ª0¢M ¯e^c± e6 O
# =,x / e¥ ©c¤ 0
²cp ,÷ ª0o j^ ø4 ª0Td9±h c?¤ ¡V0D
hp¬h=f,¡ô(¨=ª0ªC
f 6« V e𢠦cCϑ(x) .eÅhc^¤,¤,== c d9^ ^¤ ª0c ¢ D
¬ |ϑ(x)| l¯^± <e6Lx6∈ ¬D c e^ ¤,=¥
ce6¬Ve^c± e6[ò^f0=6yI« ^ c f h ^¤,=^ e6 n=1
ϑ(x)(um+1 (x) + · · · + un (x)| ≤ |ϑ(x)|(|um+1 | + · · · + |un (x)|) ≤ Lε
f c=c¤,p~ d9^6Vd9^e6c[
,ôe^¥¦ x ∈ D cC
c¤ ^d9^ c O
¥ D x e¥ !0 ^ ¤0
!÷ o jø 4 ^ ¤ ^¤ T T D ¨ª0 f « c¶we6ª0d9dP ñ6c^c¤0
/ ^ ¤ ^¤ T, D S(x) = u (x) + u (x) + · · · + u (x) + . . . nµ¸l¯dº^± ce6e6ppf c6 d¬ ¤0c
=e6 ª0rd9d#(x)ªI¤ 0
7÷ o jø S(x) ¤ ¥
e6p= d nµ¸,=e6 c±e6ª0d9d9c± S (x) ÷ o; :ø S(x) = S (x) + r (x). ¾©e^0 ,ª¯¤,= cd9^¤ c±7eA¦ cC
d9ce6[¤0
4 ¤ 7Q?
p cd d9c?¡V c4ò¤,p¬,=e6cp ¬f c
cp , y¬=·7e^c¥¦[ e¥ c ncC
. cc¤ ^Àd9^c e6 pc p,cf»c¯¤0c
A
¶IªQ
,cp¶ c6e6ppc¤f07 ε¤ 0^
¤,¯pI^ ¢4e6=§ª ¦V|r
=(x)| < ε/4 ª V ¦ c
f ?¦ x ∈ D x x = x + ∆x D e^ ¤,=¥
[c« ^ f 1
2
n
n
n
n
n
n
1
|rn (x + ∆x) − rn (x)| <
ε ε ε + = . 4 4 2
ó
¤ª0^c±we6c¤ c e6ª0d9dP Vc=C ¯c= e^ce6c¶¯f c ^ c^cV e¥ e¥ =¥=^d9§¦w e¥ ¥
c=p¥ ¬ Sc (x) p , 6eÅw ^ ¤ ^¤ Tr (x) c±! D 6 =^cC
=¤ hñ6cd#ªô
, n
n
=Ë ß6ØQÒ Ñ6ÞÍ Ù0Ò ÑIÎ¯Ñ Û0Ú+V4Ö ÍÅÎ Ú&£PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔòÍ9ÙÚpÛ0Ô IR7 ¤ cCcp ¬ c,±ºf0=fôc f cp ¬xf c D d9cQ¡V ¾*c¤ 6T^ªQ ¤,¬Qp=p¬©¯p= f0cpc , ª0δ,=>^dþ0« P^ cc f|Sªô (x^¤,+=^∆x) e6 − ¡ +¢
Sn (x)| < ε/2
n
|∆x| < δ
|S(x + ∆x) − S(x)| = = |Sn (x + ∆x) + rn (x + ∆x) − Sn (x) − rn (x)| ≤ ε ε ≤ |Sn (x + ∆x) − Sn (x)| + |rn (x + ∆x) − rn (x)| ≤ + = ε, 2 2
f c=c¤,p~ôT¤,p¡=6[ ^ ¤ ^¤ T ce6¬V¨ª0 f « S(x) l¯cf0pQ= c(e^c± e6cñ^f ? ^ cVe^cc= c=·7^ ¢ ♦ lim
x→x0
∞ X
un (x) = lim S(x) = S(x0 ) =
n=1 ∞ X
=
0 ~f c¤ c=f c
x→x0
un (x0 ) =
∞ X n=1
n=1
lim un (x)
x→x0
÷ o oø C c7c=C,Q,=6?0cI
,y¤,= cd9^¤ c¯eA¦cQ
,7^^ceð¶¤0
y÷ o jø9c=Cd9cQ¡[^ c0 ^ T± ¤ ¥
¥ ¬ T±h ^¤ ¥¦cQ
c[QQ,=e6=ª0¢ ªC
c c[ ¤,=f ^e^f ¦h ¤ 0 cQ¡^ 0¦; p6=eðfy¡Ç =
d9cd9e66=cQp ¡dT6cô TòcVdTª¬ô,e¥ cc^ ¤ 7^ ¤ ¤,^T=cC ¦ccQc±
d9 ^d9¤ °M T
cdT,±w=f eAó&¦c¶ª0cC^d9
e¥ dP d9c e6e6^ª0¤,wd9=dP
¶ , ôceAd9¦
cC^=
,¤ 7c[c^eA^¦^cyccCe^eÅ
, c7¤± 06e6^
cyeÅ~ ^¤p 0 ,¤
0¥ µµ ¤ cT¤ hd9 § ^¤ ¢4¦þ co±tùDg7chf ¨¤ o0ª0^ù e6nf ø¥ « c e6±º ¤ ñ66c^±¤, ^c =f 6¤¤,0p
þC¤ T¶»d9©cQ¡ ^6f c=eA¦ ccC¤
c±þ¬eðcº f ¤,w=ñ6 ccd9±^c¤ p cþ=e6÷|e^dT <~ ' (X xEQ x ?e^e¥ ¥
cp¬V, ^ ¤ ^¤ T ce6¬Ve6ª0d9d#ªô¤
lim
x→x0
∞ X
un (x) =
∞ X n=1
n=1
∞ X
lim un (x).
x→x0
1 . n4 + n 2 x 2
÷|, @9 e^^±~ x. e¥m( pc¡[c
±~Tc±ôe^0 ø¥^ô#0
w= ¤, =c^ ccd9¤^¤
c[^e6eð¦¬VcC
¨ª0eÅ ôf «
, ~ e^^¥ ¦ ¤ ^¤ ,T c,e^f pcp~
¬ ,f~ª e^¥¦ n=1
x
x
1 1 ≤ 4, 2 2 +n x n
e¥ ¥
cp¥ ¬ c
,¶
= c^c¤0
[e6ª 76e^=ª6ydPp¡[c¤ ¤ª0¢M7 ±~¤ 0
n4
∞ X 1 , n4 n=1
^ef c=c± ce6¤ T=±ªh g0p ,e6ª0 d96dPeð7º
c= c =c7^^c ¤ 0T
d ¥^=e6¤ ¬Vd9c¨ ª0 f ^« e^f dÀ ¤^ 0
¤ ^c¤ d Te ,αph= ¤ 4~>e^1¥P¦ ó§xc A =e^ c O H x c/ ñ6e¥ c=ºV0¤ 0^
h d9öc?¡V¤0
cº ÷co0 jøI^ ^c ¤ ^¤ T^^¤ ,¤ c pD¬V# ,[xc= ¤ 6xCf 4 ¢4òy ?
=
e¥ [ D# [x , x] Z Z nX o XZ ÷ o ø u (x)dx, u (x) dx = S(x)dx = 0
0
x
x
∞
∞
x
n
n
x0
x0
n=1
n=1 x
0
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô =¤ ñ6cd cd cp ,ª0 ^ T±»¤ 0
»eA¦ cC
eð»¤,= cd9^¤ c~c= ce^¥ ¬ c x ¤ ¨( f0e^ ¤ cpµ l¯^± e6x6 ¬ c eA¦cQ
,»ªe¥ c ^ ¤ ^¤ T ce6¼¨ª0 f « ± u (x) e^0 ,ª ¤ ¥
§
,ª 7C^cVe^c± e6[d9cQ¡V c7ª ^¤ ¡[
p¬ ,ce6ª0d9dP[¤ 0
~÷ o jø p , 6eð ^ ¤ ^¤ T0
(c±~÷ o0;:ø¥DÇ ¨?ª0
==f ·7« ^e^± ¬ m( =e¥fô côd ¤ ¶
c=f0pe^Q0p ,ª7¥ ¤,¬=e6 cId9e^^¤ c ± ce6±eAV¦cCS(x) g0
d9¤ c¥e6
e6wp=÷ o jd ø S(x) e^c^e^A¬[
¶^d9¤,=cQ¡V^ cye6Tc!÷|¤,e^pdT,¬ n¤ ,d9=^e6¤wcpo ùn¬øf0cô=εcp> ¬=0·7 dT c
,!e^¥¦ x ∈ D T cp 0µ 1
IRB
0
n
÷ oúqQiø ,¹=c7±0
^^cd ¤ ^d9Ice^¤ ¥
^dTC^e6 c±~I ^^¤,? ¬ c^c e^ e¥ ^ ,eª0 6cd¼÷ oúqQiø |S(x) − Sn (x)| = |rn (x)| < ε.
Zx Zx Zx [S(x) − Sn (x)]dx ≤ |S(x) − Sn (x)|dx ≤ ε dx = ε(x − x0 ).
÷ oúqq?ø
ó
¤ª0^c±[e6c¤ c = ^^¤ ¤,c= Tf c ^ c±e6ª0d9d9 d9cQ¡V c=T cp ¬ c0 ^ c ,bcA
¯ ^=ª0¢ ,=e6¬ ^¤,=^ e6w÷ oúqq?ø§d9cQ¡V c7SQ=(x) eCp¬ x0
x0
x0
n
Zx Zx Zx [S(x) − Sn (x)]dx = S(x)dx − Sn (x)dx = x0
x0
x0
Zx Zx nX n n Zx o Zx X = S(x)dx − uk (x) dx = S(x)dx − uk (x)dx . k=1
¹cC
e6p= w÷ oúq?gø§h÷ oúqq?ø¥ cp ,ª0 d x0
x0
x0
k=1 x
÷ oúq?gø
0
Zx n Zx X S(x)dx − < ε(x − x0 ). u (x)dx k k=1 x
kP¨(e6c¤ ¤ d#6ªCdP , ª ÷ εo fhø¥ ªQm¼ ¢ñ6c÷ εd#ªy→e¥ 0¥øT
,hª6[^d*
ceC==d9Td ¬ nfhc[^ e^cf0eÅc ¥^
^cIe6 ^º¤,=÷ n=^→ e6∞=cø¥ Tcp ,cpª0 0d µ 6eðô
,ôe^¥¦ x D cC
c¤ ^d9^ c ñ6cc=C,Q,=6?c¤ 0
x0
0
x
n Z X k=1 x
uk (x)dx
¤ ,V¨(c ôf e^ ¤ ¤ ^cc= ? ccde^¬x
cIfpª0Qf0ppQ¬ = c±ycp =e6yeA¦cQ
eð¶¤,= cd9^¤ c¯c= ce^¥ ¬ c x d9^¤ ♦ cl(± ,eð¦cC¤,
= e^d9e^cd9e6c=©¤ ^ , c=Cpc© , e^6ceð± e6 ^cC9¦fcC=
fº d9»T
dT ,ºó&e^ª 7c± ^e6e6=ª0©¢Mg0w# ¤ ^¤,^=c =c d9 ^~¤ ¤,cô=eð ¦cpcpµµ
,^^7¤ ^¤ eðcw=¤ 0
,c
cªe^ªfe^=f0¢M=¢MV7~ 7^f c=c0c ¤ ^T I c¤,¯=eð ¦cC
,^^7¤ , ¤ ^ceÅh=¤ 0
= 6 ,b#cp =^f ( ,c=^ c ¤ pd9=^f ¤ c¯¤ 00
µ 0
0
∞
X 1 = (−1)n xn , 1 + x n=0
=Ë ß6ØQÒ Ñ6ÞÍ Ù0Ò ÑIÎ¯Ñ Û0Ú+V4Ö ÍÅÎ Ú&£PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔòÍ9ÙÚpÛ0Ô IWG ¦ cC¤
¥
e6peŲ=p ,© ¢Mc7 f ±yxe6ª0=d9d#1ª[ò^°M^c
d9,6=f0¤ c» ^ceC0f c^±y ¤ c!c=^ ¤ ^e^e^^ ^.¤ ¤,eA¦ ccC
= eðh
= ,6 |x| ¤,<=10 ¬¤, T=e¥± µ ¤ 6^ªC ¬Q=p ¡ +¢
Z1 0
1 dx = ln(1 + x) 0 = ln 2 = 1+x =
∞ X
(−1)
n
n=0
¾*e^ ¤,=¥
ce6w¤,p6 cQ¡^
Z1
Z1 X ∞
x dx =
0
ln 2 =
∞ X (−1)n n=0
(−1) x dx =
n=0
0
n
n n
n+1
.
∞ X (−1)n
n+1
d9¤,À¢ ø¥ª0 ¥
d9eðt ^e^f0cp ¬f c! c=^¡!÷|e^dT¤,p6
=p6 cQ¡^ y¨ª0 f « ±h¤ 0
b^±0 cpµ n=0
<~ ' (X xUT;x(¹cfpQp¬ cV¤0
∞ X
1 n2 + x 2
d9cQ¡V c c0 ^ cV ^^¤ ¤ cp¬ @ 9 xm(p¡[
T±w0 ^~
= c^cy¤0
[õ ^ ¤ ^¤ T,p!¨ª0 f « ~
,we^¥¦ x ¤0^¤,
y=eA¦^cQ
e6eðc V¤,= cd9^¤ cI,e^^±V e¥ c=c±Vc=e^ ==ff0=f
,e^¥¦ x T cp 6eð n=1
1 1 ≤ . n2 + x 2 n2
ó§¥c A? ¦ô=e^ Ic7ce^p c=± e6e¥~=ªyeð¦cC=
=d9 c e6T±y ¤ ,0=
¶ d9¤ cQ¡Vd9^ ¤~cI~ c 0[0, ^ x] c¯¹ ¤ c ^^¤ ^^¤,¤ c ¤ pc¬7=( ¢4 cp ,§ª0¦V d ¤ ¥µ Zx nX ∞ 0
n=1
∞
X 1 o dx = x2 + n 2 n=1
Zx 0
∞
∞
X1 1 x x X 1 x dx = arctg arctg . = 2 2 x +n n n 0 n=1 n n n=1
O ¤,= cd9 ^ ¤ ¥ c[Jeðx ¦/cC
,e¥ 7 ± ¤ eðw ¤c00
w ^ ^ ¤ cd*^¤
T ¨( §¨(¦h^¤ ¨^ ª0« f ¤ « c ± = ¤
÷ o jøM cp ,ª0,=6eð X ÷ oúqQø u (x) = T (x), ∞
0 n
cþe6ª0d9dPºñ6c^c¤0
¤ ¥
e6==p , 6*e^cc± ¤ cCcQ
ª0¢íc=e6ª0d9d9ä¤0
÷ o jø¥ ? = T (x) = S (x) n=1
0
∞ X n=1
u0n (x)
=
∞ nX n=1
o0 un (x) .
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô e¥ ¥
cbpcA
¥ ¬pµ c l¯^^^c± e6d9cQ¡V 6c[ ¬ c0c 0¤^ 0
c÷ o úqQ^ø^ ¤ p ,¤, c6peðV¬V¤,,=¯ c¢4d9^c¤ dþ c¯ eA¦cC^
,¤ 7? dP]xeðy, x[∈ D 1
IRI
0
Zx
T (x)dx =
x0
x0
=
∞ Zx X n=1 x
u0n (x)
Zx nX ∞
=
∞ X n=1
n=1
o u0n (x) =
[un (x) − un (x0 )] = S(x) − S(x0 ).
ñ6c¤,=^ e6 ccVe^ ¶¤,=¤ ¥^
c ? cIc
e^ ,¬y
cfe^¥p¦ Cpx¬ ∈ D c7^^cI
¨(¨(^¤ ^ « ¤ c?µ
¹ ce^f
cp= 6¬ fTª(x) = S (x) <~ ' ( X Ex X x4l7=~¤0
0
0
∞ X sin n3 x
n2
.
° ¤ ¥
¥ ¬²q?ø[eA¦cQ
eð *ñ6c=¤0
¤,= cd9^¤ c 5 gø[d9c?¡ c ^^ct c0 ^ c
¨(¨(^¤ ^ « ¤ c?¬ d9^@ ¤ 9 c p= f~x4fq?pø7f~ó§cc~A dP=pe^¡ cc¤ ¤ ¤ªC6,=eðfhªeA¦ ¾§cC^
,± ^7¤·¯ d9eð¤,h=e^ eC e¥#
c= T dT±º¤0¤
0c
þd eA¦ cC
eðþ¤,= cpµ n=1
∞ X 1 . n2 n=1
cC
g øò§l¯¦ ¨(¨(^¤ ^ « ¤ cp¬¶ c0 ^ cV
= T±!¤0
¥ ¬=^ p=f!f0=f!¤0
¤ c6µ ∞
∞
X n3 cos n3 x X 23 cos 23 x 33 cos 33 x cos x + + + · · · = = n cos n3 x 2 22 32 n n=1 n=1
¤,=eA¦cC
eÅ <~ ' ( X xE` x(¹cfpQp¬ cV¤0
∞ X sin nx
d9cQ¡V c c0 ^ c[
¨(¨(^¤ ^ « ¤ cp¬ @ 9 x #0
n=1
0 ∞ X sin nx n3
n3
=
∞ X cos nx
n2
,
f0e^c=f(cc=±7e6d9ª0^d9,d#? ª¯c e^^¬M ¤ ò^¤ ·7T=Q ¤,ò=¦ ¨cd9ª0^ ¤ f « cÉ eA± ¦ cCóP
¥
eð¯c,pT¥ e^¬^±( c e¥c 0c ^c ±I cce^&(
( ¨(¤ ¨(¥
^¤ e6p^ =p« , ¤ 6cp µ = ¤,=cd9^¤ c n=1
n=1
pË QÍ
¬ ÍÐÒÒÔòÍPÙÚpÛ0ÔMËYXÍÅÑðÙÍ Þß*¨±«QÍÐ×Ú § }|A
IRL
) +-'³ '(Ha%&!«ªÉ
©±
b^c¤ ©e6^ ^ §¦t¤0
ch d9^6!c=·7 ¤ c ¤ dP6 ^ T e¥ ^ yñ¥ ^d9^0µ p = ¤ e¥ §^ ¦¯ ¨Iª0c f ¤ « ¥
ű7 e9^ Q?
ò=¦~ c±I^^c¤, ? cce6 ¬ ¢ 5 ^c^e6¤ ¤ ¤ cc^ = ò ¤
p¨( ¨(¡^¤ ^^ « §,¦¯? ¨(¬c ¤ §d#¦ôªC ª05 ¤,=§pµµ 5 ^ ±~¶
¤ ®¯ª0 f « c,? ¬ T±h¤ 0
h0
X ÷ úq?ø a (x − x ) , A
a x ö¥ Q= eð7 Ic= x ,pCT=6eÅhe6^ ^ ÉdT ¹cp cQ¡V x = 0 , cp ,ª0 dþ¤ 0
♦ X ÷ ùgø a x . #0
Kó&÷ ^úq? ø§^ » ÷ c±wùg¤ø§0e^
=÷ CTúq?ø§=¥e^V^A¨(
c[¤ eAdP¦ ?cC
¬,peð¶wQ =¤ d9 ^, x −x¤ →0
tx ÷ 0 ùg ø x → ¤ x − x ♦ x
= x cf0pCT=6e¥ 4¥
,ª0¢Mxp¯ =0^cpµ & ó ¤ 0 ª f 0 ª ¤ ¯ ª c p = 6 e ¯ A e ¦ C c
9 d c 6 e 7 6 e ^
^
ò 7 ¦ ¤ 0 c M ¤ ^dP %& ' ` >x =¶Ay +- ·?¸x ò
] N*]6Z[Y6`Y6D,D>QF@ ,<©÷ úq?øô] >CQ\ _0Y x = x G Z[>[>QD!] >Q ºTZD>RN,D,Z[Y¸@RpV: Y ∞
n
n
0
n=0
n
0
0
∞
n
n
n=0
0
0
0
1
N@ :5IRp:D>Q>]ap(>A@Y¸@>QDZ¯>[GPD,Y^:y]
>QNZ4@@,Y]<[p_ ÷ Yú|xq?øP−@ :x]|>Q≤Q\0x_ |Y 3 G#Z[>¯>QD[@ :]>QQZ[>A@E |x − x | > |x − x | 3 U -+
x#¹ªe6¬w¤0
P a (x − x ) eð¦cC
eÅ bcA
ô^^c~c=7 ±©0 ^ e6¤ ^d9eÅ~f¶ªQ ¢ë÷ ø& ¤ ¤ ∞d óPT ªC¥
ccp p6¥ c¬¤ c¢M,7cô ©c^ª¤,e¥= c 6¢ ) →ø¥É 0 =e^e^d9nc=→ |a (x − x ) | < ak (x÷ k −=x const x ý ÷ ^ þ d C e = 9 d T þ d
p c = ¥ = ^ T d , c |x −¹xcC
|¥< |x − x | x − , |x − x | c,=e6h ^¤,=^ e6 cp ,xª0 6= d 0ø¥ |x − x0 | < |x1 − x0 | 0
1
0
1
0
1
∞
n
n
1
0
0
n
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n=0 n
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x−x 0 = q < 1. x1 − x 0
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?tñ6c^c* ^¤,=^ e6*Q=f0 ¢4,=^dTc* ¤
0 ^ ¤0
P ,Q ,p¶e ^f c=c¤ c^c[ cd9^¤,0e6== c=eðôd9^ ¬=·7e^cc==6e^=ªµ |a (x − x )| ¢M70¦~0 ^ c=^^cd96¤ ,^e^f0c±h ¤ c^¤ ^e^e^ ∞
n
0
n
|x − x0 | < |x1 − x0 |
n=0
k + kq + kq 2 + · · · + kq n ,
× ß6Ø=ß Ë}
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d9ce6h^^cd96¤ ,^e^f0c±h ¤ c^¤ ^e^e^ h ¤ q < Q=f0 ¢4,=^dTòc¤0
P a (x − x ) eA¦cC
eÅ =e^cp ¢M c òcþ
cfpCTp6 1 ^¤ =ª0¢²,=e6¬(ª ^¤¡[
^ óP¦ cC
,7 ± eðV e¥ cc±¤0
P a (x − x ) p , 6eð dPp¡c¤,=c±»¤0
t÷ úq?ø
,»e^¥¦ x ªC
cp 6c¤ ¢M70¦tªe¥ c ¢ |x − x | ≤ r < |x − l¯cxf0p|QpóP ¥¥
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c÷cCú
q?ø§eAeð¦tcC
d96eÅcCh
c¤,d= ¬Écc=d9w^¤ ¤ c c= cpµ ^cD®(¹¤ ¥
cp cQ¡V dTc! ¤ |x − x | > |x − x | ¤ 0
P a (x − x ) eA¦cQ
eð c~cA
f0=f
cf0pQ= chò·7=eA¦ cC
eðt»¤0
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cp¥ ¬ c &¤0
P a (x − x ) ¤,=eA¦cQ
eð &b#=f d c¤,l(pC ,c dTe¥^^c ¤ C ^dP c^
ccf0¤ p0Q
=,e6ª 7^e6=ª6* cp c?¡V¥ ¬ c e¥ c e^¥¦ x ªQ
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weð¦cC
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ªeC(eA¦cQ
d9ce6 0=e^e^d9c=¤ d©¤ 0
P |a (x − ?p=f7f=f¯^^c ^¤ ? eð¦cC
d9ce6[e^c,?
=6e§ ^¤ ? cdeA¦ cC
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x) | P a (x − x ) ¹¤ d9^ df¯C,=f c cp cQ¡V¥ ¬ cd#ª7¤0
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∞
n
n
0
n=0
∞
n
1
0
n
n=0
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0
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0
1
0
n
n
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n
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0
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0
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n=1
n
n
0
n
n=1
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n
0
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n=1
a (x − x )n+1 a u n+1 n+1 n+1 0 = lim = |x − x | lim lim . 0 n n→∞ n→∞ n→∞ un (x − x0 ) an an
l( ,ôeA¦ cC
d9ce6h¤ 0
¯
cp ,¡V c[ò¬T cp ^ cªe¥ c a n+1 |x − x0 | lim < 1. n→∞ an
bcA
R=
a 1 n = lim . lim |an+1 /an | n→∞ an+1
¾§ce^ cp ¬=Cc==·7 e^¬ ¤ C,=f0cdmc=·7
,~¤,?
ªeC[eð¦cC
d9ce6w cp ,ª0 d n→∞
R=
1 p . lim n |an |
n→∞
¹¤ ~ñ6cde¥ ¥
,ª6¶=ò e^ ¬y c¥
^ Ie6^ ^ cCcV¤
x ±R 0
∞ P
n=1
an (x − x0 )n
¤
x=
pË QÍ
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d9ce6w¤0
§ }|A
∞ X
(−1)n−1
LZM
xn x2 x3 xn =x− + − · · · + (−1)n−1 + . . . n 2 3 n
@9 xqó§ce6p= d¤0
w¯=e^cp ¢É §¦h¥ n=1
x2 xn |x| + + · · · + + . . . 2 n
ô ¤ d9^ d ¤ C,=fôl7? =d9^¤,
n |un+1 | = lim |x| = |x|. n→∞ n→∞ |un | n+1 lim
e¥ ¶ñ6c=[ ¤ ¥
¥ hd9^ ¬=·7¥
« ¤0
~eA¦cQ
eð °Me^¢ò
1 #g00,
w¹¤,¤ = eA¦cQ
eð f0c¤A0
|x| > 1 /
|x| < 1
0
−1 < x <
x=R=1
1 1 1 + − + ... 2 3 4 x = −R = −1 1 1 1 1 −1 − − − · · · = − 1 + + + . . . 2 3 2 3
eA¦cQ
eðh c ¤ C,=fª¶áI^± «,[ô ¤
¤0
1−
¤,=eA¦cC
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§
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=6 ∞
4
∞
n
n
n
2
n=1
n=1
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C
°4d9^e^6¢§V
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, ª^6 p T±hc¤,?¤
0
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cC
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øò¹¤ d9^ ¤,?
f? ¬ T±~ ¤ C,=fwmc=·7 , d9^^d lim
r n
nn |x − 1|n = |x − 1| lim n = n→∞ (n + 1)2
0, x = 1, ∞, x = 6 1.
∞
n+1
C °4ce^¢§¤,
?
he¥ª eI¥
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? #d9ce6cw¤h0¤,
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ªeIeA¦cC
d9ce6we6^ ^ ò¦h¤0
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n
2n
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n=0
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,c ¤ ¥
¥ 6 ©¤,?
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d9ce6 1
LRS
|an | , n→∞ |an+1 |
R = lim
,?¦cC
d
2n+1 = 2; n→∞ 2n 2n+1 n R = lim n = 2; n→∞ 2 (n + 1) 2n+1 (n + 1) = 2. R = lim n→∞ 2n n
a) R = lim
ø ø ,ª0,=6?
eðºªe^ö¶e𤠦0cC
! d9ø(ce6 cþ0 ^e^ ¥¦t Td¼¤ ¥
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cCª
e^c¤ 7ceA ¦c cQ
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p ¡Vd9c e6T4 e^cò± e6^f0=¯¢Me67^^ ±^ ¯ò¦¶^c¤¤ 0^
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=d9þ0
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õcp ^hc=7^^c»0
÷ úq?ø7 ¤ cQ
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¤ !ccd9S(x) 6^¡0ª
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f t ÷ [−R, e^e¥ Å
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©=ªC
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n
n
n=0
∞ X
an x n = a 0 + a 1 x + a 2 x 2 + . . . = b 0 + b 1 x + b 2 x 2 + . . . =
∞ X
bn xn
¤ eA¦ cQ
x c==^c0cQe¥¡[ ¥
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n=0
0
0
a1 x + a 2 x 2 + . . . = b 1 x + b 2 x 2 + . . . ,
f c=c¤ c(d9c?¡V c7Q= eCp¬Vf0=f
x(a1 + a2 x + a3 x2 + . . .) = x(b1 + b2 x + b3 x2 + . . .).
pË QÍ
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d96c e^¤, =cp y¬=C ccp ===¥·7p e^¬ô¬6
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6 CªfeIªh eA ¦y cC¯
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4fI¤0µ § }|A
1
1
2
X ∞
an x
n
0
=
∞ X
2
nan xn−1 .
¹ ¤ d9^ ôf cp ,ª0 ^ cd#ª!¤ 0
,ªw¨(c¤ d#ªQ ,ªhl7? =d9^¤,V
,c ¤ ¥
¥ ^ ©¤,?
ªeC eA¦cQ
d9ce¥. ¤ 0¦cQ
dþf~¤,=^ e6=ª n=0
n=0
n|an | n |an | |an | = lim = lim = R. n→∞ (n + 1)|an+1 | n→∞ (n + 1) |an+1 | n→∞ |an+1 |
R¯(°v± = lim
c0b# =^ f Tdtd c
¤, p¨(Cc¨(dT^0¤ ¤,^?
« ª¤ eceð¦=cC
^dTd9c9e6e^c ,RQ
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¥
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V X X ÷ ø u (x). u (x) = S (x) = ∞
0
0
∞
0 n
n
l( ,t
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,ª0¢M ±w¤0
n=0
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n=0
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n=0
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,¶ ^^c[¤,?
ªeeA¦cQ
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dºfô¤,Qµ 0
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R = lim
b#c=0f ^d´ cT¤,döpC cdT§^^¤,¤ ?,
¤ cªe= eA¦ cQ^
dT Éd9e^cce6, ?
R=6ºe6e^ ¤,^ ?
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eð¦cCò
cpd9 ,cª0e6 ^ Rc6cº eA¦ cCþ
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¤ d9¤,cce6=¬[ hcp , ª0eA ¦^cC
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S(x)dx =
Zx X ∞ n=0
∞ Zx X un (x)dx, un (x) dx = n=0 0
|x| < R.
÷ 3 ø
d9ce6Ç =d9x6= dTR0 cVx¨(c=¤ d#−RªC w0^÷e¥ ¶3 ø§e^cp ¤,¬=f c¥
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eA¦cQ
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,»f c f ¤ 6 c^c!¤0
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1
∞
Z1
∞
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Z1
∞ ∞ n+1 1 X X (−1)n n x (−1) , x dx = = n + 1 0 n=0 n + 1 n=0 n
ce^f cp ¬fª ce¥ ¥
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, x = 1 p , 6eð¶eA¦cC
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T
T ] − l¯ R,^±
R[e66 ¬ c Q
,¯¤ 0
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d9ce6 ¾»¾*e^0 ,ñ6ªc¤,±~=cf ¤ c^d9e6^¤ δcce6±heð¦¤cC0
h d9eA¦ ccCe6
V¤eð0h
¤,[=÷ ùgcpøµ ÷|0 ~e^c± e6Vø§^^cd9cQ¡V c[ c0S(x) ^ c
¨(¨(^¤ ^ « ¤ cp¬ ¤ ^d 0
0
S 0 (x) =
∞ X
nxn−1 .
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&e^cA =e^ cte^c± e6=ª q&¤,=T ccþd9c^ ¤ ^ ¤,c»=« eð¦ cp¢ µ
c0 eð ^ º cª0^f0cIp
Q = ¨( ¨c^± ¤ ^c f « ¤ ^¤ e6c c=e6 ¶d9f cQ ¡V^ cI¤ ^ ¤ cT=c ¤c±¬7¨ ª0^ cCf
« c f ¤,Sp(x) c c=fªC
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cdT d9¤0
,cQ♦ª¶¡^¹e60c
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^ w^ ÷ cj^ø¥c~ ¤cy0
ñ6l(c±w ,!cp, ?=¦e6cQ¡[w
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d9c ce6cc=7c=^ cce^6c©e66 ^¬ ^c c ^c»¤0
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d9c n=0
0
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n
n
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n
n
n=0
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x
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¥ ¬Vcp =e6¬^^ceA¦ cC
d9ce6h~T e¥ ¬V^^ce6ª0d9d#ª F
∞
n=1
n
∞
n=1
4n−3
pË QÍ
¬ ÍÐÒÒÔòÍPÙÚpÛ0ÔMËYXÍÅÑðÙÍ Þß*¨±«QÍÐ×Ú @9 xqó§ce6p= d¤0
wId9cC
,ªQ 6± § }|A
LR7
∞ n X x n n=1
ô ¤ d9^ d ¤ C,=fôl7? =d9^¤,
|un+1 | |xn+1 ||n| n = lim = |x| lim = |x| < 1. n n→∞ |un | n→∞ |n + 1||x | n→∞ n + 1 lim
°4e^¢§
−1 < x < 1 ô¤,?
ªe(eA¦ cC
d9ce6 R = 1 ?e^e¥ ¥
,ª^d*¤0
w,eð¦cC
d9ce6¬ ¤ ¤0
x = −1 eA¦cQ
eð~ªe¥ c= c 5 ¤
x=1
¤ 0
−1 +
1+
1 1 − + ... 2 3
1 1 + + ... 2 3
¤,=eA¦cC
eÅ´ ÷|¥=¤ d9c ^e^f ± ¤0
;ø¥±?p=f Écp =e6¬ºeð¦cC
d9ce6 ¤ 0
−1 g0≤,# x 0<
1
[−1, 1[
0
x2 x3 xn S(x) = x + + + ··· + + ... 2 3 n
−1 ≤ x < 1
d9cQ¡V c c0 ^ c[
¨(¨(^¤ ^ « ¤ cp¬ bcA
¤0
S 0 (x) = 1 + x + x2 + · · · + xn−1 + . . .
^e6¬h^e^f c ^ c~ª0T=¢Mp^^cd96¤ ^eCf0p© ¤ c^¤ ^e^e^ e6ª0d9dPôf0cpc¤ c± ^^f c T e¥ , 6eð ÷ ; :ø 1 S (x) = . #0
e^ce6=¹=¤ ?c ^ ^ò^¦!¤ ¤ c =¤ !c÷ C; :cCø§
[§c¦;p =e6d9^~61¶eð−¦cCxc=
y ¡[d9c¯e6eC= d9T cp±! , ª0 6d ¤ ? ©eA¦ cC
d9ce6 ] − 1, 1[ 0
Z
0
S (x)dx =
Z
dx +C 1−x
0 − x) + C. ° ¤ ¥
¥ d ce6cQ ª0¢ C 0 ¹ S(x) ª e6¬ =x−=ln(1 ý ÷ c f0ôcp =e6©eA¦cC
d9ce6ø6bcA
0ª0d9dP[¤0
& ó S(0) = 0 = − ln(1 − 0) + C C=0 S(x) = − ln(1 − x) = ln
1 . 1−x
l( ,ôe¥ ,ª0,ph ø§=ªC
^d eA¦cC
¬y0¦ô¤0
[eC^e6 c±we6ª0d9d9c± ∞ X n=1
x4n−4 =
1 . 1 − x4
1 ×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô LRB C c=[e6^ ^ ,c±ô¤ 0
~¤,= cd9^¤ ceA¦cQ
eð~I ¢4cd ^¤ ? = 6¡[p7^dcp ?µ
e6©eA¦cQ
d9ce6 . cñ6cd#ªw
cªe^f0=6w c0 ^ c ^^¤ ¤ c= V,ô ¤ cpµ d96¡ª f (0, x) x|x|< <1 1 x
∞ Z X
°4e^¢§
x
4n−4
dx =
n=1 0
Z1
dx . 1x4
0
Zx Zx ∞ X 1 x + 1 x4n+3 1h dx dx i 1 arctg x + ln = + = . 4n + 3 2 1 + x2 1 − x2 2 2 x−1 n=1
<~ ' (` xEP x((=±he6ª0d9d#ª¶¤ 0
0
0
∞ X 1 . nen n=1
@9 x a ¸F]6`>?]^>Qa?=e^e^d9c=¤ d*e^ cd9c¥p¥ ¬ T±he6^ ^ ,c±h¤0
∞ X 1 n x nen n=1
eP¤,?
ªe^cdweA¦cC
d9ce6 Ie6ª0d9d9c± ?¹ce^f0cp ¬fª eA¦cQ
T±¯¤0
7 cp ,ª0,=6pµ ðe ¯òe^ cd9c¥p¥ ¬ c^cR ¤ = ex = 1 7ñ6pMS(x) c f04 ¤ ,?
6¡Vcp =e67eA¦cQ
d9ce6 ch¤ cCe6
ª0 d9¨(dP¨(h^ ¤ eð^¦ cC«
¤cª^^cwd¤ ¤,0=
~^ ce6 ¤ ¥c
¥ eÅtf0=f S(1) l( ,©,?¦c?¡[
^ ©e6ª0d9d9 S(x) ∞ X xn . S(x) = n ne n=1
bcA
[e^0 ,ªôe^c± e6e6^ ^ ,§¦h¤0
c c? ,ª0 d 0
S (x) =
X ∞
xn nen
0
∞ ∞ n ∞ X x 0 X xn−1 1 X x n−1 = = . = nen en e n=1 e n=1 n=1
d9¹ ^c e¥e^ ¬=f ·7cp ¬¬dþff0ª¥=
ft© e6cª0 pd9« d#=e6ªt;óP ^eAe^¥¦ f
cCcc
^p d9 ¥c ce6¬±º ^c ^|x/e| c d96¤ < 1C9e^f cc»±þ c¤ e¥ cÅ^
¤ ^e^¢4e^¢ ºe6e^ª0c!d9d#C,ªº=d9d9^c?,¡Vp cÅ ^§dTµ n=1
S 0 (x) =
1 1 e1−
x e
=
1 . e−x
¤ 6Cf0 [0, x]b cA«
¥ f cd 6¡[p7^dº[ 0µ ¹^c¤ e¥ ?¥
I ^eA¦(cC¤,
= d9^c e6e6 c[|x| ¤
Zx 0
x dx = − ln(e − x) 0 = e−x
= − ln(e − x) + ln e = ln
e , e−x
pË QÍ
¬ ÍÐÒÒÔòÍPÙÚpÛ0ÔMËYXÍÅÑðÙÍ Þß*¨±«QÍÐ×Ú e^cc=6e6^ ,c e6ª0d9dP[ eA¦ cC
c^c¤ 0
§ }|A
LWG
∞ X 1 e . = S(x) = ln x=1 n ne e−1 n=1
c ¸F!]6`>?]^>Qa?=e^e^d9c=¤ de^ cd9c¥p¥ ¬ T±w¤0
f c=c¤ T±~Q=d9^ c±
∞ X 1 , n+1 x n=1
y = 1/x
e^cQ
eðhfôcT cd#ªôe¥^ C cd#ª~¤0
,ª ∞ X
y n+1
e7¤,?
ªe^cd²eð¦cC
d9ce6 R = 1 !cp =e6¬¢ÀeA¦cQ
d9ce6 |y| < 1 .°4e^¢§
¶e¥ ¥
,Cª6? c=ctCcpc ,c =67»^ c e^T cp±* ¬=e6Cc^ p^ ¬ eðc±²
¤ ,0
² eð^¦^cCc»
e^eÅc²± e¥¤,=c d´cd9^&¤ b cc»ð
»c,p© = e6 ^¤ |x|Q > 1 c (e, ∞) « ¥ f cdþ ¤ ,?
6¡p76dcp =e6h¤,== cd9^¤ c±~eð¦cC
d9ce6 d9^^d n=1
Z∞ X ∞ n=1
e
1 xn+1
∞
dx =
∞ Z X
n=1 e ∞ X
∞ X
∞
X dx lim = A→∞ xn+1 n=1
ZA
dx = xn+1
e
∞ 1 A 1 1 1 X 1 lim =− = lim − . =− A→∞ nxn e n A→∞ An en nen n=1 n=1 n=1
¹¤ [c[ dP= =,c
∞ ∞ X 1 X 1 1 1 1 = 2 = 2 n+1 n−1 x x n=1 x x 1− n=1
,=±0
^d Z∞ X ∞
Z∞
1 x
1 ∞ 1 1 dx = dx = lim ln 1 − = A→∞ xn+1 x e 1 − x1 x2 n=1 e e e 1 1 e = ln , = lim ln 1 − − ln 1 − = ln 1 + ln A→∞ A e e−1 e−1 1
e¥ ¥
cp¥ ¬ c
∞
X 1 e ln . = e − 1 n=1 nen
<~ ' (` xEQ x((=±he6ª0d9d#ª¶¤ 0
∞ 1 + x n X (n + 1) 1−x n=0
ôe(^( cd9c=7¬¢¼ ¤ ce6ª0d9d9 ¤ cp¬V e¥ cT(¤ 0
∞ X n=0
(−1)
nn
+1 , 2n
∞ X n+1 n=0
2n
.
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô e^ @cC9
eðh f~xc?eðT¦ cC
cTd#ª¶±te6c^c =^7 ^ c d#Tª~±t¤0e6
,ª^ ^ c±¤0
tQ=d9^ c± (1 + x)/(1 − x) = y 1
LRI
(n + 1)y n ,
f c=c¤ T± Éf0=fd9cQ¡V ctª0¥
¬eð É,= ¤ d9^¤ Ée cd9c=7¬¢Ä ¤ C,=f0»l7? =d9^¤, eA¦cQ
eðw[cp =e6 |y| < 1 =e^e^d9c=¤ d*e^ c=dPc=6p¥ ¬ T±he6^ C c±w¤0
P y p=f¡(eA¦ cC
,7 ± eðhcp =e6 |y| < 1 fôe6ª0d9d9 ∞
n
n=0
S(y) =
∞ X
y . 1−y
yn =
, ¹ª0¤ cC
d ¨(¨(^¤ ^ « ¤ c=Vñ6cV¤,=^ e6=cyc[e^¥¦ôc f?¦~ ^¤ ? eð¦cC
d9ce6 ; c?µ n=0
∞ X
e¥ ¥
cp¥ ¬ c
ny
n−1
n=0
=
∞ X
ny
n−1
n=1
∞ X = (n + 1)y n = n=0
1 (1 − y)2
÷ oø 1−x m(=fôª ¡Ic=d9^,? ce^¬ ñ6cV¤,=^ e6cVe^ ¤,=¥
c[
,~e^¥¦ x ªQ
cp 6c¤ ¢Éµ 70¦ô ^¤,=^ e6=ª ∞ X
(n + 1)
n=0
1 + x n
=1
.
1−
1 2 1 + x 2 1 1− = . 1−x 4 x
1 + x < 1, 1−x
f c=c¤ c(e^cc=6e6=ª^
=ª0d
¤ª0^ d 6·7^ ^d* ^¤ c^c ^¤,=^ e6
−1 <
1+x > −1 1−x
0 p , 6eð~d9 c?¡^e6c 0 d9 cQ¡^e6c
1+x < 1. 1−x
E1 :] − ∞, 1[ 5
2 >0 1−x
¤ 6·7^ ^d*=c¤ c^c 1+x <1 1−x 2x <0 1−x
9°=³¥
wch¤ 6·7^ # cp ,ª0 d ^¤ ?
4eA¦cQ
d9ce¥wcEc2=7:]^− ∞, 0[∪]1, ∞[ c ^ c 6 e ^
^ i c ¤ ce6ª0d9d9 ¤ cp¬V e¥ cc^Tcy(¤¤ 00
E,¤,:]=−e^e^d9∞,c=0[¤ dc7ª0¤,= ^ 1 1+x =± , 1−x 2
pË QÍ
¬ ÍÐÒÒÔòÍPÙÚpÛ0ÔMËYXÍÅÑðÙÍ Þß*¨±«QÍÐ×Ú ¤ 6·7b#^= fô f0d9=hf f c=c¤ §¦ô p ,c ,¢M cpeÅ c?¡Vx =w−1/3 ÷ oø § }|A
1
x1 , x 2 ∈ E
LRL
x2 = −3 x = x1 = −1/3
,,=±0
^d
∞ ∞ 1 − 1/3 n X X 1 1 (n + 1) n = (1 + 3)2 = 4. = (n + 1) 1 + 1/3 2 4 n=0 n=0
ó§cc=6e6C c cp cQ¡V [h÷ oø ∞ X
(n + 1)
1 − 3 n 1+3
=
<~ ' (` xUT;x((=±he6ª0d9d#ª¶¤ 0
n=0
x = x2 = −3 ∞ X
(n + 1)
n=0
, cp ,ª0 d
1 1 2 4 (−1)n = 1 + = . 2n 4 3 9
÷ ø (2n + 1)! @9 xó§cA =e^ c¶ ¤ C,=fªôl7? =d9^¤,¤0
eA¦ cC
eð,Ve^^±w e¥ cc±!ce^ § ë ¾ c p = 6 e ð e ¦ C c
9 d c 6 e ² 6 e ^
^
c ^ t c ¤ 0
÷ ø = c C 9 d Q c V ¡
c ^ ^ c ¤ c0 ^^ d cS(0) = 0
¨(¨(^¤ ^ « ¤ c= = S(x) =
∞ X (−1)n x2n+1
.
n=0
S 0 (x) =
∞ ∞ ∞ X X 0 X (−1)n 2n (−1)n (−1)n 2n x =1+ x , x2n+1 = (2n + 1)! (2n)! (2n)! n=1 n=0 n=0
e^c¤ c= ^c=d ·7S^ (0) ¢ = 1 / 7IcC
c7
¨(¨(^¤ ^ « ¤ c= I cp ,ª0 ^ c^cV¤0
[ ¤ cC
f 0
0 X ∞ ∞ X (−1)n 2n (−1)n 2n 0 S (x) = 1 + x (x ) = = (2n)! (2n)! n=1 n=1 00
∞ ∞ X X (−1)n 2n−1 (−1)k 2k+1 = x =− x = −S(x). (2n − 1)! (2k + 1)! n=1 k=0
b#=f dºc¤,pCc=dÉ ¥ ª
d9cQ¡V c[,=±hf0=fô¤ 6·7^ IQ?
Q hmc=·7
S(x)
b#=f¶f=fôc=7^I¤ 6·7^ (ª0¤,= ^ h d9^6V0
S 00 (x) + S(x) = 0,
S 0 (0) = 1.
S(0) = 0,
cþce6^^cQcþ cC§
¦;e6= = cf0º þ,Q,? ¬óP T¥
cªe¥p c¥ ¬ c ¤ ¤ 0
cQ
÷ ø§feA¦e¥cQ
¥
,ª0eð¢Mwf~ e6dõª0d9Cd9, Q ^ d S(x) = ? = C = 0 C = 1 sin x S(x) = C1 cos x + C2 sin x,
1
2
∞ X (−1)n x2n+1
= sin x.
c¤,=^ e6cVe6 ¤,=¥
cV,e^^±~ e¥ cc±~ce^ <~ ' (` xEX x((=±he6ª0d9d9 e¥ ¥
,ª0¢M0¦h¤0
c X (x − x ) ø X(−1) (x − x ) , a) , C
n=0
∞
0
ø
n=0 ∞ X n=0
(2n + 1)!
∞
n
n!
(x − x0 )2n , (2n)!
0
n
Qø
n=0 ∞ X n=0
n
n!
(x − x0 )2n+1 . (2n + 1)!
1 ×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô MONRN @9 xøó§cA =e^ c ,= ¤ d9^¤ ¤ C,=fªhl7? =d9^¤,;¤0
eA¦ cC
eð,ye^^±
e¥ cc±ôce^ ¹cp c?¡V
÷ úqQiø n! n! ¤ ^d S(x ) = 1 ;ce^ c? ¬=^ª^d9eð!c=Cd9cQ¡V ce6¬¢ c0 ^ c^cV
¨(¨(^¤ ^ « ¤ c?µ ôe6^ ^, c^cV¤0
[[cp =e6~eð¦cC
d9ce6 bcA
y÷ úqQiø§,=±0
^d S(x) =
∞ X (x − x0 )n
=1+
n=0
∞ X (x − x0 )n
,
n=1
0
0 X ∞ ∞ ∞ X (x − x0 )n−1 X (x − x0 )n (x − x0 )n = S (x) = 1 + = = S(x). n! (n − 1)! n! n=1 n=1 n=0 0
b#=f dºc¤,pCc=dÉ ¥ ª
d9cQ¡V c[,=±hf0=fô¤ 6·7^ IQ?
Q hmc=·7
S(x)
° =7^¤ 6·7^ (ñ6c^c7ª0¤,= ^ ¶f=fVª0¤,= ^S(x ô)e4=¤,p1.6
¥ , ¢M7 d9 eÅô ^¤ ^d9^ §µ d9y ^^f0c,?¦cQ
eðh~ d9^¥y0
S 0 (x) = S(x),
0
¹ cC
ce6± p= ^^c77 ,Q,? ¬ c4ªe¥ c = ¤ 0¦cQ
dfye¥ ¥
,ª0¢M7Cd#ªVª0¤,= ^ ¢ ce6cpµ b#=f Cdº=ce¤,pCcdT X (x − x ) ÷ úqq?ø . =e n! ½4,? c^ c,?¦cC
d X ÷ úq?gø (x − x ) (−1) =e . n! C c¤,=^ e6cp=f¡(e^ ¤,=¥
c,[e^6±h e¥ cc±~ce6 óP cQ¡V [¤,=^ e6w÷ úqq?ø§©÷ úq?gø¥ cp ,ª0 d S(x) = Cex .
−x0
∞
0
n
x−x0
n=0
∞
0
n
n
−(x−x0 )
n=0
c=fªC
∞ ∞ X X 1 (x − x0 )2n n n [1 + (−1) ](x − x0 ) = 2 = ex−x0 + e−(x−x0 ) , n! (2n)! n=0 n=0 ∞ X (x − x0 )2n
=
ex−x0 + e−(x−x0 ) = ch(x − x0 ). 2
¾§TQhI¤,=^ e6h÷ úqq?ø§¤,=^ e6cw÷ úq?gø¥ ,=±0
^d n=0
c=fªC
(2n)!
∞ ∞ X X (x − x0 )2n+1 1 n n [1 − (−1) ](x − x0 ) = 2 = ex−x0 − e−(x−x0 ) , n! (2n + 1)! n=0 n=0
÷ úqQø
÷ úqV3 ø ¾pº=¤ Q= f0§ ¦~¢4¨ ^ª0 f « c=± d96 (2n dT, +c1)! ¤ e6 ^x ^= ,0Td9d9cQw¡V¤ 0c7
= d9cp ,~ª0, ¬e^^ ±~¤ ¥
e6e¥= =cp c^±~ ce^( ñ¥ ^d9^0µ e sh x ch x ∞ X (x − x0 )2n+1
=
n=0
0
x
x
e =
∞ X xn n=0
ex−x0 − e−(x−x0 ) = sh(x − x0 ). 2
n!
,
∞ X x2n ch x = , (2n)! n=0
sh x =
∞ X n=0
x2n+1 . (2n + 1)!
Ê?²?Ë}¬ Í Ù,ß[ÖÖÒßÅÛ~Î]QÍ
¬ ÍÐÒÒÔPÞ ÖÙÚpÛ,ß^Þ Ö MONZM ÿ ³± !´d´!- +-'% !% ¾ñ¥cd»¤,p6
¥ Md9þf ¤,pf cIce6=p c d9eÅ, ^f0c=c¤ §¦c ^¤,=« 0¦,?
Ve6^ ^0µ Td9¹~c¤0
=c d9 c7ª0d9 cQ¡^ e6^ ^, c^c[¤0
¯,7 ce6cQ T±ôd9 cQ¡V¥ ¬ ¤ c6µ cC
eð ,f0=fy
,ôeð f0c6c[eð¦cC
,7^^ceðh¤0
ª ^dþª0d9 cQ¡^ ôf0p¡[
c^ce¥ =6=¥µ d9c^c[,[ñ6cpyd9 cQ¡V¥ ¬ X X ÷ÐqQiúq?ø (λa )x . a x = λ ¹¤ ¥
cp cQ¡V dT cV¤,?
ªe^ eA¦ cC
d9ce6h¤ 0
c X ÷ÐqQiùgø a x , ∞
∞
n
n
n
n
n=0
n=0
∞
n
n
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R 4 d9^ ¬=·7 ±h¯ñ60¦~¤,?
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,h? ^^¤,p ^e^f 0¦h cp cd9c X X X ÷ÐqQi 3 ø (a ± b )x ; b x = a x ± n=0 ∞ X
bn xn
n=0
n
n
A
n=0
n
n
n
an x n ·
∞ X
n
bn xn =
n=0
∞ X
n
÷ÐqQiùnø
n=0
n=0
n=0
∞ X
∞
∞
∞
c n xn ,
n=0
Ð÷ qQi jø (= cd9 dT;c¶ d9^ c÷ hcp ¬f0c øM ¤ cC¥
6 [¤ 0
cy cômc=·7º÷ g0 jøÉ ¥µ ♦ ¤ ^^ cCe6
cw[÷Ð qQ¤ icùnø¥C ¥
^ (e6^ ^, §¦~¤0
ce^ c7e6^ ^ c±~¤
c^e^ ^ pô¤,Qµ °4d96 dTc,=e6 cI ¤ ¶
¥ ^ he6^ ^, §¦h¤ 0
cp=f¡(dPcQ¡V c ¤ ¥
e6=?µ ¬[0
(e6^ ^ c^cy¤0
X X X ÷ÐqQi; :ø c x , b x = a x ^e¥ ¶cp ¬f c[f0cñ^¨(¨( « ^ c=Q ^~c=[ªQ , ,¾ ¤ c= cdþe¥ ,ª0,=¤ 0
»÷ÐqQi ø& c mf cñ^x¨(=¨(0 « c ^¤,p =6eÅe6~6 Cb ªC ¬ c^cV¤ 0
~÷ÐqQi; :ø§d9cQ¡V c[ cp ,ª0 ¬V(ªe¥ c c cn =
n X
ak bn−k .
k=0
∞
∞
∞
n
n
n
n
n
n=0
n=0
n=0
n
0
k
∞ X
n
an x =
∞ X
n
bn x ·
∞ X
c n xn ,
c¤ cQ¡[
=¢M7^^c ,e^cA =e^ cw÷ÐqQi jø¥ e¥ ¥
,ª0¢M¯ª¢ ¤ 6fª0¤ ¤ ^ ª0¢ e^ e6^d#ª~¤,=^ e6 n=0
n=0
a0 = c 0 b 0 ;
n=0
1
MONRS
a1 = c 0 b 1 + c 1 b 0 ; ...............; n X an = ck bn−k ; k=0 ..................
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô ÷ÐqQi0 oø
?V ^¤ c^c~ª0¤,= ^ »,?¦cQ
d .y=c¤ c^c ?
òb^ ^¤ ¬ò eA¦cQ
,* cp ,ª0 ^ §c¦²=f cañ^¨(/b¨( « ^c Td9cQ¡V4 cc»= e^(ae¥ ¥
−ccpb¬)/b, eA^¦^cycQ
eA ¦d9cQ
ce¥ d9¬¶ce6¨(c ¤ .dP~? ¼¬ c[¯ Q=¤ eCcC=
d*Tc±w=e6¯ª0^¢ ^ ¨(,cc¤ ±wd#¤ªQ 0
º ¤ ÷ÐcqQif; :ª~øòh^cc¤ ^¤ d9¥¼
¥ cV ¬¶c^p¤ =?e6 ¬ eAeA¦¦cQcQ
d9eðc e¥ ^²e¥ ¤ [0
cp ÷ЬqQf ic ; :ø¥§¾´
Q ¬
^ ± 7 · ^ ë d , = d
c 6 e p p c
» c C , p
¬ § » c ¤ 0
Ð ÷ Q q i ; :ø §c¤,p=6eŪQ ¬ x¹¤ 7Cd9ñ6^c dw6
eÅ¥V (d9 c^òf ¯c=
c¥¤ cd©¥dP ?¬
ccpd , ¡V t^¤ ò? =¬ eAA¦
cCò
,
7¥ d9 ¥ e𬠤0
=d9 <~ ' # =@h.>x =,x((=±h ¤ cC¥
6 ¯¤0
c 0
@9 x ¹cp c?¡V
0
0
1
1
0 1
∞ ∞ X 5n (x − 2)n 4n (x − 2)n X (−1)n · . n! n! n=0 n=0
∞ ∞ ∞ X 4n (x − 2)n X (−5)n (x − 2)n X Cn (x − 2)n , · = n! n! n=0 n=0 n=0
an =
4n , n!
bn =
e^cA =e^ cc ¤ ¥
¥ ^ ¢õ÷ÐqQi jø¥,=±0
^d Cn =
∞ X
(−5)n , n!
ak bn−k = a0 bn + a1 bn−1 + a2 bn−2 + . . . + an−2 b2 + an−1 b1 + an b0 =
k=0
(−5)n 4 (−5)n−1 42 (−5)n−2 + + + ··· + n! 1! (n − 1)! 2! (n − 2)! 4n−1 (−5) 4n 4n−2 (−5)2 + + ·1= + (n − 2)! 2! (n − 1)! 1! n! 1h n! n! = 1 · (−5)n + 4(−5)n−1 + 42 (−5)n−2 + · · · + n! 1!(n − 1)! 2!(n − 2)! i n! n! 1 (−1)n + 4n−2 (−5)2 + 4n−1 (−5) + 4n · 1 = (4 − 5)n = . (n − 2)!2! (n − 1)!1! n! n! =1·
°4e^¢§
,ô ¤ cC¥
^ h¤ 0
c[ cp ,ª0 d
∞ ∞ ∞ X 4n (x − 2)n X (−1)n 5n (x − 2)n X (x − 2)n · = . (−1)n n! n! n! n=0 n=0 n=0
<~ ' #[email protected] x4l( ,~« ¥ §¦~ cp cQ¡V¥ ¬ §¦ X ∞ n=0
xn n!
m
m
.
T e¥ ¬
0
?Ê ²?Ë}¬ Í Ù,ß[ÖÖÒßÅÛ~Î]QÍ
¬ ÍÐÒÒÔPÞ ÖÙÚpÛ,ß^Þ Ö @9 x ¹¤ m = 2 ,e^cA =e^ cc ¤ ¥
¥ ^ ¢õ÷ÐqQi jø¥0
, X ∞
¹¤
2
∞ h X 1
d9^^d
1 1 + + ... + n! 1!(n − 1)! 2!(n − 2)! n=0 n=0 ∞ ∞ 1 1 i n X (1 + 1)n n X 2n xn 1 + + x = . x = + (n − 2)!2! (n − 1)!1! n! n! n! n=0 n=0 m=3
∞ h n X n=0
=
+
e^cc=6e6^ c X ∞ n=0
=
xn n!
x∈R
MONQà
xn n!
3
X ∞
=
n=0
n−1
xn n!
n−2
2 X ∞ n=0
∞ ∞ xn X 2n xn X xn = = n! n! n=0 n! n=0
2 2 22 2 1i n 2 + + + ··· + + + x = n! 1!(n − 1)! 2!(n − 2)! (n − 2)!2! (n − 1)!1! n! =
∞ X (2 + 1)n
¾*c=7^dþe¥ ,ª0,=I=ªQ
^dþ d96¬ X ∞ n=0
n!
n=0
xn n!
m
=
n
x =
∞ X 3n xn n=0
∞ X m n xn
n!
n=0
,
n!
.
x ∈ R.
<~ ' #[email protected] x(¾§T e¥ ¬ ∞ ∞ X 5n xn . X 2n xn . (−1)n n! n! n=0 n=0
d9 @eð9t ,h e^x#^DZ
^ e^ ¬Ve¥
cÅ c ±tdPc=ce^ h
¥¤ ^¥d ¬ô
¥ p , ¥¢M ¬eð!he6c^ f0^ , Td9¤c=0C
= d9 ^;eðc=¦!cC
,ªQ7 ,0 µ x = 0 ¹cñ6cd#ª= cp cQ¡V ∞ ∞ X 5n xn . X
n=0
n!
an =
∞
(−1)n
n=0
5n , n!
2n xn X = C n xn n! n=0
bn =
(−2)n , n!
e^cA =e^ c[c ¤ ¥
¥ ^ ¢÷ÐqQi oø¥ d9^^dþe^ e6^d#ª¶ª0¤,= ^ ±¶
,~c= ¤ ¥
¥ ^ hf cñ^¨µ
MON 0
1
¨( « ^c C
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
n
1 = C0 · 1, 5 7 + (−2) −2 = = C0 + C1 · 1, 1! 1! 1! i (−2)2 2! 1h 2 −2 52 2 7 + = 7(−2) + (−2) == C0 + C1 + C2 · 1, 2! 2! 1!1! 2! 1! i 1h 3 3! 53 3! 2 = 7 (−2) + 7(−2)2 + (−2)3 = 7 + 3! 3! 2!1! 1!2! 3 2 (−2) (−2) −2 = C0 + C1 + C2 + C3 · 1, 3! 2! 1! ......................................., n! 5n 1h n n! 7 + = 7n−1 (−2) + 7n−2 (−2)2 + . . . + n! n! (n − 1)!1! (n − 2)!2! i n! 7(−2)n−1 + (−2)n = + 1!(n − 1)! (−2)n−1 (−2)n−2 −2 (−2)n + C1 + C2 + . . . + Cn−1 + Cn · 1. = C0 n! (n − 1)! (n − 2)! 1!
0ª?¤,V= ^¤ ^ = c^=ch ª0cp¤, ,=ª0 6d »e¥ ¥
,ª6pc c=fªC
, c=ª0f ªC
d
? = C
C0 = 1
¹cQ
e6p= hñ6chC,Q ^ Vc~=c¤ c
7 + (−2) (−2) = + C1 , 1! 1!
C1 = 7/(1!)
¹cC
e6p= ñ6cC,Q ^ I¤ 6¬Iª0¤,= ^ =,=±0
^d
72 1 (−2)2 (−2)2 1 + 7(−2) + = + 7(−2) + C2 , 2! 1!1! 2! 2! 1!1! C2 = 7/(2!)
¹¤ cQ
cp ,¡V ô cC
c Td c¤,pCcdT
, ¤ cCcp ¬ c^c
Cn
c?µ
7n 1 1 + 7n−1 (−2) + 7n−2 (−2)2 + · · · + n! (n − 1)!1! (n − 2)!2! 1 (−2)n + 7(−2)n−1 + = 1!(n − 1)! n! 1 1 (−2)n = + 7(−2)n−1 + 72 (−2)n−2 + · · · + n! 1!(n − 1)! (n − 2)!2! 1 + 7n−1 (−2) + Cn , (n − 1)!1! n
= 7n /(n!)
óP ¥
cp¥ ¬ c
∞ ∞ X 5n xn . X
n!
(−1)
n2
n n
∞
X 7n xn x = . n! n! n=0
¹ cp ,ª0 ^ T±h¤0
hp=f¡(eð¦cC
eÅh,e^^±~ e¥ cc±~ce^
,ôbd9c=w c¡^cy0 ¤ ^6 ^ªQc ¬Q pp©d9cQ¡V c! cp ,ª0 ¬ #^e¥ t
¥ ^ ô ¤ cC^e6}?ª0A cd¢?9f0=f n=0
n=0
?Ê ²?Ë}¬ Í Ù,ß[ÖÖÒßÅÛ~Î]QÍ
¬ ÍÐÒÒÔPÞ ÖÙÚpÛ,ß^Þ Ö <~ ' #[email protected]x(¾§T e¥ ¬
MONR7 1
x3 x + + ... 1+x+ 2 6 2
d9 @eð9t ,h e^x#^DZ
^ e^ ¬Ve¥
cÅ c ±tdPc=ce^ h
¥¤ ^¥d ¬ô
¥ p , ¥¢M ¬eð!he6c^ f0^ , Td9¤c=0C
= d9 ^;eðc=¦!cC
,ªQ7 ,0 µ x = 0 ¹cñ6cd#ª= cp cQ¡V ∞ ∞ ∞ .X X xn X = C n xn 1 n! n=0 n=0 n=0
a0 = 1,
an = 0,
n = 1, ∞;
bn =
1 , n!
n = 0, ∞,
e^¨(c A « =e^^ c[cc ¤ ¥
¥ ^ ¢÷ÐqQi oø¥ d9^^dþe^ e6^d#ª¶ª0¤,= ^ ±¶
,~c= ¤ ¥
¥ ^ hf cñ^¨µ Cn
1 = C0 · 1, 0 1 + (−1) 1 0= = = C0 + C1 · 1, 1! 1! 1! i 2! [1 + (−1)]2 1h 2 1 1 0 1 + = = 1(−1) + (−1)2 = C0 + C1 + C2 · 1, 0= 2! 2! 2! 1!1! 2! 1! h i 3 3! 2 0 [1 + (−1)] 1 3 3! 1 + 0= = = 1 (−1) + 1(−1)2 + (−1)3 = 3! 3! 3! 2!1! 1!2! 1 1 1 = C0 + C1 + C2 + C3 · 1, 3! 2! 1! ......................................., [1 + (−1)]n 1h n n! n! 0 = = 1n−1 (−1) + 1n−2 (−1)2 + . . . + 1 + 0= n! n! n! (n − 1)!1! (n − 2)!2! i n! + 1(−1)n−1 + (−1)n = 1!(n − 1)! 1 1 1 1 = C0 + C1 + C2 + · · · + Cn−1 + Cn · 1. n! (n − 1)! (n − 2)! 1!
?V ^¤ =c^chª0¤,= 6 »e¥ ¥
,ª6pc ª0¤,= ^ = cp ,ª0 d
c=fªC
c=,=f±0ªC
^d
C0 = 1
¹cQ
e6p= hñ6chC,Q ^ Vc~=c¤ c
1 (−1) 1 + = + C1 , 1! 1! 1!
C1 = (−1)/(1!)
¹cC
e6p= ñ6cC,Q ^ I¤ 6¬Iª0¤,= ^ =,=±0
^d
1 (−1) (−1)2 1 (−1) + + = + + C2 , 2! 1!1! 2! 2! 1!1! C2 = (−1)2 /(2!)
¹¤ cC
cp ,¡V y cQ
c òdc¤,p^cdT
,! ¤ cCcp ¬ c^c
1 (−1) 1 (−1)n−1 (−1)n + + + ··· + + = n! (n − 1)!1! (n − 2)!2! 1!(n − 1)! n! 1 (−1)n 1 (−1)n−1 = + + + ··· + + Cn n! 1!(n − 1)! 2!(n − 2)! (n − 1)!1!
Cn
MONRB
0
Cn = (−1)n /(n!)
1
óP ¥
cp¥ ¬ c 1
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
∞ n X nx (−1) = . n! n! n=0
∞ .X xn n=0
¹ cp ,ª0 ^ T±h¤0
hp=f¡(eð¦cC
eÅh,e^^±~ e¥ cc±~ce^ C c=7¡M¤ 6^ªQ ¬Qpp[d9cQ¡V cI cp ,ª0 ¬ ^e¥
¥ ^ ¤ cC^e6f¬Tª0ð cd^®(0f0=f
,ôd9 c^c0 ^ c <~ ' #[email protected] x((=±h ^¤ T(¬0 ^ c,=e6 c^c a
.h
3b2 c 2 3bc d 5b3 3 b − − − x + 1− x+ x + 2a 8a2 2a 4a2 2a 16a2 3bd 3c2 i2 e 35b4 4 15b2 c x + ... . + + − + − 4a2 8a2 2a 16a3 128a4
6e @cQ97 ^^cô Vx(C°,=c=d9C^,,Qp ¥º = .^,¤ =6±0
A^d² ÷ n¤ 6¡[=
0,7∞e^^øy^c¶f c^ñ^^¨(cy¨(f ?«
¤,^p?;äf0ce6ñ^¨(^ ¨(^ « , c^^cþ¤ 0f0
c?µ c¤ c^c ,[e^c¢¼c ^¤ ¥
¬ ,cc=C,Q dþ ^¤ 6 b ÷ n = 0, ∞ø¥? = n
n
X ∞
An x
n
n=0
2
=
∞ X
bn xn .
n=0
bcA
e^cA =e^ cc ¤ ¥
¥ ^ ¢õ÷ÐqQi jø¥ d9^^dþ¬ ^¤ §¦~0 ^ c b0 = A1 A1 = 1, b b1 = A 0 A 1 + A 1 A 0 = − , a b2 = A0 A2 + A1 A1 + A2 A0 = 2A0 A1 + A21 =
c b2 + , a2 a
b3 = A0 A3 + A1 A2 + A2 A1 + A3 A0 = 2(A0 A3 + A1 A2 ) = 2
bc d b3 − − , a2 a a3
b4 = A 0 A 4 + A 1 A 3 + A 2 A 2 + A 3 A 1 + A 4 A 0 = bd c2 e b2 c b4 = 2A0 A4 + 2A1 A3 + A22 = 2 2 + 2 − − 3 3 + 4 . a a a a a
b ^^¤ 6^ ¤ ¬~ ^¤ ^±0
^d f,?¦ cQ¡[
^ ¢´,=e6 c^c f cñ^¨(¨( « ^Kf c=c¤ c^chcc=C,Q d Cn
∞ X
an x n
∞ .X
bn xn =
∞ X
C n xn .
¾§ce^ cp ¬=Cc==·7 e^¬Mc ¤ ¥
¥ ^ Cd÷ÐqQi oø;( ¤ McÉ dP= =Qc
,¯e^¥¦ n ≥ 1 pd9cQ¡[^dhQ= eCp¬¤ ^fª0¤ ¤ ^ª¢e^ e6^d#ª¯ª0¤,= ^ ±¯a
,7=,a?¦acQ¡[=
¥0µ n=0
n=0
n=0
0
n
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß ô ^¤ §¦~hf cñ^¨(¨( « ^c C
MONWG
n
a = C0 , b 0 = C0 − + C1 , a b2 b c 0 = C0 2 − + C1 − + C2 , a a a bc d b3 b2 b c 0 = C0 2 2 − − 3 + C1 2 − + C2 − + C3 , a a a a a a bc d b3 bd c2 e b2 c b4 0 = C0 2 2 + 2 − − 3 3 + 4 + C1 2 2 − − 3 + a a a a a a a a b2 c b +C2 2 − C3 − + C4 . a a a C0 = a 0 = −b + C1 C1 = b
0ª?¤,= ^¤ ^ c^=c~0 ª0cp¤, ,=ª0 ^d »e¥ ¥
,ª6p c=cfªQ
¹cC
¹e6=cCp
e6 ph=ñ6 7ch,C=,±0Q
^^ TVf c~c=ñ^¨(c¨(¤ c0 µ « ^ ¤ 6¬(ª0¤,= ^ =,=±0
^d b2 b2 0= − c − + C2 . a a
°4ce^p¢§
¥ ¬C c = c .¹¤ cC
cp ,¡V ¶ cC
c Tdc¤,pCcdT; cp ,ª0 d
C3 = d C4 = e
2
∞ X
Cn xn = a + bx + cx2 + dx3 + ex4 + . . . ,
.óP ¥µ
cô¤ ^c=? ce^¬,=± ÿPÿ ò!µ A ¶($Êedz ·az ! ¾§ò·7¶d9´ªe6=p c0 9ceð f ±e6^ ^ c±º¤ 0
ºª ¤ e^c^^c ^¤ ? el¯A¦¤cQª0
^ d9d9c¶e¥e¥» ceð¦=cCd9
0eÅQ©?
f= ^T ¤ ±ô^¤ e6T^ ^c ±© c±ô^e^¤ f 0c
~ ^ ¤ ¥cô
e6
p =¨(p ,¨( ^6¤ ^ ^« f c=¤ªc^¤d9ª0c¢¼±»¨¨ª0ª0 f f « « ¢¯ e6^e6^ c~¤,=e^e^d9c=¤ 6¬~c¤,p c±!C?
?ª cy¤,p6 c?¡^ !QQ
= c±¨ª0 f « /e6^ ^ c± ¤0
Tb#=f0pQ?
?,tñ^f ? ^,ºT e¥ ^ ¢ f cñ^¨(¨( « ^cþe6¥µ eA¦^cQ
c^¬=ceŶ¤0f
Q»?
²= c ¤ c±y¥
¨¥ ª0^ f « ¢Ä c0pf0 c==e6c¤²ª0¢ ^^cC,eA¦ QcC,
? d9¬ cc7e6d9 cQÉ¡V¹ ¤ c¯²¤,ñ6=e^ce^dë dPp¤0¤
²
pcp¬ ,¡f0=^f e6ª0d9d#ªô e6f0cd9c^ce6^ ^ c^cV¤ 0
= =,>x =,x ! '1+©%§ + Qw
¤° ª0c^e^T0= ¦; 7c40^ ¨( cc¤ ôd*d#¤,ªC p ,e^6ªIe¥
b¥¥ ^
±0=c c =¤,c= Ée^ =¢¤,=7e^¨e^^ª0dP pf c« ¤ d ±! =¤ eI¢Mc CcMd9cC¤,c=
7p 6¬T
¢ ¥d ¨(§TcdP¤ e6pd#·7^ªQ0dP ¦pm c=c ·7¤ ^0 eC
f f cá¯c^ =c6=¤,°M,=
? ,¡0? µµ f Tc , ¥
ce^^f dþcp ñ6=¬ªôfªô¨( cc¤ d#ªQ , ª¶I¤0ªQ
c[b ^c±0d
c¤, ,[ôc, = e(¤,=06
=eðh ,[¨(c¤ d#ªQ ,ªôb^±0 c¤,0, ¡[(d9 ¨ª0 f0µ (= cd9 dT cI
,y ^ ¤ ^¤ T c¯
¨(¨(^¤ ^ « ¤ª^d9c±¶,¯c=¤ 6^f0 « f (x) e^ ¤,=¥
¨(c¤ d#ªQ á¯=^¤,=¡[=0 ¨(c¤ d#ªC f0c= ^ c^cI [x¤ , ¤,x]p7^ ÷Ðqqúq?ø f (x) = f (x ) + f (ξ)(x − x ), e^=CT=¢M?C,Q ^ ¨ª0 f « ,¶^¤,= «,?¦!c=¤ 6Cf [x , x] ce^¤ ¥
e6cd ¤ cpµ CcC°
cc=±~7ñ6^ c ±~¨¨(ª0c ¤ f d#« ªQ h [ᯠ=^^f0¤,c==c¡[¤ cô±ô ¤ c f0cC
ξ w∈]xf¨(, x[c¤ d#ªQ b^±0 c¤,;
,©TcC
f c=c¤ c±~,=dþ cp¤ ^=ª6eÅhe¥ ¥
,ª0¢M[pô ^d9dP n=0
0
0
0
0
0
0
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô Qh ''¥= =,x>=,xò]N²OD,_N,N,N DY6`=@Y¸@EP`Rp>AD,@EÀ,Q_WZ4 @ºTY]\0p_ N,YZ[[a,Y b]DGI>^GÉ:
MONRI
(k)
lim X (k) (x) = lim F (k) (x) = 0
G&Z[>[]br¹[Y^]6Z[RY Y6ZZ:=_:R!Z[>Q\0_: ξ GP\0Z[> x→a
k = 0, n
x→a
Ð÷ qqùgø U¤ ^d9 c±h m c= ·7+-
!c= ^0
c¶we¥ ¥
,ª6~7« ^ c f !¤,=^ e6 ; c¤ cQ¡[
=^d9§¦h^cpµ F (b) F (n+1) (ξ) = (n+1) . X(a) X (ξ)
F (b) − F (a) F (b) F 0 (ξn ) F 0 (ξn ) − F 0 (a) = = 0 = 0 = X(b) − X(a) X(b) X (ξn ) X (ξn ) − X 0 (a) F (n) (ξ1 ) F (n) (ξ1 ) − F (n) (a) F (n+1) (ξ) F 00 (ξn−1 ) = . . . = (n) = (n) = , = 00 X (ξn−1 ) X (ξ1 ) X (ξ1 ) − X (n) (a) X (n+1) (ξ)
A
a < ξ < ξ < · · · < ξ < b %& '>Qa]= :==,:=,Y6xZ ò]
`=@Nô>QNO=R=D,>Q DY¥`=@Y¸@`EP>A@Rp,D,Q_WZ4 ºT@Y]\0pN,_ Z[Y Y
[a, b]D>^GG§:¯
n
f (k) (a) = lim f (k) (x)
G&Z[>[]br¹[Y^]6Z[RY Y6ZZ:=_:R!Z[>Q\0_: k = 0, n x→a
f (b) =
n X f (k) (a)
k!
(b − a)k +
ξ ∈]a, b[
G&\0Z[>
f (n+1) (ξ) (b − a)n+1 . (n + 1)!
÷Ðqq ø
;8
x;¹ce6¤ c de^ cd9c¥p¥ ¬ TI¨ª0 f « k=0
(0)
(
n X
(x − a)l , l! l=0 X(x) = (x − a)n+1 ,
F (x) = f (x) −
f (l) (a)
÷Ðqq 3 ø
ªC
e^¥c¦ p 6c¤ ¢M7 he^^d ªe¥ c d¼ ^d9d9 qqúql¯^± e66 ¬ c Pcpµ¸ ^¤ §¦;#
, k = 1, n
¤ ^d
^e¥ ^e¥
X (k) (x) = (n + 1)n · · · (n + 1 − k)(x − a)n+1−k = 6= 0, x ∈]a, b[; = = 0, x = a, X (n+1) (x) = (n + 1)!.
÷Ðqqùnø
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß ¾§cpµ¸=c¤ §¦;, ce^f cp ¬fªy
,ôe^¥¦ k = 1, n F =f
(k)
(k)
(x) = f (x) −
MONRL n X l=0
(x − a)l f (a) l! (l)
(k)
=
x−a (x − a)k (x − a)n (x) − f (a) + f (a) + . . . + f (k) (a) + · · · + f (n) (a) 1! k! n! k! k! = f (k) (x) − f (k) (a) + · · · + f (n) (a) (x − a)n−k , k! n! 0
(k)
c ¤ x = a F (a) = f (a) − f (a) = 0, ¤ ^d Ð÷ qq jø F (x) = f (x). ÷Ðqq¹jø&¤ ,d9=±0^
^ d yfw¨ª0 f « d ÷ q=q 3 øT¨(c¤ d#ªQ ,ª»÷Ðqqùgø§ ^d9d9´qqúqeIª0 6cd ÷Ðqqù nøT (k)
(k)
(k)
(n+1)
f (b) −
n P
l=0
(n+1)
f (l) (a)(b − a)l /(l!)
=
f (n+1) (ξ) , (n + 1)!
c=fªC
~e¥ ¥
,ª6y¨(c¤ d#ªQ ~÷Ðqq ø¥ Ç ^¤ =d9?6 ]b,dT.a[,¾chd9c= 7¤,^dº=e^e¥e^ ,d9ª0c=,=¤ =¥ ^©e¥ ¶Q^=¤ ¨( ? f0e^]a, ¤ cb[p=,¬[?cC
c^ªy (c~^¤,¤,p= e^e^ dP«~p ¤ ^¤ =6? eÅ 7c f x ,I
¤ª06ª0¢ e^ pp¬V ^¤ ^d9^ c± x 0c¨(c¤ d#ªC ,ª÷Ðqq ø&d9cQ¡V c7Q= eCp¬ [0
X f (x ) ÷Ðqq; :ø f (ξ) f (x) = (x − x ) + (x − x ) . k! (n + 1)! áI^^f c[ª00
6¬ cV ¤ n = 0 y÷Ðqq; :ø§e¥ ¥
,ª6y¨(c¤ d#ªQ á¯=^¤,=¡[~÷Ðqqúq?ø¥ ó§cc= cp·7^ ,[÷Ðqq; :ø9,pCT=¥eŶ¨(c¤ d#ªC c±yb^±0 c¤,
,V¨ª0 f « f (x) cpµ cd X f (x ) ÷Ðqq oø P (x, x ) = (x − x ) k! cp cd9cdºb^±0 c¤,e¥ =¥=^d9c 4 ÷Ðqq ø f (ξ) r (x, x ) = (x − x ) (n + 1)! ce6ppc Td*0 ^ cdþ¨(c¤ d#ªC b^±0 c¤,[¨(c¤ d9á¯=^¤,=¡[ 4 ?d9^ c7Ip=f c±y¨(c¤ d94¯e^0 ,ª^4 ¤ ce6c=,= cp ^M,=e6c7Q= e6T=¢M[c=e6=?µ c T± 0 ^ r (x, x ) M°M
,=f cººc=Q
¥ ¬ §¦ e¥ ,ª0,p0¦ñ6pº¨(c¤ dPcf0pCT=6eð ¨(^c ¤ ¤ dP =^dTcC
0 Ic ±yd9
^ , ¶ c¯^^ c[c^« ^^¤ ?f ¬ ¶c±¶ ¤ ¨(0c¦ ¤ cC
d9MVeÅô¨( c¤ ¤ d9^m6pc=·7¬[ fV
Ç ¤=ª0d9^6 dT dTd9^ ^c¯ ¥¤ c e6 Td ª ,c ¤ ¥
¥ , ^d#ª0¢ ¨(c¤ d#ªQ c±»÷Ðqq ø¥ d9cQ¡V cQ= eCp¬=0
(b − a)n+1
0
n
(k)
0
0
(n+1)
k
0
n+1
k=0
n
n
(k)
0
0
0
k
k=0
(n+1)
n
n
0
0
n+1
0
rn (x, x0 )
f (n+1) (ξ) rn (x, x0 ) = n!
Zx
x0
(x − t)n dt,
÷ÐqqúqQiø
=
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô f c=c¤ T±hd9cQ¡V c[¤,=e6e^dPp¤ p¬yf0=f~ ¤ ¥
e6p=p ^ ¯ ^^¤,? Z ÷Ðqqúqq?ø 1 f (t)(x − t) dt r (x, x ) = 1
MRMON
x
n
(n+1)
0
n
n!
e cd9c=7¬¢¼cc=7^ c±ô^c¤ ^d9 ce^¤ ¥
^dT be¥^° ±0ô e6pcp¤,^ c^ [¤ ¬Tfô±~^ 0^ ¤,^?^ ^¬¤,r ?c(x, ± ,wª©¨(x÷ q=c)q¤ ú÷Ðd9qqq?=qø& úq q?¤ øP ,d9p^C T=¬V6ceð~T ce6pª0p¢c ^ cT¤ ^dþd#ª¶0 c[^ e^c¤ d¥
¨( ^cdT¤ d#ªc µ ce6pp/ c T±w0 ^ r (x, x ) d9c?¡ cQ= eCp¬Vf0=f ÷Ðqqúq?gø f (ξ) (x − ξ) (x − x ), r (x, x ) = n! A
ξ 4 ^f0c=c¤,pôc f0¤,=e^ c? c?¡^ ,pôd96¡[
,ª x x be¥^° ±0¶ e6pcf p¤,ccc¤ [
¨( T,c±~p¤ =d90ªV I^m cc= r·7f (x, 0x¤,)=e^÷Ð qcpq úq?cQg¡[øP^, p CcT±y=d966¡[eð~
,ª ce6pp c 0 T¤ dþ¥
0e6 =^= cd¬[¨(¯c¤ 0d#
ª µ / ξ cIce6ppc T±¶0 ^¨( xc¤ d#xªQ b^±0 c¤, A
ξ[= x + (x − x )θ 0 ≤ θ ≤ 1 ¨(c¤ dP?¦yá¯=^¤,=¡7hmc=·7~d9cQ¡V c e^cA =e^ cw÷Ðqqúqq?ø§©÷ q=qúq?gø¥Q= eCpr¬V(x, f0=f x ) x0
n
n
0
0
(n+1)
n
n
0
0
0
n
0
0
0
0
n
0
÷ÐqqúqQø ÷ÐqqúVq 3 ø ¾§c7C6¡= ¥
c¤,p^ª0d9^ ~ cC
^¤ f ^dT,cñ60¦ô¨(c¤ d#ªC ?¦¶c7dP cQ¡V¥ MCñ6^0e6¦7 cM¤ ¥
¥·7 ¬4?¦¯cpd9 cQ¬¡[f 6cM4d9c ^pc4¬ceð¯7 ¤ ¤ ¯ CdPd9=^6 4^C , ? ^ 7(d9
6p¡¡
,PªI ¤ ªCc e6^d!c ¯¤ ¥7
^¤ Å« ¦ ^cC±
θ c=cC
c±~¨(c¤ d9ce6=pc c=^cV0 ^,w÷ÐqqúqQø&f¶
¤xª0^nc±t÷Ðqq=úqV3 ø¥ ¹¤ c=7(e^^^c¨(c¤ d#ªQ [b^±0 c¤, P (x, x ) r (x, x ) TA ,0
,?^e¥ x = 0 X f (0) ÷Ðqqúq?nø x + r (x), f (x) = P (x) + r (x) = k! A
r (x) d9cQ¡V cQ= eCp¬¨(c=¤ dPá¯=^¤,=¡[ ÷ÐqqúqQjø f (ξ) f (θx) r (x) = x = x (n + 1)! (n + 1)! 0 ~[¨(c¤ d9Imc=·7 ÷ÐqqúWq :ø f (ξ) f (θx) r (x) = (x − ξ) x = (1 − θ) x . n! n! m♦ (=fyª ¡c=d9^,? ce^¬ Q=d9^, x = x − x e^cQ
[¨(c¤ d#ªC ,ª÷Ðqq; :øPfô¨(c¤ d#ªC c ÷Ðqq¤,¡úpq?=n ø¥TPQ±I= ^ ^¤ ¤ e^= ¥T¦d9wcCc
e6 c=cpV7 ,ª0c=÷Ðq q^Qú q?? n
cøQ± fy ÷Ðhqd9q=;f0þ: ø=ccªC¤ e6
^ª ^ 7d©c^dK e6cpp ,¬= Ccc6eð eðp7¯¬6ceð±²[¤,^f0p6=ct fhc ÷б(d#qqQ =;ò:d9ø¥°^p p =c^f± 0?h
¹ ÷Ðcqc ñ6q&úcq?d#nø¥ªc ¾§C e^ycôe^Qf0=pd9Q^=, = c ¶c=¤,= e^c e^¤ ce6¥ ¤,¬== c 6cC
eð ©c^pc!=f¤,¡p6[ ,cQô¡^¨( c¤ ©d# ªC ^^ f0c!÷Ðq q^ú¤ qQ^ø¥ 9ce^÷ q=qúeðqVº3 ø4,~
÷Ðq¤qª0ú^qQcj=ø¥ ÷ÐqqúWq :ø¥ f (n+1) (x0 + [x − x0 ]θ) (x − x0 )n+1 , (n + 1)! f (n+1) (x0 + [x − x0 ]θ) rn (x, x0 ) = (1 − θ)n (x − x0 )n+1 . n! rn (x, x0 ) =
n
0
n
n
n
(k)
n
0
k
0
n
k=0
n
(n+1)
n
(n+1)
n
n+1
(n+1)
n+1
(n+1)
0
n+1
n n+1
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß MRMRM e6==9pªQ ,
c6p[
6 ,~c ¤¤ ¢Mp¯ ª0¡¢ ^ª e¥ §c¦~ §d µ e¥♦ ^° e^± c=Tª °¢¼,« ^ c= Cce6cp ,¬V ¨(6c¤ d#¢4ªQ= ª0¢£b^¨±0ª0 cf ¤,« ¢ ¤ ¥f
(x) ^c¤ ^d9Àqqúq Q=d9^¬ cp cd9cdþb^±0 c¤,[e(c=·7 f c± c ¤ ¥
¥ , ^d9c±hce6ppc0µ Tdþl( ,0 ^·7 c dþ¤ c¨(f cc^¤ cId#f0ªQ =e^eCb^±0¨ ª0c ¤,f « ± ,= ¤ d9^¤[¥¦;ª7f0c=c¤ §¦ µ ¤ cCcQ
µ , cp©=÷| e^c[cp f ¢M¤,= ± c ±²^±hd9¥ ^ ¤ =, ¤ !~ Ce¥d9 c^ d ^ !§ =¤ c^6¤ ª0d96·7^ pce6d9¬ 6T¡[c
, ª¶¤ Å
ªQ¥ ,^ (ndþ^dP+ p1)xø§cce6^p¤,p= c ^T,d 0 ^ cdT ªQ
c? 6c¤ 6Vc« ^ f0 M ÷ÐqqúqQoø M |x| . |r (x)| < (n + 1)! óP ¥
,ª6c=d96¬ ?cò
ce6ppc c4^¤ cd9c=6
f ±(TcQ
I¨(c¤ d#ªQ b^±0 c¤,I÷Ðqq; :ø d9cQ¡V ce6ª 7^e6=C cª0 ¤ ce6¬ &^e¥ º,¨ª0 f « ¢ f (x) ,? c?¡µ¸±¶¬ ¤ cpc ^C¶cC¡
^ e6cp± µ f cc^¤,= ^ 4 c=¤ ^cp¬[ 4cp ¬f0ce6ª 7^e6c= (n + 1) cô^( ^ ¤ ^¤ T ce6 bcA
e^ ¤,=¥
^c¤ ^dP f (x) %& ' = =,Ex D x ò
] N»O D,_ N,uN f (x)GTuN (bY º¹R: tR¶Z[>Q\0_ Y x DY6`=@Y @EPRpD,EPY`=@>QN Y R=L >QY6A@ Y:
n
(n+1)
0
n
(k)
0
n
0
k=0
x
n
0
n (n+1)
n!
U +-
x;#=e6e^d9c=¤ dcQ¡[
^e6c x0
f (x) − f (x0 ) =
f c=c¤ c(d9c?¡V c7Q= eCp¬Vf0=f A
Zx
f 0 (t)dt,
x0
÷Ðqqùg0q?ø
f (x) = f (x0 ) + r0 (x, x0 ),
÷ q=qùggø ¹¤,==ª0¢ ,=e¥¬©ñ6c^c©¤,=^ e6 ¤ c ^^¤, ¤ª^dK c,=e6? dT#Q=¨( f0e^ ¤ c= e¥ cp¥
cQc¡V p¥U ¬= =c f (t) dV = dt = −d(x − t) cA
dU = f (t)dt V = −(x − t)x r0 (x, x0 ) =
Zx
f 0 (t)dt.
x0
0
00
x r0 (x, x0 ) = −f (x)(x − t) x0 + 0
= f 0 (x0 )(x − x0 ) +
Zx
x0
Zx
f 00 (t)(x − t)dt =
x0
f 00 (t)(x − t)dt.
MRMOS
¹cC
e6p= ñ6cT¤,p¡[^ (h÷Ðqqùg0q?ø¥ ,=±0
^dT,c
1
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
f (x) = f (x0 ) + f 0 (x0 )(x − x0 ) + r1 (x, x0 ),
A
÷Ðqqùg=ø
Ð÷ qqùgY3 ø ?^^¤,? ~ ¤,=c±,=e6²÷ÐqqùgY3 øM==f¡T e¥ d c~,=e6Q dTQ=¨( f e^ ¤ c= cp cQ¡V U = f (t) dV = (x − t)dt cA
dU = f (t)dt V = −(x − t) /2 x e¥ ¥
cp¥ ¬= c r1 (x, x0 ) =
Zx
f 00 (t)(x − t)dt.
x0
00
000
1 (x − t)2 x r1 (x, x0 ) = −f (t) + 2 2 x0 00
= −f 00 (x0 )
(x − x0 )2 1 + 2 2
Zx
Zx
2
f 000 (t)(x − t)2 dt =
x0
f 000 (t)(x − t)2 dt.
¹cC
e6p= ñ6c¤,=^ e6cVh÷Ðqqùg=ø¥ cp ,ª0 dþ¨(c¤ d#ªQ ,ª x0
A
f (x) = f (x0 ) + f 0 (x0 )(x − x0 ) + f 00 (t)
(x − t)2 + r2 (x, x0 ), 2!
Ð÷ qqùgnø ¹¤ cC
cp ,¡V É ^^¤ ,¤ c= § cM,=e6? dT?d9» ¤ 0¦cC
d!fI¨(c¤ d#ªQ &b^±0 c¤,¯÷ÐqqúqQø 1 r2 (x, x0 ) = 2!
Zx
f 000 (t)(x − t)2 dt.
x0
f (x) =
n X f (k) (x0 )
k!
ece6=pc Td0 ^ cdþ0
V÷Ðqqùg=iø
k=0
1 rn (x, x0 ) = n!
Zx
+ rn (x, x0 )
f (n+1) (t)(x − t)n dt,
cô¤ ^c=? ce^¬
cfpQp¬ c4 e6♦pp¹¤ c d9 c^ d# ªºI0f ñ6^cªºd#ª[ ¨(c¤ ^d9^¤,h?m ,ª[c=·7^ c ¤ / ^d#e¥ ªþc(¡e^¤ ô¥
fº ñ6^dTc0d#e^ªº¤,p ^ª[ ^¤ ^¤,0¦? ,cC
ªþ d»¤ fw d9÷Ðq^ qúq?g¬ ø cc=7^ ª0¢^c¤ ^d#ªôce^¤ ¥
^dT,c cp ,ª0 d x0
f (n+1) (ξ) rn (x, x0 ) = n!
Zx
(x − t)n dt = f (n+1) (ξ)
(x − x0 )n+1 (x − t)n+1 x , = f (n+1) (ξ) (n + 1)! x0 (n + 1)!
=c fªC
~e¥ ¥
,ª6÷Ðqq ø 4 ¨(c¤ d#ªQ ce6=pc c=^cy0 ^,[[¨(c¤ d9á¯=^¤,=¡[ c c^c~0 ^,¶¨(c¤ d#ªQ b^±0 c¤,÷ÐqqúqQoø
<~ , ôñ¥' ^d9 #^= ==,=x>=,¤ x( §¹¦hcp ,¨ª0ª0 f « ¬~ c± « f^ (x)fªw=ce6epp sin ln(1 + x) x cos x (1 + x) x0
x
µ
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß @9 xq f (x) = e ¹ce^f cp ¬fª (e ) = 1 , d9^6d
MRM¥à
x
x (k)
x=0
ex =
n X xk
+ rn (x).
k!
÷Ðqqùg=jø
b#=f¶f=fôce6=pc T±h0 ^h[¨(c¤ d9á¯=^¤,=¡[ k=0
c ,,= ¤ d9^¤ , ¤
x>0
¾*,=e6 ce6. ¤
eθx xn+1 , (n + 1)!
|rn (x)| <
ex xn+1 . (n + 1)!
c^¤ 6·7 ce6¬Vc« ^ =6eðhfpf
|x| ≤ 1
0g f (x) = sin x ¹ce^f cp ¬fª (sin x) e¥ ¥
cp¥ ¬sin c
rn (x) =
(k)
c
sin 0 = 0, sin(2m) 0 = sin mπ = 0, 0 = sin(mπ − π/2) = (−1)m−1 , m = 1, ∞,
(2m−1)
sin x =
m X
(−1)
k−1
¾*ñ6cde¥ ,ª0,=(ce6=pc T±h0 ^h¤,=^ k=1
ec« ^ f0c±
3 . (n + 1)!
|rn (x)| <
= sin(x + kπ/2)
r2m (x) =
÷ÐqqùgZ:ø
x2k−1 + r2m (x). (2k − 1)!
÷Ðqqùg=oø ÷Ðqqùg=ø
x2m+1 sin(θx + (2m + 1)π/2) 2m+1 x = (−1)m cos θx (2m + 1)! (2m + 1)! |r2m (x)| ≤
= cos x ½4,f?(x) c^ c ¤ ¥
§
,ª ^d#ª~e¥ ,ª0,=¢ cos x =
m X k=0
(−1)k
|x|2m+1 . (2m + 1)!
x2k + r2m+1 (x); (2k)!
r2m+1 (x) = (−1)m+1 cos θx
x2m+2 ; (2m + 2)!
÷Ðqq iø ÷Ðqq q?ø
Ð÷ qq gø 3 ÷|e6^ ^ ¬ ce^f0cp ¬fªyñ6cdþe¥ ,ª0,=¤,p6 c?¡¥µ ( d9f^(x)6V=0
w(1+ x)cdP[¬¢Mc,µø¥6= 0, 1, 2, . . . |r2m+1 (x)| ≤
|x|2m+2 . (2m + 2)!
µ
[(1 + x)µ ](k) = µ(µ − 1) · · · (µ − k + 1)(1 + x)µ−k ;
MRM 0
1
f (k) (0) = µ(µ − 1) · · · (µ − k + 1); µ(µ − 1) · · · (µ − k + 1) k (1 + x)µ = 1 + x + rn (x), k! k=1 f (0) = 1,
A
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
n X
÷Ðqq ø
µ(µ − 1) · · · (µ − n) (1 + θx)µ−(n+1) xn+1 , (n + 1)! µ(µ − 1) · · · (µ − n) (1 + θx)µ−n−1 (1 − θ)n xn+1 rn (x) = n! rn (x) =
h¨(c¤ d9á¯=^¤,=¡~tmc=·7 e^cc=6e6^ c 9m(=f©ª ¡Vc=d9^,? ce^¬
, ce6ppn0 c T±w0 ^~¨(c¤ d#ªQ b^±0 c¤,¤,=^~ªQ ¢¯ ? = r (x) = 0 f (x) = ln(1 + x)
n = µ
µ
(−1)k (k − 1)! , (1 + x)k f (0) = 0, f (k) (0) = (−1)k−1 (k − 1)!; n X xk ln(1 + x) = (−1)k−1 + rn (x). k k=1 f (k) (x) =
÷Ðqq d3 ø
Ç = ·7^dþe^,Q,? [ce6ppc T±w0 ^~[¨(c¤ d9á¯=^¤,=¡[ cA
7
,
rn (x) =
0≤x≤1
(−1)n+1 xn , (n + 1)(1 + θx)n+1
e^ ¤,=¥
[c« ^ f0
|rn (x)| ≤
1 , 1+n
pñ6=cf©^cf=d9f d9cQ¡V c?¡V¥ ,~¥ ¬ x/(1 ,e6 c +,~θx)d9cQ¡V c[ ¤ c^e^ ccpeA¦ cQ¬=
Ccwp¥
¬ eÅ~ ¨(« c¤ #d9¹c±~¤ mxc=·7< 0 c¥
^ rn (x) =
bcA
¤ ^d
,
(−1)n xn+1 (1 − θ)n . (1 + θx)n+1
|rn (x)| ≤ −1 < x < 0
|x|n+1 1 − θ n , 1 − |x| 1 + θx
|rn (x)| ≤
|x|n , 1 − |x|
pñ6=cf¶dþf0=e¥ ,fyª0d9, =cQ ¡V¥ ¬ |1 − θ|/|1 + θx| 4=ªQ
6 ¤ ^ceA¦ cC
¬[¥
« ce^f0cp ¬fªV b#=f dºc¤,1p+CcθxdT0>
,1 −|x|θ < 1 e^ ¤,=¥
[c« ^ f0
1 , x > 0; n + n1 |rn (x)| ≤ |x| , x < 0. 1 − |x|
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß MRMO7 cp ♦=e6¹ cd9D d9!cVf ¤,c==e^ce6¤ dPccp±¤ ^cp , c ±!cd¼òb·7^¯±0 d9cc?¤,¡Vh Qc=¤,d9=^e^e^ d96c=¤ 6eA¦¬¶cQ
QQª0
¢ ?¨ª~ª0c Vf c« ¤ ¢ ¥
¥¤ »^ Q ? µ
= cdº e¥ Ie¥ =¥=^d9§¦ n e( c^¤ 6·7 ce6¬=¢ r c^¤ 6·7 ce6¬0¢þ d9^ ¬=·7&i iiq e^cA (==e^ ¤ c ÷Ðd9q^q¤ Qi ø¥¤ ,
ncp ,=¡V 1cy
,(T ccp^ c Q¬ceðw ªsin e¥ cx ≈ x|xeP/6| ÷|cþe^cc=
,[6e6c^=c ª6²c ¤ d9^¤ c qQi ø¥M¹¤ eT ce^±[ cp¡[ ò¬=Ccc = c <e6²¬0,
¢¯=001 ª 0 0^ c^C ¦ cCc
|x|± d9¨(
(n ≈ x− r:^ =ø§ ¯ô2)ª?e¥
c |x |/(120)sin< x0,001 0 x /6|x| < 0,6544 ÷|cye^cc=6e6=ª6~ ¤ d9^¤ c ?p=f ;¨(c¤ d#ªC Vb^±0 c¤,[
cªe^f0=6~= ¤ cf0e6 d&p« ¢À¨ª0 f « ycf ¤ ^e¥pµ f c=ce6c²¤ T±ôc d9f cQ ¡V xc¯ e¤ ¥
ce6^=¤ p6·7 c¬[e67¬¤,¢¯pÉ6 c ¤ ¥
§¥¦¶ , ¨(^cd9¤ cdP± ?¦;c e6p¾ºpc¥¦¶ e¥T ,ª0d,p00 ¦;^f (x) cf cd A
r¯(x,¤ ^x=ª)µ 6eðy= ¤ cf e^ dP=« ¨ª0 f « f (x) ^ ce^¤ ¥
e6^ c7c f x ÷ý==f,pCT=^dPp Q cf0? ¬,p¯= ¤ cf e^ dP=« ¢??f cð
ø¥Cc4 e^ cp ¬p^ª6eÅ767&cC
,ɨ(c¤ dP x−x → 0 ce6ppc c^cI0 ^, ¨(c¤ dP¹C= c e¥ µ ¤ c=CcQ
,p f (x) c^¤,= ¥µ ,Te6 ·7¤ ^ ^cVx →c¤x0
f0[c c4 c[e6pe^p¤,=c ^T ±h¢ 0 e ^ / r (x,n+1 ¤ ¡V xcQ^=e6 ¬ eCp^e^¬f c ^ 0
c dP? p x )~^^cxd9cQ→ n
3
◦
3
5
◦
0
n
0
0
0
(n+1)
0
n
0
rn (x, x0 ) = o((x − x0 )n ).
ó§cc=6e6C c f (x) =
(= ¤ d9^¤
,
0
(x − x0 )n
n X f (k) (x0 )
k!
k=0
f (x) = (1 + x)−1
¤
(x − x0 )k + o((x − x0 )n ).
n=2
÷Ðqq nø ÷Ðqq jø
x0 = 0
1 = 1 − x + x2 + o(x2 ), 1+x
A
x3 r2 (x) = o(x ) = , 1+x 2
p=f~f0=f o(x2 ) =
1 x3 − 1 + x − x2 = − . 1+x 1+x
l ( , ¥ ñ ^ 9 d ^ = =
¤
§ ² ¦ ¨ 0 ª
f
«
² ± h e 0 ª
6 c d Ð ÷ q q ù = g j ø Ð ÷ q q d3 ø7¨(c¤ d#ªQ b^±0 c¤,»e 4 ce6ppc Td0 ^ cd[¨(c¤ d9I¹C= c[ d9^¢MV0
x2 xn e =1+x+ + ... + + o(xn ); 2! n! x2n+1 x3 + . . . + (−1)n + o(x2n+2 ); sin x = x − 3! (2n + 1)! r: x2 x2n cos x = x − + . . . + (−1)n + o(x2n+1 ); 2! (2n)! µ(µ − 1) · · · (µ − n + 1) n µ(µ − 1) 2 x + ... + x + o(xn ), (1 + x)µ = 1 + µx + 2! n! n x2 x3 n−1 x n + + . . . + (−1) + o(x ). ln(1 + x) = x − 2 3 n x
÷Ðqq ø
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
1
MRMOB = =,xED x
( ;"%§ + e6ôc¹ ªf e6 ¬7x ¨ ª0b cf ð«
¤f0
(x) ^e^f c ^ c¯
¨(¨(^¤ ^ « ¤ª^d&7¯ ^f c=c¤ c±¶cf ¤ ^e¥ cpµ X f (x ) ÷Ðqq oø (x − x ) n! #Q= ce^ f0d9 ce6¯ c=M^^cÉeð¦cC
d9ce6¯Ie6ª0d9d9©,pCT=6e爨0
cd~b^±0 c¤,T¨ª0 f « f (x)¹¤ x ¤0
÷Ðqq oø&,pCT=6eÅh¤0
cd*h=f0 c¤ 6, x = l( ,ò e^ 0^ [ªe¥ c= ± ¤ [f0c=c¤ §¦[¤0
Vb^±0 c¤,[÷Ðqq oøeA¦ cC
eð f0=f[e6¥µ ^ c±¤0
[ d9^6¯e6ª0d9d9c±eC=d#ª¨ª0 f « ¢ f (x) c¤,p d9eðf[¨(c¤ d#ªQ Éb^±0 c¤, [0
X f (x ) ÷Ðqq ø f (x) = (x − x ) + r (x, x ). k! 0 ^¹, ¤ ¥r
(x,cp xcQ¡V) e¶dT ¤ ce6c(c
d , n xe6¤ 4^ d9^eð¤ þ?fº ªC]x ¢¯− R,C xcc=+C,R[Q,=¥6 ? P ,cIc^e6e^pf pc c^ cT^±c ¤0
0
∞
(n)
0
0
n
n=0
0
0
n
(k)
0
0
k
n
0
k=0
0
n
0
0
∞ X f (k) (x0 )
k!
(x − x0 )k
eA¦cQ
eðh~ d9^6Ve^c^±~e6ª0d9d9c±~ e¥ c f (x) b#=f dºc¤,pCcdTC,Q ^ I¨ª0 f « f (x) p , 6eÅ~e6ª0d9d9c±~e6^ C c^cV¤0
X ÷Ðqq 3iø f (x) = c (x − x ) ef0cñ^¨(¨( « ^p=d9 ÷Ðqq 3 q?ø f (x ) c = , k! ? =,e6ª0d9d9c±h¤0
b^±0 c¤,h÷Ðqq oø¥ kPe¥ c ¤ Vf c=c¤ §¦Vce6=pc T±¶0 ^ r (x, x ) e6¤ ^d9eÅyfVªQ ¢eMc=C¤,=e¥µ p= ^d cd9^¤, n c ¤ ¥
¥ , 6 %& ' = =,Ex Hɶ¸"# / ô1;=+ t ·?x ò] Nôb< uôapY^]6_ >QDY6\0D>(QF!D,:N,D,Z[Y¸@RpV: Y O D,_ N,N,N `>?] Y6
x;¹cªe¥ c ¢^c¤ ^d9 d9^^d ÷Ðqq 3ø R max |f (x)| ≤ M. n! ¾§T ·7^dce6ppc T±w0 ^ r (x, x ) [¨(c¤ d9á¯=^¤,=¡ k=0
∞
k
0
k
k=0
(k)
0
k
n
0
0
n
(n)
x∈D
n
(n)
x∈D
n
0
rn (x, x0 ) =
f (n+1) (ξ) (x − x0 )n+1 (n + 1)!
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß ôce^ cp ¬=^ª6dPeðt÷Ðqq 3ø6bcA
e^ ¤,=¥
[c« ^ f
MRMbG
|x − x | n+1 |x − x0 |n+1 (n + 1)! 0 |rn (x, x0 )| ≤ , M =M n+1 (n + 1)! R R
If c=c¤ c±~~e¥ ¥
,ª6
|x − x | n+1 0 = 0, n→∞ R
lim |rn (x, x0 )| = M lim
^e¥ x ∈ |x − x | < R ,c¶¤ ^c? ce^¬
cfpQp¬ f dôIc¤ 0¤,
ypCbc^dT±06 ªce¥¤, c0° M¤,÷ q=pq 3d»ø p ,dP =6 eð É
,ce6pc p c c¯Tñ6d~c¯
¤, ,p(6¤, pcQ¡[6 ^cQ ¡ ^T ,?( ,¨,¥ª0 eÅf0 µ « b#f=(x) ¥
e6^ Éd[e^0 ,ªôe^c± e6Vge¥^ C §¦h¤ 0
c = =,Ex H x +0s w}ô1. $ * » ;" %§ + l( ,º,Q,? cp ,ª0 d ¤,p6 c?¡^ ºce^ c §¦ºñÅ ^d9^==¤ ò¦þ¨ª0 f « ±þ¤ 0
b c^¤,±0 ~c÷Ф,qqùg=l(jø ,÷Ðqñ6q cd^3 côøPôªQ
ªce¥= cc~ ^cpd¼ ¬=÷ÐCqc=q p3¬ø¥eÅ cp ,ª0 ^ Td9©¤,= ^[¨(c¤ d#ªC =d9b6±0µ
·£<~ 4 -+ *}ô1. $ f (x) = e ce¥ ¥
cp¥ ¬ ce6¬h÷Ðqq 3gø# p , 6eð ¹
¤ 4 ¢ c d
, y ^ e ¥ ¦ R > 0 x ∈] − c^¤,= ^ c± ,p=f~f0=f~e^ ¤,=¥
c R,^¤,R[=^ e6c n→∞
0
x
Rn R n eR max |f (n) (x)| ≤ , n! x∈]−R,R[ n!
n = 0, ∞,
¤,=p,=e6¬!f c=c¤ c^cwe6¤ ^d9eÅtf©ªQ ¢ö ¤ » ^c^¤,= ^ cd¼c=C¤,=e6p= n b#d#=ªôf ¨ dª0 cf « ¤,p CcdT.¤,ª?e¥6 c=6=©6eð!^cV¤ ^¤d90´
hcô=
f0c e6pcp¤ ^c, V ,cd¼ ¢4¤ cCd,=f0f cV ^ T c=cpd* ^ ^#¤ ¹?c ñ6= cp µ C,Q ?,e^^e±~67^e6^ c±hce^ Xx ÷Ðqq 3Z3 ø e = . n! ·y¸P) ,+/ 0 }ô1. $ f (x) = ch x f (x) = sh x ?y÷Ðqq 3Z3 ø& cp ,ª0 d x
∞
n
x
n=0
°4e^¢§
e−x =
∞ X
(−1)n
n=0
xn . n! ∞
X x2k 1 x −x ch x = (e + e ) = , 2 (2k)! k=0
÷Ðqq 3 nø
Ð÷ qq 3jø ¡¾^e^ 0 ,·7 ª[%Ich¥
x L= e6sh^x', [c ¤e6 y
h/¤,b 0p^6±0 cQc¡ ¤,^}ô ,1.[V ¨e^$ª0^±~ f « 6 7V^e6Ie6^^, ^c ±h Éce^4 ¤ 0
cp ,ª0½4 , ?d» c¤,^p 6 cpc µ e¥ ,ª0,=¢¼ø9
,ô¨ª0 f « sin x cp ,ª0 dºe¥ ¥
,fª0(x)¢M=^Isin¤,px6 fcQ¡(x)^ = cose6x^ C c±~¤ 0
X ÷Ðqq 3@:ø x . sin x = (−1) ∞
X x2k+1 1 . sh x = (ex − e−x ) = 2 (2k + 1)! k=0
∞
n
n=0
2n+1
(2n + 1)!
1
MRMOI
b#=f¶f=fô ce¥ ¥
cp¥ ¬ ce6¬
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
Rn Rn max | sin(n) x| ≤ n! x∈]−R,R[ n!
c^¤,= (^, !cp ,
ª0 , ^ ¢4T±¯c¤^c0
¯ReA¦9cQ
c©ceðe6¯=p,òc e^ ^T±I±* 0e¥ ^c=þc¤ ±I0
ce^Vb÷ ^±0 c¤, r (x) → 0 ¤ ø¥ n →¾§∞ ce^ cp ¬=Cc==·7 e^¬Me^c± e6cdô
¨(¨(^¤ ^ « ¤ª^d9ce6¯e6R^ =^,∞ §¦¯−∞ ¤ 0
2
0
4
6
n
2n
µ
∞
n
µ
n=1
f (x) =
∞ X
c n xn
¢MVe eð^þC,^e6ce^ Tcd9=º c=ºf h^f f cc=ñ^¨(c¤ ¨(§ ¦þ« ^Cp^=e6d9 §c¦9e^Ç cp± e6^d¼f ñ6cñ^c¨(±þ¨(¨ ª0« f ^« §c¹cce¥ ¤ ô¥
ñ6¥ ,c^0c µ
¥
c f0 pCe6T^= 6 ceðte6VeA¦¤,cQ
p6 d9cQc¡e¥^¬! ¤[0e6
~^ ©^ c c¤ ^¥c¯
¥¤ ,0
6eð tc? ,^ª0^ ch^¤, ?
ɱªeV¤0eA
V¦cQ
pd9 ,c e66 eð#¾ ¤0e^
0c ,d ª b^±0 c¤, ?p=f
,~¨ª0 f « cp =6=^d f (x) = (1 + x) X ÷Ðqqùn=iø f (x) = (1 + x) = c x . °4d96 dTcVñ6p[¨ª0 f « hcp ?
=6e¥ ¥
,ª0¢M7 d*e^c± e6cdT (1 + x)f (x) = (1 + x)[(1 + x) ] = ÷Ðqq=ùn0q?ø = (1 + x)µ(1 + x) = µ(1 + x) = µf (x). ¹cQcC¡
e6^p = ô÷Ðqqùn=iø#7 ¤,==ª0¢ ^=ª0¢ ,=e6¶ñ6c^c7e^cc= c=·7^ cp ,ª0 dt0¦V¤,p6µ n=0 n
n
µ
∞
µ
n
n
n=0
0
µ 0
µ−1
0
(1 + x)f (x) = (1 + x)
X ∞
µ
cn x
n
n=0
=
∞ X n=0
ncn x
n−1
+
∞ X n=0
n
ncn x =
0
=
∞ X n=0
[ncn + (n + 1)cn+1 ]xn
ÊQÊ?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß
MRMOL
µf (x) =
∞ X
µcn xn .
?(ñ60¦h¤,p6 cQ¡^ ±~[e^0 ,ª©÷Ðqqùn0q?ø§ cp ,ª0 dþ¤,=^ e6c n=0
∞ X
n
[ncn + (n + 1)cn+1 ]x =
∞ X
µcn xn .
f¹ ª0¤ ¤ ¤ ¤,^= òh¦~f0ª0c¤,ñ^=¨(=¨( ^ « ± ^ë ¤ tcC
,=f0c=§¦»e6^ ^ 0¦ x ¤ 0
^d¼f»e^ e6^d9ô¤ ¥µ n=0
n=0
c1 = µc0 , c1 + 2c2 = µc1 , ..............., ncn + (n + 1)cn+1 = µcn , ........................,
c=fªC
d9c7^6VV÷Ðq0q
ùn=iøP ¤
c1 = µc0 , (µ − 1)c1 µ(µ − 1)c0 c2 = = , 2 2! ..............., (µ − n)(µ − n + 1) · · · µc0 (µ − n)cn = , cn+1 = n+1 (n + 1)! ................................. x=0
,?¦cC
d cn =
c0 = 1
bcA
¯c=7 ±ô0 ^ô cd9,? ¬ c^c¤0
µ(µ − 1) · · · (µ − n + 1) , n!
ô d9 ¤ 0¦ cC
dfô¨(c¤ d#ªQ V÷Ðqq 3ø¥
d9ce6?ô(l7É?^ ¤,==d9 «,^¤,?¦I cf^p¤ Cò? = 6?c¤ 0eð
h¦cCeA
¦ cCd9
ce6eð¬wc ¤ ¤ ¥
|x|¥ ,< 61eÅ ¤C,,=QeA¦ ¹cC^¤
C^,deÅ=7f~ ceA¤ f0¦ pcCQ|x| > , 1 x = ±1 p ¥ µ g0q,,^^e¥e¥ µ−1><0 µ
n n
n=0
∞
n
2
n=0
n
1
MOSRN
T¤0
÷ µ = −1/2ø √
3 T¤0
÷
µ = 1/2
ø
√
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
1·3 2 1·3·5 3 1 1 x − x + ··· = =1− x+ 2 2·4 2·4·6 1+x ∞ X (2n − 1)!! n =1+ (−1)n x ; (2n)!! n=1
÷ÐqqùnY3 ø
x2 1 · 3 · x3 1 · 3 · 5 · x 4 x − + − + ··· = 2 2 · 4∞ 2 · 4 · 6 2·4·6·8 (2n − 1)!! n+1 x X (−1)n x . =1+ + 2 n=1 (2n + 2)!!
1+x=1+
Ð÷ qqùnnø ¹÷ ce¥ ¥
^Pø¥¤,pl¯66 ± c?e6¡^ P¥ ¬cp ,c ª0 ^ cM ^^¤ ¤,c= ^dw¤0
¯÷Ðqq=ùnY3 ø;É ^¤ ? ]0, x[ µ = −1/2 Zx 0
√ dx √ =2 1+x−1 = 1+x =x+
Zx
1+
0
∞ X
(−1)n
∞ X
(−1)
− 1)!! n x dx = (2n)!!
n (2n
n=1
(2n − 1)!! xn+1 , (2n)!! n + 1
=c fªC
¹^~¤ e¥ T¥7
,ª
6y÷Ð0q q!ùnnø¥¤ e¥ =¥=^d9§¦ñ60¦¤0
cy
=¢Mô,= cp ^,=e6c~ e^ cp ¬pµ ^ª^d9♦TI ¤ p ¡[^ TI¨(c¤ d#ªC l( ,º« ¥ §¦ ÷ ø7¤0
÷Ðqq 3ø¯T¤ c?¡[
=¥eÅþf c ^ ª0¢£e6ª0d9d#ª ♦ µ¸^c[e¥ =¥=^d9c^µc µ = cnd≥ 0¢Mc, ¬ 4 n+1 n=1
(1 + x)n = 1 + nx +
n(n − 1) 2 x + · · · + xn . 2!
l( ,!¤,=« c,? ¬ c^c e6ª0d9dP¶ cd9,? ¬ c^c¶¤0
ye^^A
V
=6~=¤ ¨(d960µ ^e^f ♦c(C,Q ^ I¤,?
f0? µ "¾§·¥ce^Qh cp} LQ¬= ^ ª^d9}~eð'w¨/c 0¤ d# ªQ c±~}ôe61.ª0d9 d9$ ^ ^cfd9(x)6¤ = ln(1 ¦ ^eCf c+±wx) ¤ c=|x|^¤ ^<e^e^1 e^ ¤,=¥
c±~ ¤
1 = 1 − x + x 2 − x3 + x4 − . . . 1+x |x| < 1
ln(1 + x) =
Zx
,¹¤ c ^^¤ ¤ c=V^(c=ªQ ,¶
c x ,=±0
^d
n dx x2 x3 x4 n−1 x =x− + − + · · · + (−1) + ... 1+x 2 3 4 n
C¹ ,¤ = f cx ^=¤ ^−1 ¤ 0
, ¤ = A e ¦ C c
ð e 9 h
¤ d9^ T±w¤0
0
x = 1
eA¦cQ
eð 9¾ eC=d9cd
¥ = ¤ ∞
X 1 1 1 1 (−1)n ln 2 = 1 − + − + · · · = 2 3 4 n+1 n=0
x = 1
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß ªC
cp 6c¤ 6Vªe¥ c dþ6c¤ ^d9 áI^± «, ?p=f ,^e¥ ? =
ln(1 + x) = x −
−1 < x ≤ 1
MOSZM
xn x2 x3 x4 + − + · · · + (−1)n−1 + . . . , 2 3 4 n
÷Ðqqùn=jø
∞ ∞ n X X xn+1 n−1 x = . ln(1 + x) = (−1)n (−1) n n + 1 n=0 n=1 L f (x) = arctg x f (x) =
t·º[ u[ À =' / 0 K}ô1. $ arcsin ¾§ xc¬[ ¤ d9^ d¨(c¤ d#ªC ,ªôe6ª0d9d9 ^^cd96¤ ^e^f0c±h ¤ c^¤ ^e^e^ w ¤ |x| < 1 1 = 1 − x + x 2 − x3 + x4 − . . . 1+x
bcA
1 = 1 − x 2 + x4 − x6 + x8 − . . . , 1 + x2 x
¹¤ c ^^¤ ¤ c=c=ªQ ,¶
c ,,=±0
^d arctg x =
Zx
|x| < 1.
2n−1 x3 x5 x7 dx n−1 x = x − + − + · · · + (−1) + ... 1 + x2 3 5 7 2n − 1
C y ^¤ ? #ó§c=ð =e^ c ¤ c=C!,¤=0f
ª¶áIeA¦^cQ±
eð«,t ,¤ »¤ e^x¥¦©= C±1,Q ¤^0
h0¦ cQ¡x(eA¦ cC
eð ;¹¤ −1x =<1x < 1 0
arctg 1 =
? =,e6ª0d9dP[¤ 0
1 1 1 π = 1 − + − + ..., 4 3 5 7
Ð÷ qqùnZ:ø p6 c?¡ 1/√1 − x c©¨(c¤ d#ªC t÷ÐqqùnY3 ø7þ ¤ c ^^¤ ¤,c= ^=ª0¢£þ ¤,?µ = ª0cp¢ ,ª0, = e6d hñ6c^cy¤,p6 cQ¡^ h, ^¤ ? [0, x[ ,f=f~hV ¤ ¥
§
,ª 70¦we¥ ,ª0,p0¦; ∞ X (−1)n−1 n=1
1 π = . 2n − 1 4
2
Ð÷ qqùn=oø f cpc ñ^¨(♦^w¨(#e¥ 0 «
cQ ¡Võ^ b§c^¦±0 ¨cª0¤,~ f c÷Ы « q ^q± 3Zòf 3 I ø 4wc÷Ðe6cqpqp¤,ùn=pf0o ø7p ^e^,¤ ¬tm¥
f¤ f cc»¨(d9 c¤ c=Cd#cªQ^cp c ,w ¢Mb^»±0^ ¤ cpc ,¤,? !ª0 eA
¦ ,cC¬t
¤,d9Tpc6 e6 cQe¥w¡[ ¤^^ 0
b¨^ª0±0 f c« ¤,w*¨cª0p f ?
« =c6 »f (x)^ ¤ d9^¤ cQT¡V c!É,d9=²± r¤ cf0=CfcCdP
= f Te^ d9dPþ? ¢4¬ Tc^±ºc» c¤ ^0¤
f0? ò9°! f ^c=0c
¤ cc d c¤,?
ªeôeA¦cQ
d9ce6 R ¤ ñ6cd =ªQ
6e^c,?
p¬e¶¤,=e^e6cQ ^d c=« ^¤, ¤,p6 cQ¡^ ? =,c f x
cp ¡[=±·7^±~f~ ^±~ce^cc±¶c f x ÷Ðqqùn=ø R = |x − x |. ¾§e^(ñ6cd9 ~ ¤ c0 ¢4e6¤ ¤ªCde¥ ¥
,ª0¢M7,d9h ¤ d9^¤,=d9 ∞
X (2n)! 1 3 1·3 5 arcsin x = x + x + x + ··· = x2n+1 . 2n 2·3 2·4·5 2 n!(2n + 1) n=0
n
n
0
1
0
1
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô <~ ' #= =,xED x((=±h¤,p6 cQ¡^ (¤0
wb^±0 c¤,[¨ª0 f « ± c[e6^ ^ d (x + 3); a) f (x) = ln(2 − 3x) ø f (x) = x − 2x + 7 ce6^ ^ d x; (2x + 3)(x − 2) ø f (x) = ln(1 + x) ce6^ ^ d x. 1+x @9 x øÉó² c=dPcp7¬¢cQ¡[
^e6^ ò¦h ¤ ^c¤,pCc= ±~,?¦cC
d 1
MOSRS
2
2
ln(2 − 3x) = ln(2 − 3[x + 3 − 3]) = ln(11 − 3[x + 3]) = i h i h 3 3 = ln 11 1 − (x + 3) = ln 11 + ln 1 − (x + 3) . 11 11
§¾ ce^ cp ¬=^ª^d9eð¤,p6 cQ¡^ ^d÷Ðqqùn=jø
, ln(1+t) = cp c?¡V t = −3(x+3)/11 b=f f0=f~¤,p6 cQ¡^ V÷Ðqqùn=jø§e^ ¤,=¥
c, ^¤ ? −1 < t ≤ 1 c¤,p6 cQ¡[^ ln(2 − 3x) = ln 11 +
∞ X (−1)n−1 h
e^ ¤,=¥
c, ^¤ ? n=1
n
∞ in X 1 3 n 3 (x + 3)n − (x + 3) = ln 11 − 11 n 11 n=1
−1 ≤
3 (x + 3) < 1, 11
? =,C ,[c= h¡[^[¤ ¤ ?6 ^ªC −20/3 ¬?pp!d9≤c?¡Vx
1
1 1 x2 − 2x + 7 = + . 2 (2x + 3)(x − 2) 2x + 3 (x − 2)2
bcA
e^cA =e^ cw÷Ðqq 3ø¥
∞
,
∞
2x n 1 X 2 n 1 1 1 1X (−1)n (−1)n = xn = = 2x + 3 3 1 + 2x/3 3 n=0 3 3 n=0 3 |2x/3| < 1
0
|x| < 3/2
∞
∞
1 1 1 1 X x n 1X 1 n =− =− x =− x−2 2 1 − x/2 2 n=0 2 2 n=0 2n
÷Ðqq jiø
, ¹|x/2| ¤ cC
<¨(¨(1^0¤ ^ « |x| ¤ c<=2 ÷ q=q jiøÉ0 !ce^ cp ¬=Cc==·7 e^¬¶^c=cTd¤,p6 cQ¡^ ^d ÷ÐqqùYn 3 ø¥ d9^^d ∞
∞
1 X n n−1 1 X n + 1 n 1 = x = x . (x − 2)2 2 n=1 2n 2 n=1 2n+1
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß b#=f dºc¤,pCc=dÉ
MOSQà
1 1 x2 − 2x + 7 = = + 2 (2x − 3)(x − 2) 2x + 3 (x − 2)2 ∞ ∞ ∞ n n 1 X n + 1 n Xh 1X n + 1i n n n 2 n 2 x + = x = + x . (−1) (−1) 3 n=0 3 2 n=0 2n+1 3n+1 2n+2 n=0
c¤,p6 cQ¡^ ¤ ¥
e6==p , 6[e^cc±ye6ª0d9d#ª
=ª¦y¤0
c7ecp =e6Q d9¶eA¦ cC
d9ce6 e^cc=6e6C c #»e^ ¤,==¥
c!wcp =e6»0¦» ^¤ ^e^^ ^ |x| < 2 ? =C<
,c=3/2 [|x|¡[ <|x| ¤ 63/2 ^ªQ ¬QppVò^f=6V(¨(c¤ d#ªC K÷Ðqqùn=ø¥ ce^f cp ¬fª¶
=ª¦¶ce^c§¦ ¤,?2
pª eTeA¡[¦ cC=
± ·7d9^c±e6ft « ^¤ªt¤,p6 cQ¡^ z=e¥ =¥
0c ?p ,¥ 6¬ eðc −3/2pbcAx
4= cc f0^f xx ==−3/2 R = |0−3/2| = 3/2 cp =e6¬eA¦cC
d9ce6 |x| < 3/2 øPl( ,ô¤,p6 c?¡^ C
1
2
0
1
f (x) =
ln(1 + x) 1+x
ce^ cp ¬=^ª^d9eð ¤,=0 cd ª0d9 cQ¡^ e6^ ^ ò¦ ¤0
cÀ÷ÐqQiùnø~÷ÐqQi jø¥bcð
ª0d9 cQ¡^ c6=¤ ¨(d9 ^e^f c^cV¤0
∞ X
xn+1 (−1) ln(1 + x) = n+1 n=0
,^^cd96¤ Ce^f ±
n
∞
X 1 = (−1)n xn 1 + x n=0
=6
1 1 3 ln(1 + x) 1 2 x + 1+ + x − =x− 1+ 1+x 2 2 3 1 1 1 4 x + ..., |x| < 1. − 1+ + + 2 3 4
<~ ' #= =,xEH xI®¯ª0 f « ¢
ø f (x) = √arcsin x ø 1−x ¤,p6 cQ¡V¬[¤0
h=f0 c¤ ^, @9 xl( ,~¨ª0 f « f (x) cp =¥=^d F
2
f (x) = ex sin x
Ð÷ qq jq?ø A
^=,C ¤ 4 ^Cc[^e¥ ÉdP=¯ c=f,[f c cñ^¨(¨( « ^¼¤0
Vwpf0 c¤ ^,Vñ6c±h¨ª0 f « ,l7?µ ÷Ðqq jgø 1 x arcsin x f (x) = + ∞
X arcsin x f (x) = √ = C n cn , 2 1−x n=0
n
0
1 − x2
(1 − x2 )3/2
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
Qf (0) hm=c=·70 d9cQ¡^d*ª ^¤ ¡[
p¬ .cô eð¦cC
,p¨ª0 f « ! p , 6eð¤ 6·7^ ^d Q?µ ÷Ðqq jø (1 − x )f (x) − xf (x) − 1 = 0, f (0) = 0. ¹cC
e6p= ©6 ^¤ ¬tª0¤,= ^ t÷Ðqq jø7¨ª0 f « ¢ f (x) ©0
~e6^ ^ ,c^c»¤ 0
÷Ðqq jq?ø¥ cp ,ª0 d MOS 0
1
2
0
2
(1 − x )
0
∞ X
X ∞
Cn x
n=0
nCn xn−1 −
∞ X
n
0
−x
∞ X n=0
nCn xn+1 −
C n xn − 1 = 0
∞ X
÷Ðqq jd3 ø
Cn xn+1 − 1 = 0.
¹ce^f cp ¬fª¶f0p¡[
,ª0¢¼e6ª0d9d#ªôy÷Ðqq jd3 ø&d9cQ¡V c[ ¤ ^c¤,pCcp¬fô0
,ª n=0
∞ X
nCn xn−1 =
n=0 ∞ X
nCn xn−1 = C1 + 2C2 x +
nCn xn+1 =
∞ X
∞ X
(n + 2)Cn+2 xn+1 ,
n=1
nCn xn+1 ,
n=1
Cn x
n+1
= C0 x +
∞ X
c0¦ô cC
e6p= cf[h÷Ðqq jd3 øP
=6 n=0
n=0
n=1
n=0
∞ X
∞ X
n=0
Cn xn+1 ,
n=1
(C1 − 1) + (2C2 − C0 )x +
∞ X
[(n + 2)Cn+2 − (n + 1)Cn ]xn = 0.
¹ ¤ ¤,= ºf cñ^¨(¨( « ^ä ¤ cC
,=f0c§¦e6^ ^0¦ x T ¤ 0¦cQ
df e^ e6^d9 ¤ ^fª0¤ ¤ ^ §¦~ª0¤,= ^ ± n=1
C1 − 1 = 0, 2C2 − C0 = 0, ............, (n + 2)Cn+2 − (n + 1)Cn = 0, ........................,
f c=c¤,pV¤,=e^,?
=6eðy,
M 6Q= e^ d9T4e^ e6^d9(
, 6 §¦
n = 2k k = 0, ∞
2C2 = C0 , 4C4 = 3C2 , ............, 2(k + 1)C2(k+1) = (2k + 1)C2k , ,
÷Ðqq jnø
C1 = 1, 3C3 = 2C1 , ............, (2k + 1)C2k+1 = 2kC2k−1 ,
÷Ðqq jjø
^^^^^^^^^^6^^^^^^^^^^6^^^^^^ y
,~ ^ 6 ò¦ n = 2k + 1 k = 0, ∞ ^^6^^^^^^^^^^^6^^^^^^^^^
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß MOSR7 b#=fwf0=f ;e^cA =e^ c÷Ðqq jq?ø¥ c¶e^7f0cñ6¨(¨( « ^ e7 6 Td9 0µ
^0f
eC^=f d9eC=d9=hªQ
,ª y(÷Ф,q=q j jtø&,ªQ= ±0¢¯
^fd C(0) == 0C k==00, ∞ Ql( ,7f cñ^¨(¨( « ^c(eò ^ 6 Td9 0
2k
Ð÷ qq jr:ø ¹f0 cCc
¤ e6^p,= »÷Ðqq jr:øM©÷Ðqq jq?ø¥; cp ,ª0 d e^f cd9c[¤,p6 cQ¡^ [¨ª0 f « ©ô¤0
th?µ X (2k)!! ÷Ðqq joø arcsin x √ , = (2k + 1)!! 1−x eA¦cQ
d9ce¥¬7f0c=c¤ c^c¯ ¤ |x| < 1 ò^f0=6 ^ ce^¤ ¥
e6^ c74 ¤ C,=f0(l7? =dµ ^¤,~÷|0 ôªe¥ c ±h ^ ¤ ^¤ T c±ô
¨(¨(^¤ ^ C « ¤ª^d9ce6w¨ª0 f « ±ø6 l( ,ô¤ 0
ø§ce^ cp ¬=^ª^d9eÅ~¨(c¤ d#ªQ c± ±0 ^¤, d9^ c C2k+1 =
2k(2k − 2) (2k)!! 2k C2k−1 = C2k−3 = . 2k + 1 (2k + 1)(2k − 1) (2k + 1)!! ∞
2
k=0
x
x
x ix
e sin x = Im[e (cos x + i sin x)] = Im e e = Im e
(1+i)x
= Im
kQ ,cV c[¨(c¤ d#ªQ ¯ôª=¤,
n=0
(1 + i)n =
,ô¤,p6 c?¡^ ~pªQ
^dþ d96¬ x
e sin x = Im
∞ X
√
n!
.
√ n nπ nπ 2 cos , + i sin 4 4
n
∞
2 (cos nπ/4 + i sin nπ/4)xn X = n! n=0
¾§ce^ cp ¬=Cc==·7 e^¬[¤ ^c cd96¤ Ce^f d9we^cc= c=·7^ d9 n=0
∞ X [(1 + i)x]n
√
n
2 sin(nπ/4)xn . n!
π sin lπ = 0, sin lπ − = (−1)l+1 , 2 1 1 π 3π = √ (−1)l+1 , sin lπ − = √ (−1)l+1 , sin lπ − 4 4 2 2
e^f cd9c(¤,p6 cQ¡[^ (d9cQ¡V cQp eCp¬V[0
ex sin x =
∞ X 22l−1/2 (−1)l+1
(4l − 1)!
x4l−1 +
<~ ' #= =,xKJx((=±he6ª0d9d9
∞ X 22l−1 (−1)l+1
(4l − 2)!
x4l−2 +
∞ X 22l−3/2 (−1)l+1
(4l − 3)!
x4l−3 .
¤ 0
c[I ¤ d9^¤,ôq j c=C @ 9 , =¢M eð&wm( =^ff c=ª ¡[c¤ ~c±hc=^d9¤ ^c,d9?c= ¥c
e^f0¬»ce6¤,=¬ ¢¯6=& ¤ c6e^·7f cp^ ¬f ªô§c ¤ ô c¥e^
^c TTw=¢M© eð~¤ d9cp ^¤ ¬©f cq, j c ¤ ¥
¥ ^ e6ª0d9d9í¤0
f0=f ¤ ¥
¥ * ce¥ ¥
cp¥ ¬ ce6 ,=e6 §¦¼e6ª0d9dT ¹¤ ¥
^dþ ¤ ce6T¯¤ ¥·7^ ce^ c= T(,e^c± e6?¦~e6^ ^ ,§¦h¤0
c l( ,ô ^¤ c^c¤
0, ¤ cC
¨(¨(^¤ ^ « ¤ c=VC^e6 cI¤,p6 cQ¡[^ =,,=±0
^d l=1
X ∞ n=0
l=1
x
n
0
=
1 0 , 1−x
|x| < 1,
l=1
1
MOSRB
0 ¹cp cQ¡V
∞ X
nx
n−1
∞ X
=
nxn−1 =
×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô
1 . (1 − x)2
¤ 0
^dfô cp ,ª0 ^ cd#ª~[ ¤ d9^¤ £fWfWfI¤,==^ e6=ª x=q n=1
n=0
∞ X
nq n =
1 , (1 − q)2
|q| < 1.
¾*e^c¢¼c ^¤ ¥
¬
=ª0f ¤,p c
¨(¨(^¤ ^ « ¤ c= (c^c¡ eA¦ cC
c^c¤ 0
¯
=6 n=0
X ∞
0
nx
n−1
n=1
∞ X
0
=
0 1 (1 − x)2
n(n − 1)xx−2 =
2 . (1 − x)3
^e¥ cQ¡V c ¤ ^c¤,pCc= ^c±~,=e6h ¤ ¥
6Vf~e¥ ¥
,ª0¢M7Cd#ª~¤,=^ e6=ª n=1
∞ X
2 n−2
nx
−
∞ X
nx
n−2
∞ ∞ 2 1 X 2 n 1 X n−1 . nx − nx = = 2 x n=1 x n=1 (1 − x)3
¾§ce^ cp ¬=Cc==·7 e^¬e6ª0d9d9c± T e¥ ^ c±ô
,~ ¤ ¥
§
,ª 7C^cV¤ 0
,=±0
^d n=1
n=1
∞ 2 1 X 2 n 1 1 nx − = , 2 2 x n=1 x (1 − x) (1 − x)3
c=fªC
∞ X
n2 x n = x 2
h
i x(1 + x) 1 2 + . = (1 − x)3 x(1 − x)2 (1 − x)3
¹cp cQ¡V x = q ¤ 0
^dfô¨(c¤ d#ªQ =,,=±0
^ c±h[ ¤ d9^¤ q j (=f c ^« ;
,¤ 6¬^^cw¤0
ôc=e^ cp ¬=^ª^d9eð©¤,p6 cQ¡^ ^d ñ^f e^ c ^Kô¤ 0
h=f0 c¤ ^, ? = n=1
¹cp cQ¡V
, d9^^d x=1
ex =
∞ X xn n=0
n!
=1+
∞ X xn n=1
n!
.
∞ X 1 e=1+ , n! n=1
c=fªC
∞ X 1 = e − 1, n! n=1
c~e¥ ¥
c=? c cp ,ª0 ¬ e¥ cQ
[ɪ0fpQ= §¦e^ ce^cc(¤,p6 c?¡^ [ M ¤ d9^ dTc
,§µ d# ªC e¥♦ c^±þ/ b[^±0f0 ccñ^¤,¨(þ¨(÷Ð q« q 3 6q?ø¥9c¹c¤ º ñ6^ccQdÀ¦cC
e¥ ¥d9
,c(ª6tce^ ccpd9 ¬=Cc¬ pP¬eð[c© ¤,^p ¥c e6c?¤ ¡¥
^ e6 ~^¨, ª0cI f ¨(« c ¤0 µ c=Cd9cQ¡V c ·7¬t ¤ *^e^f c ^ c±
¨(¨(^¤ ^ « ¤ªCd9ce6²ñ6c±*¨ª0 f « ² f (x) n
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß MOSWG « c ± f x 0 ^f0√c=c7¤ ¤ c±[0
~^T ccf e6¤ ^^e6 ^ c e6d 0¹ ccñ6e^f c=cpd9 ª7¬f ªyT0e6¦¶ª 7 ^¤ e6c==Cª^cQ¯
¤, pT6 cQc¡¤,^p [=¢M¨ª0eð ôf0 µ x ^e^f cln x^ ce6¬Vx ¤ x = 0 c¤ Twɤ 0d9
º^b¢M^I±0 e^cɤ,h ¤ wcñ6CccC±»
cT Éf0(=9 (^f =c= ¤ c ¤ d9c±[^¤ c
,f0 x ♦ cwó& ª 7y^d9e6c6=ª ª0¢4¯ò¨ª0¬! ¤,f p« 6 cQ¡[f0^c= ´ ¨ª0 f « ¤ = 0 c f x = 0 e6ª 7^e6=ª0¢M¯ ¤ cCcC
µ T¯ ¢4cf^c¶(x) c=¤e
f0.°M
,=xf0c 6=; 0ce^f fcp(0) ¬fªwe^ f (x) = 0 e^cc=6e6^ c e^ c = 0
, k = 0, n cce6ppc ɱþ0 ^ºe^^A
¤,=^ºe^=d9c±º¨ª0 f « º e¥¤, ¢p¥6
eIccQ¤ ¡cp^e6 ¥ c~d¬= [nc .c f C ¤ ^Acye6
hh cc=÷ýQe6Cy,~? ,e^=f0c6 Q¢4f ; ^ c Q^?d
=eC =,d9pch±w¨cª0 f f « xw= ¯0 øMd9 ^76Ve6¤ ^^±0d9 c¤ eÅce^ff c^ªc µ 0
3
0
−1/x2
(k)
x=0
k
x=0
= =,xKJx
~z u[ h) +0s þ ;;"# t%§ + T§ ¦w# 0e¥
*^ ^^·7¤,w ?¤ eIccQf ?c
= , ¤ w cd9¤ ±~^6·7 c^¢M eðc!ôe6c7¬¢T ¤ f C ,pcQ ^^ ¡ ^§ ±w¦! ¤,§~p¦¶6
T¨( ¨( §^e¥¦w¤ 6^¨ « ª0,0 ¦;?f ,« ¬¯ ±!,§=h¦!e6cª0 ¤,c¤ =e6¥
y¥^
^ , 0± µ ¶?
?e^ cp ¬=Cc= ~¨(c¤ d#ªQ b^±0 c¤,!
,þT e¥ 6 C,Q ^ ±¨ª0 f « §µ Tc~d ¤ ce^0 cp ¢4¬=Ce6c¤ = ¤, cV=, =che6 ò ·7 §=#¦t°e6ª0Td9 d cô¤0
,hbñ6^±0c ±»c¤,« ¥ ©C ccwfpcC=T³. =e^6 eð»6eÅ©cp ^^[dTªC
cpc µ
c^b¤ ^6±0·7 cc¤,e6,¬ Tf c==c6V¤ª0d9¢À^ d9¬=cQ·7¡V= c ^Qdº?
Cp,?¬ ;6c ^ ¤,¯=c e6 p p cp e^c¬y^cV ^0¤ ^T,d9!e^ce¥c= =¥6=^e6d9Tpªd9¢M!¤^0± µ ¨(c¤ d#ªC Àb^±0 c¤, C checC
c±©e¥c¤ c #ó
¤ª0^c±©e6c¤ c ¤0
¤ ¥
e6p=p ,0µ ¢M7 ±w
= ª0¢K¨ª0 f « ¢¯.d9cQ¡[6¶ò¬y
ce6=pc c~ ¤ ce6cô ¤ ^c¤,pCc=!ô c± Te6¤ ^Ieð¦cC
,7 ± eð.¤0
=e^e^d9c=¤ d* 6e^f0cp ¬f c ¤ d9^¤ c <~ ' #= =,xEP x(¾§T e¥ ¬C,Q ^ I e¥
e
e( c^¤ 6·7 ce6¬¢
δ = 0,01
c @± 9p =f~ ôx¤i0
ccdº b,^=±0± c~¤,C,®7Q c^¤ d# ªCI [ be¥ ^±0 ec,¤,d9ôcQ÷СVqq ùcg=jø&ce^ ¤ cp ¬=Ccp
¬=eð6~fe¥p f~¥
,¨(ª0c¢M¤ 7d#Cª µ x=1 ¤ ¥
e6p=p ^ n X 1 e≈ k! k=0
ec« ^ f0c±~ce6ppc c^cV0 ^, |r | = 3/(n + 1)! < δ c[
, e¥ ô
,ôT e¥ ^ e d96¬V[0
,ª¶^e^¬¤0
wb^±0 c¤, / n
n ∞ X X 1 1 e= + , k! k! k=0 k=n+1
δ = 0,01
cc=·7 f0[ ¤ p ¡^ c^c[C,Q ^ e =ªQ
6V¤,=,ce6ppfª~¤0
∞ X 1 rn = , k! k=n+1
=6
n=5
× ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô c« ^ fª¶f0c=c¤ c6cVd9cQ¡V c ¤ c^e6he¥ ¥
,ª0¢M dc¤,pCcdT 1
MOSRI
∞ X 1 1 1 = + + ... = k! (n + 1)! (n + 2)! k=n+1 h i 1 1 1 1+ = + + ... < (n + 1)! n + 2 (n + 2)(n + 3) h i 1 1 1 1+ + + . . . = < (n + 1)! n + 2 (n + 2)2 1 n+2 1 = . = (n + 1)! 1 − 1/(n + 2) (n + 1)!(n + 1)
rn =
°4e^¢§
,
δ = 0,01
,?¦cC
d
n=4
r4 =
ce^f0cp ¬=fª
6 6 1 = = . 5! · 5 120 · 5 100
b#=f dºc¤,pCc=dÉ d9cQ¡V cª ^¤¡[
p¬ ce6ª0d9dP 1+1+
1 1 1 + + ≈ 2,71 2! 3! 4!
bc=C^ ±0 ,c =¤,6 eÅd9ô*cpd9 cQ¡ e¥^ d» ec ¤ª0 cp ¬^eð4¶ C^Id=,ª ¡δ É=c0,01 c0e6c¬7A
cp7 f0¬=f f c¯0 ccpA
¬=^ªf ce^A¬
¨(¯c=¤ c=d9C¬ªCd9 ^cd ± n =l75= T±² ¤ d9^¤0 ¢4e6¤ ¤ª^þ¤,p6 tñ^¨(¨(^f ce6 e^ cp ¬=Cc= ¨(c¤ d#ªC þ¤ 0
^e6bª ^7±0C e6c¤,^§ ¾§ c cPc=óP7 ~¥
,^ª0c¢M7c¤ ±P ¤
=d9 ^ ¤tcdK ce¥f0 ,pª0¡,6=?w¤,pc!6 h ^wf0c=d96c¡[¤ §
,¦ª n e¥ ,=ª0,p40¦ôñ6ncV=¤,5p6 Id9cQ¡6
ce6 ¥p¬ye^c=^h¶Teð w¤,p= <~ ' # = =,Ex Q x(¾§T e¥ ¬C,Q ^ I e¥ ln 2 e c^¤ 6·7 ce6¬¢ δ = 10 @ 9 x ¾§ce^ cp ¬=^ª^d9eðh¤0
cd÷Ðqqùn=jø −5
∞ X
ln(1 + x) =
¹¤
c¶
=6 x=1
n=1
(−1)n−1
xn , n
−1 < x ≤ 1.
∞ X 1 (−1)n−1 . ln 2 = n n=1
70c=¦eð¤ 0¤
º0
eAc¦ cC,
,!eźTc 6 e¥¬ ^d9 ¥
^ c e¯?c f0=cfte6¬e¥ ¢ ¥
,ª6©¶e^cª ± ¡Ve6 !c¶C=,^=f c¬ ^¤ ¥
,ª0¢Éµ d9c^e^ f ^cp qQ¬i fª ,|r?,?| ≤¬ §n¦ye¥ ==610=^d9§9¦¶lnl¯¤¤20ª0
6 dPb#=f e¥c ce6=ª0d9d9δ d9& =¤ e¥ c10¥
,=ª 6 º p¤ ¤ce6ª0ª0 d9d9ª0¢ ¤ ¢[cn ¤,p≥=¬»f10 0 µ ^e^f ^c=Cd9cQ¡V c ó§ ¤,=¥
ce6y¤,?
cpdP¥ dÉ c(p=f0c=±¡T¤ 6^ªC ¬?ppI
=6 ô¨(c¤ d#ªQ [b^±0 c¤, ^dT&°M
c=,C=c?f ,cI,
¢É ,7y ¤ ±*0
e6¯ª 7b^^±0e6 c^¤, 0c»¯ª0c=QdP 6 ¬= ·7c=¬t¨( c ¤ e¥ d#c»ªQ e¥² =b¥^=±0^d9 c§¤,¦*0e6ª ª 7^dK^e^ª=e^ªf 6c¤ ^ ¤ 0 µ QeA¦=cQd9
^ d9 cd e¥w,¤ 0
, ¤ ¥
e6p=p , ¢M7C^cy e¥ c ln 2 l( ,ôñ6c^cV¤,p6 cQ¡^ t÷Ðqqùn=jø C
−5
5
x
−1
n
−5
−x
ln(1 − x) = −
∞ X xn n=1
n
5
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß ôT^d cp ,ª0 ^ cI¤,p6 c?¡^ (y÷Ðqqùn=jø¥ bcA
¹¤
∞ X x2n+1 1+x =2 , ln 1−x 2n + 1 n=0
x = 1/3
÷Ðqq j=ø
|x| < 1.
¨(c¤ d#ªC h÷Ðqq jø9
=6
ln 2 = 2
MOSRL
∞ X
1 1 . 2n+1 2n + 1 3
k X
1 1 2n+1 2n + 1 3
°« ^ f0 kµ¸,=e6 c^cce6=pf0¤0
7
, kµ¸,=e6 c±he6ª0d9d9 n=0
ln 2 ≈ 2
=6
n=0
h
i 1 1 1 1 + + . . . < 2k + 3 32k+3 2k + 5 32k+5 1 2 1 1 + < + + . . . = (2k + 3)32k+3 32 34 1 1 2 = . = 2k+3 (2k + 3)3 1 − 1/9 4(2k + 3)32k+1
|rk | = 2
4° e^¢§
Md9cQ¡^dh,=±¯f cp ^e6c4e¥ =¥=^d9§¦;p ¤ ¯f c=c¤ cdhc=·7 f0M P ¤ ^=Cc±0
6 l( ,¶ñ6c^c7
cp ,¡V c7ò¬ + 3)3 ≥ 10 0áI^^f0c7ª00
6¬ c[ ^¤,=^0µ 10 e6óP ¥c
=cªQ
p6[¥ ¬=T c cp ¬eŶª ¡ 4(2k ¤ k = 4 ce^f cp ¬fª 4(8 + 3)3 = 8,6 · 10 > 10 −5
2k+1
5
8+1
5
5
4 X
1 1 = 2k+1 2n + 1 3 n=0 h1 1 1 1 i 1 ≈ 0,693144. =2 + + + + 3 3 · 33 5 · 35 7 · 37 9 · 39 ln 2 ≈ 2
#b =f dc¤,pCcdTd9^e6c!qQi e¥ =6=^d9§¦w¤0
!÷Ðqqùn=ejø&
c±ce6¡=pc c c¶ c=e6^¬¢ ¬y¬ye¥ ?µ l¯¥=¤^ª0d9^§ ¦d9¤ 0e¥
©c÷Ð=qd9q j ,ø¥ =d*ªQc
?À c,e^=¬ô±~g=Ci,iQi iV^¤, p ªlne^f 2c¤ ¬heA¦cC
d9ce6¬h¤0
δ =¾10e^0 ,ª cpp ¡V^ (ce6e6¤ ycñ66c c ±ô ¤ c« ¥
,ª0¤ *
,¶ ¤ p ¡^ §¦¶T e¥ ^ ±~ ¡d9²c ·7^d^ l7? ^(d9 =ªC
^d¤,=c=pp¬[cp ¬f ce(¤0
=d9~b^±0 c¤, ¡^ !°McQ
©c±!¤0!
t,cp±!CT¡[¯=6ceð t ce6Te6¤ d9^cQy¡VeA ¦c¶cQ
,c^7¤,= d9 eð ¬^eðd
ô¤ ª0^6¤ c± cd²^e¥¤ 0
,d9^
¬=c·7e6 0d µ ¤0^ M
e¥ cTde6,¤ Qc7,?eð¦ cC¬
, §7¦~ e¥± eÅ=y6=,^pd9CT§¦;=6 ^eðdyª0cT=e6c¤ ¤ ^c dT ;Cóòd =÷|d90c ¤ ªC^ ,cª0¤,·7pCC c =^ d& ø&IeA¦ ¤cC0
d9ce6cp µ Q7¤,~=d9ëf ¶ ¶
¤,= pe^ e^cdP^c[pf¤ ª0 ¤ e^=^ø¥d ,cQc
=,7=f c¶7d96
ccC
cpë ªC^ , ª0 (·7fôC ¤ º d9eA^¦¤cQª!
qd9qc j7e6¤, =e^÷|e^ñ6d9c=c¤ § d¦cC6
7 cC
<~ ' # = =,Ux T;x(¾§T e¥ ¬ √3 ec ce6¬¢
c δ = 10 5
−5
−3
1 ×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô M¥àRN @9 x¾§ce^ cp ¬=^ª^d9eð cd9,? ¬ Td¤0
cdþ÷Ðqq 3ø¥=m(=fe¥ ¥
,ª6I¯÷Ðqq 3ø¥
^e f c¤ ce6¬^^c[eA¦ cC
d9ce6~ª0d9^ ¬=·=6eÅ~e ¤ p ¡^ ^dºC,? ^ ± |x| fô¥
« = ¹cñ6cd#ª! ¤ ^c¤,p^c= =. ¤ cC
,7^f!ª0Te6¤ ^, ¢ëeA¦cQ
d9ce6 .Q=f0 ¢4,=6eð [Tc¤ |x| p=f dTc cô§ wf=fôd9c?¡V cd9^ ¬=·7I¥
« ,= ¤ d9^¤ √
3=
r
7 48 49 = 16 49 4
r
bcA
e^cA =e^ cw÷ÐqqùnY3 ø¥ d9^^d √
7 1 −1/2 48 1 . = p = 1,75 1 + 49 4 1 + 1/48 48 h
i 1 1 3 1 = 1,75 1 − + − ... . 3 = 1,75 p 2 48 8 482 1 + 1/48 1
°4e^¢§
I0
c cIª ¡T¤ 6¬4e¥ =¥=^d9c= ¤,= cMi iiig=o=dP6 ¬=·7i iiq=k §µ p¤ ^Qò·cp¤0¬¯
7ñ6C,c=^f0cIcC ,^Q¤ ¥^
, ª0¢M 70,± ceð cðd9
cQ¡V
ccÉe6=ª ?c^ ¤ ¡[c7
=?^¬ =¬
cce6e¥= p=6c=f7^d9¤ §0¦
M¤
§0=ªC
6c
=6 √
1 3 ≈ 1,75 1 − = 1,732. 96
<~c f ' x #== 0=, xEX ^e¥x( ( =±!¤,p6 cQ¡[^ ¯¨ª0 f « y(x) y¤0
b^±0 c¤,Vycf ¤ ^e6 ce6 ÷Ðqq;:=iø xy − e + e = 0. d96@ 9cC
! ^c= xɤ ¹¥
¤ ¥ Cx =§¦h0 f0chñ^¨(ª0¤,¨(= « ^^ c e¥ ¨¥
,ª0ª 6f p« § ¢ c y =70 &^d¹c[ñ6c0d#
ª(&¤ 0e^
cp ¬=^ª x
y
y(x)
y(x) =
cC
e6p= cf0f c=c¤ c^cª0¤,= ^
=6
∞ X
an x n ,
n=1
1 2 a2 (a1 − 1)x + a2 + 1 + a1 − x + 2! 2! 1 3 a31 − + a3 + a 2 + a 1 a2 + x + · · · = 0. 3! 3!
ce6c=V¤0c
h f f0 =fôe¥^ C c±~eA¦ cC
eðw=e^cp ¢É c¶¤,= cd9^¤ c[ ^f c=c¤ c±~cf ¤ ^e6pµ fye6ª0d9d9 = 0 ¹cñ6cd#ªf cñ^¨(¨( « ^ ¤ ¶=e^¥¦ye6^ ^0¦
x cp ,¡ òx=¬¤,0= ªQ ¢¯y(0) C
°4e^¢§
a1 − 1 = 0; a1 1 a2 + a 1 1 + − = 0; 2 2 1 3 a3 + a2 (1 + a1 ) + (a1 − 1) = 0. 6
a1 = 1 a2 = −1 a3 = 2
e¥ ¥
cp¥ ¬ c ¤ ¥
e6==p ^
y = x − x2 + 2x3 + . . .
=6©,Q,? ¬ T±c=¤ 6Ccfþ¤ 0
b^±0 c¤,¤,=e^e^dPp¤ =^d9c±¨ª0 f « C c» ¤ ¥
µ e6p¥ =p¬ ^ª0 ¢ë ¶ c¶hdPc? cc±»e6c©f ¤ c^« e6^ fcªe6¨tª0 cf « f xy == f0 (x)d9cQ.¡ 66~
pc¶¬wQ?
=cp ª0¢KªQ
ª0c¤,p= 6 ^ c ¤ ^0d µ ÷Ðqq; :=iø¥
QÊ Ê?Ë+¢ß®ð× Ñ¾IÍÐÒÖ ÍI£PÝÒÏ[ÖÕØÙÚpÛµXÍÐÕ× ÑðÙ,ß <~ ' #= =,xE` x(¾§T e¥ ¬Vec ce6¬¢
cVi0 iq Z1
@9 x?I¤,p6 cQ¡[6
2
e−x dx.
0
ex = 1 + x +
cp ,ª0 d Z1
e
−x2
dx =
0
Z1 n 0
x2 x3 + + ... 2! 3!
x4 x6 x8 − + − ... 2! 3! 4!
2
e−x = 1 − x2 +
°4e^¢§
M¥àZM
o x4 x6 x8 − + − . . . dx = 1−x + 2! 3! 4! 2
1 x5 x7 x9 x3 + − + − ... = = x− 3 2!5 3!7 4!9 0 1 1 1 1 1 1 1 1 =1− + − + − ··· = 1 − + − + − ... 3 2!5 3!7 4!9 3 10 42 216
ðe¹ ce¥ cI¥
c e6p^p7÷|cfV cMQ ø#¤ e¥^ =c¥eA=¦^cCd9
c 71/216 ^¤ c<^cI0,01 c=¤ c=¹·7c^e^ f cp c¬^c¯fª0 ¤ ^0,
[ óPC ,¥=
f cc p^¤ ¥¥
,¬ª0 ¢Mc
,±0 µ T e¥ ^ »
= c^c ^^¤,? eVc ce6¬¢
ci iqV
ce6ppc c=^¬ 6T¤ ^¤ §¦~e¥ =¥=^d9§¦ Z1
2
e−x dx ≈ 1 −
1 1 1 + − ≈ 0,74. 3 10 42
m=fyd9*C,=^dT ^¤ cc¤,pC,py¨ª0 f « T¤,p¡[=6eÅy ^¤ 6(ñ¥ ^d9^0µ p =¤ ♦ T¨ª0 f « ¶y ^ ce^¤ ¥
e6^ ,c= ¤ d9^ ^e ¨(c¤ d#ªQ ¬=¢Mc, 4 áI^± «,
,ôT e¥ ^ w 6^¤? Z e dx ^c=Cd9cQ¡V c 0
−x2
1
−x2
:î ùï
A $ì
0
ìYî»CCuó¼C+=ó>½ ÒéØrá?8 ÝbÐb×ÐOæråèÖVéØ
y 00 − xy 0 + y = 1,
y(0) = y 0 (0) = 0.
ìYü ù ì@ó}½ ÒéÒÙrØ@ÒådâÐàRÙ@ÒÙrØrçØrÕÒ]Ógà àRØZ×uÒ±äáWÒÛuÒÙ@Ù@Ö43OÖ âYçd×Ð y(x) =
∞ X
ak xk .
k=0
4Ö 3
ÜÐVäÙ@Ö ÙuÐ]æÐbÜ8RÙrÞÓÔåWäÜuÖVàRØrçrÓÚrÙuÐßZ×uÒ]ÓÚ@æráWÖ y(x) =
a0 = a 1 = 0
∞ X k=2
ak xk .
ê2 Ö43
×tÐ
1 ×ß6Ø=ß Ë}
PÝÒÏ[Ö Ñ¥Òß6×r¿QÒÔTÍPÙÚpÛ0Ô ¥M àRS sÖ]×uäáRÐÙ@ÖVàRÏZÐHGáWÖ43OÖ âuÐÝÜuÖOÑÒÙrØrçÍà×@رÒâ@ÒÙrrØuÐbÜ8RÙ@ÖRÒ±ådâuÐàRÙ@ÒÙrØ@ÒÊ×ÐVÒá ∞ X k=2
k(k − 1)ak xk−2 − kak xk + ak xk = 1.
sÊârØrâuÐàRÙrçZà ÏZÖ4G¾ØrrØ@ÒÙráRÞ"ÛrârØÖ]×@ØrÙuÐÏZÖVàRÞãÍäáWÒÛ@ÒÙrçdã
2 ÐÏÍÏrÐÏ
a1 = 0
ÚZáWÖ
a2 = 1/2; (k + 1)(k + 2)an+2 = (k − 1)an , a2l−1 = 0
Ú+3×uÒ
l = 1, ∞
a2(l+1) =
ê+.ÊÜuç
n = 2l
x
ÚZÛ@ÖÜ@ådærØuÓ n = 1, ∞.
Ø@ÓÒ]Ò]Óâ@ÒÏrådârâ@ÒÙráRÙrå!u¿±ÖVâ@ÓåbÜ@å
2l − 1 a2l , (2l + 1)(2l + 2)
ØrÝÏZÖVáWÖVâ@ÖVß àRÞàWÖO×@Ø@ÓàRÞâuÐÑÒÙrØ@Ò×rÜ@ç ÏdÖ4G¾ØrrØ@ÒÙráRÐÖ?0VÕÒv3OÖæZÜuÒÙuиâYçd×ÐZêD2ÐÏrØ@ÓÖ?0VâuÐ ÝOÖRÓÚrØ@äÏdÖRÓÖRÒ±â@ÒéÒÙrØ@ÒØ@ÓÒ]Òá àRØZ× ∞
y(x) =
x2 X (2l − 1)!! 2l+2 + x . 2 (2l + 2)! l=1
/
ÊOË+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í Á
ÿw
&
M¥àQà
Á áI½¾§½ îïwé VìSIê
'Â#$Ê,
#0
b^±0 c¤,§f=f*§ cc=d9^ ^ cºò·7=ò p , 6eð²d9c=7 Tdö e6¤ª0d9^cd
¤ [¤ 6·7^ ·7 ¤ cf c^cIf0 =e^eCQQ
? °M
,=f c cþ¤,p6 cQ¡V¬I¨ª0 f « ¢²¤ 0
¡Cb,^ª p±0 f ¬4c=¤, e^6 &;cp ¤ ¤ ^^c=ªC6c^cCeÅc
©
T^ ,[§~ñ6^cce^ f ±7¤ c¥¨
^ª0¥ ,f ^p « w7
f M¨(c ñ^¨(^¨(f ^¨(c=¤ ^c « ¤ « c ^±¯¤ªccC d9Vf c¤=e60
C ¬h[,c4b¶^±0 ¤ ^^f =ccc=¤,[c= ¤ ^c cd&CQ¦ pcC¤
c¤ d9ªQd9
¥c µµ 6 e^ cp ¬=Cc= ò¤0
4b^±0 c¤,44Q?
Q,?¦;=A
òñ67ªe¥ c7 òT cp ¢Með ¾ ¤ cd²0 cQ¡^6 ¤ 0?¦¶ ,=e6c cf ¤ 0¶¦f cCc=
c¤ eŧô¦! d9,6=¤¬7ª ·
¥= 6ceðe¨ ª0^ ¤ f ^« ¤ T d9 c e6 d9¬ô^0¢M 7 d9^ô ¤ ,^I¤ Qò?
,=p0 µ
« ¨( ,¨(?^ ¤ ¬^ c¯« c= C¤ ª C,d9=c¢Me67¬ ¨± eŪ0¶ f c=« ¯ ¤ 0
b#I=bf ^±0M ¨c¤,ª0 f « C ¶c=d9¤ cQ0¡V
¶ ,cIp¤,CpT6 =cQ6¡Veð¶¬7¤0¯
¤c0d
®¯0 ª0¤ ¤ ¬0=µ ôf~^^c¤,=e^e^d9c=¤ ^ ¢ d9 ^¤ ¥¦cC
dT =uD x>=,xÄ<~. h ;;"#t w1.
l( ,ô ,ª0·7^Cc¶ c dP= we^d9Te¥ V¤ 0
y®¯ª0¤ ¬7c¤,p d9eðwfwe^=ª=« .;c=C 0µ f0=¢M7^±¶^fc¤ c±? ^^¤ =.b#=d*cC
eÅ c f0cc¤
,p c^côpC eC c ^fc¤,=d¼f c=c¤ c^c!dPcQ¡V ch¤,p6 c?¡V¬~ ¢4c±t,= ^¤ ¥
ºQ?
= T±^fc=¤ #b#=f0c ¤,¤ p6¥¦ cQ0¡ ^ cp ^c=[C cp ,e^ ¥6 [Ve^
¢^± e6 ¨(c¤ ©dP,=?«
© ¢^fc=cñ6¤,=cd9dþ =4^ff!c
¤ ^(± e^e6^e6 ~d fô,,?
©=,c¤=ª[c
¤,==ªd9¦; ñ60¦» e^¥ l¯¤ª0^ d9te¥ c=d9 ªe6p=,=p =6eÅt=Q= d9 cwcC
c=C,? cVe^cc=6pµ
e6¥ , T^d9§^f¦!c ¤,¤ c=77,=e^c^¤,^c¶4 f0=^f0fwc=e^fc?¤ ,§ ¦[¤ c ¯e^¥ ¤ c,pCCT¥
=^^ d9 §[¦[¤,=^e^6e^c(dPf pcc¤ ¤
=,^pd9pc=^d9c¶[^fc c¤ ¤,¥ µ ô^fc¤ cc¤c c¤ d9 ¤ c= c^cf0cc¤
,p c^cpC eC C pV¡¯0
6w¤,p6 c?¡^ w cypC e6ª¶ 6¡V¶yce^ c¯¤0
¶®¯ª0¤ ¬=;A
7y¤ cp ¢M^feð~c¤ ^ce^wf c T^e6 = ª0T,=d9¢M óP¨ ª0¥
,f ª=« h ñ6 ch±~f 0c
^^= , T^f yc=,c=¤ª0c¢¤ À¨ ª0 e^f ¥« ¢¯÷|f ,cQc?¤
= ,pª0 ¢¼øQ,=d9 ^¤ 0cpµµ dñ696c¡dwª ¤ f c d9[a,6¡b]ª f d9òcQ¡Ve^ e6cw^ d9¤ ò¥¨
e6ª0p =f « ± ¬pf =¤ f»¥
¤,e6pp6= pcQ ,¡[ ¢M^ 7 V¦7 e^c!c c^± f c=p¨cª0¤ cf ±»« Q ?c
,=? ¬c ±tT± ,¢ pC e 4 =,? c7f0cc¤
,p c^c~pC eCl( ,,?¦cQ¡[
^ pf cc¤
,pt¢ôQ?
= c± ¨« ª0± .f e« Vcd9¯c=ñ¥7c¬dt¢´f pc=C ce^¤ Mce¥^ ch¥
,d9ªcQ6¡V cô^e6^ye6 ©c c 4e^f0©? , c¤ ¤ d9 c ^c¯÷ = ,¤ ?c cC¥ô¥
d9^cC
,ªQy ,¨ª0 ^f0f0µµ c¤,øc¤c^c,? ¬ ce6*¨ª0 f « ± ò¹cp chþ ce¥ ¥
cp¥ ¬ c!¤,=e6e^d9c=¤ ^, ¤¨0ª0
!f « ®¯ª0 ¤ dT¬,¶Ç
¤ ^ e^¬7=¥d9
*^ cc^¤,w=¤, p 6
¥d9 eð?¶¦;# ^f0c=c=e^=c¤ 7T^d ,§=¦cc¤ ccdt=7 ^c T d¼±¶»ye^c ^¤ « ¥,
?¥ ¬^ T± d
cªe^f0=¢M70¦7^ª0 ^ &^c¤ ¯¤ ^c cd96¤ ,^e^f0c^c¤0
4®¯ª0¤ ¬&Mc=³^d9=Q ^cpµ ¦cC
d9cd
,~f0 =e^e^ ^e^f c^cy¥=¤ d9c ^e^f c^cy=,? Q ®¯ª0 f « ,pCT=6eðð ?
f c±þ ¤ cd96¡ª f P^e¥ þc,ñ6cd ¤ cd96¡ª f 7 ^ f¤ (x) ^¤ T cV
¨(¨(^¤ ^ « ¤ªCdP ¹cC
!^,? ^ [a, d9b] f (a) f (b) f (a) f (b) c dP=¢MeðcC
ce6c¤ c ò ¤ ¥
¥ f (a+0) f (b−0) f (a+0) f (b−0) d96¡ª ®¯cª0f d9f c?« ¡V cf (x) ¤,pC,pC¬(T,4=6f0ceð !^ f ªe^ccT cpe¥µ¸ A c?
¤ f ccd9±w6¡yª ¤ f0ccd96¡ª f0pf ¡[
[a,cdb]T^e¥f c=!c ¤ ¤ §cp¦ µ ð Q
fp~¨ª0 f « f (x) 4 bc f0 c±¶c f0c±~ñ6xc±hI¨f c=ª0 cf ¤ « c ±V ¨ª0 f ¤ « c= f (x)cdþ e¥p , ,ª0 ,6=IeðVc,ð ?,
pf CcT± =,6peðChTc=e^6ceðcy± ¤,=0 ¬pµ A ?
♦f0°p ^¾§0cp
µ¸ pc cC¤ §c ¦;Qfcpªµ¸ e6c^¤ §cpµ¸¦;ðC ^?e¥
f0 pf©(x)ô 4 ¤ fcªd9e^6c¡ ª cpf0µ¸A ¨?
ª0f0 pf « C©cÉ d9cQf¡(x) 6~4òfª¬he^c 0 µ cpµ ^^¤ ¤ªCdP[ñ6cd ¤ cd96¡ª f = 0
0
0
0
0
2
M¥à 0
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ^e¥ ~ e6®¯ª 7ª0 ^f e6« =ª 6f (x) ÷|[e^,cpCe6T^= 6 ceðd»=0 e^~cp ¢M^e^c côe6 ^ ^ ^c¤ d ¤ e^ªd9CTd9e¥c ±©Qøɶ ¤ c^d96¤6¡? ª f [a, b] 1
Zb
|f (x)|dx.
¯® ª0 f « ,pCT=6eÅf ?
¤,? c ^^¤ ¤ª^d9c±[4 ¤ cd96¡ª f ^e¥ ~ e6ª 7^e6=ª6f(x)÷|[e^ce6^ cd0 ~ ^e^ce6^ cde^d9Te¥ QøÉ ^6¤? a
Zb
[a, b]
f 2 (x) dx.
ó§f0? , ¤ Tdw ¤ cC¥
^ ^dh
=ª¦If ?
¤,p c4 ^^¤ ¤ª^d9ò¦¯É ¤ cd96¡ª pµ f [a, b] ¨ª0 f « ± f (x) f (x) ,pCT=6eðh e¥ c Z ÷Ðq?g0úq?ø hf (x)|f (x)i = f (x)f (x) dx. ?c ¤ ¥
¥ ^ *÷Ðq?g0úq?øÉe¥ ¥
,ª6he^c± e6cy ^± ce6©e^f0? , ¤ c^cô ¤ cC¥
^ ? = ÷Ðq?g0ùgø hαf (x) + βf (x)|f (x)i = αhf (x)|f (x)i + βhf (x)|f (x)i, ^f c=c¤ T[ ce6cQ T=óP ¥
,ªw ^± c±? ^^¤ = e cd9c=7¬¢Àe^f0?µ ,A
¤ αc^c[β ¤ 4 cC¥
^ hd9c?¡V c^e6h¶
¤ª0^ @ ¬É^^cd96¤ ^e^f, ! ®¼ c l¯hf ?
¤,p c ^^¤ ¤ª^d9É,© ¤ cd96¡ª f ^e¥ b] ¨ª0 f « f (x) ,pCT=¢Meðhc¤c^c,? ¬ Td9h,ñ6cd ¤ cd96¡[ª f0=[a, f (x) Z ÷Ðq?g0 ø hf (x)|f (x)i = f (x)f (x) dx = 0. ,pCT=6c¤ eðd9hc ±» e¥f c ?
¤,p c ^^¤ ¤ª^d9c±º,h ¤ cd96¡ª f0 [a, b] ¨ª0 f « f (x) a
1
2
b
1
2
1
2
a
1
2
3
1
3
2
3
1
2
b
1
1
2
2
a
v u b uZ p u kf (x)k = hf (x)|f (x)i = t f 2 (x) dx > 0.
÷Ðq?g0 3 ø
c¤ dP ¶¨ª0 f « f (x) ¤,=,¶ªQ ¢ ÷ kf (x)k = 0ø¥.cp ¬f c~^e¥ f (x) ≡ 0
, x ∈ [a, b] ¥ =u§D ¦y>x =,Vx( ¹cpc ,fª0p« Q¥p §¬ ¦y cpc! cQ¨¡Vª0 f ¥« ¬ §¦0
f (x)c¤=c^xc,? ¬ f*(x),I= ¤ cxd96¡ª ¤ f <~¢4 §'¦V « # k ¤ cnd96¡ª f0
^ c ¤ c ^ c , Q ¬
À , ¶ ^ ^ ~ c
p c c
(¶ ¤ c=dP¥¡ª f0 [−1, 1] [0, 1] 4 c¤c^c,? ¬ c=e6wT e¥ ¬V c¤ d9 ñ60¦h¨ª0 f « ± @ 9 x=e^e^d9c=¤ de^,?,? te¥ ,ª0,=±« ¥ §¦ k n Mó&cð =e^ cºc ¤ ¥
¥ ^ ¢ ÷Ðq?g0úq?ø¥ =T e¥ d ¤ cC¥
^ ¯ñ60¦~¨ª0 f « ±h, ¤ cd96¡ª f0 [−1, 1] a
k
hx2k |x2n+1 i =
Z1
−1
x2k x2n+1 dx =
Z1
2k
x2(k+n)+1 dx =
−1
1 1 x2(k+n+1) = (1 − 1) = 0. = 2(k + n + 1) −1 2(k + n + 1)
n
2n+1
OÊ Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í M¥àR7 c6±¶^ªQ¨ ¬Qª0p pf « c ô^70e^
^d9 d96 c¤ e^ f ,cp ¬§f¦ôªV ¤ T¥e^
¥ e¥? ,¦; 6?eÅô¤,c= ^¤ ¥e6
¥ ^ ªCT ¢±ô e^f0?^ ,^ ¤,¤ ? hc^c=c[ ¤ ^c 6p6µµ ¥
^ ôQ?
= §¦~¨ª0 f « ±hôe¥ ¥
,ª6y0¦ôc¤c^c,? ¬ ce6¬ b ^
^
¤ V ¬ 6 e f ? ,
¤
c I
¤ c C ¥
^
= ; ^ e c A = ^ e
V c c
¤ ¥
¥ ^
õ ¢ ÷ Q q 0 g ú ? q ¥ ø , T
¥ e
* d , ¤ cd96¡ª f [0, 1]
hx2k |x2n+1 i =
Z1
x2k x2n+1 dx =
0
Z1
x2(k+n)+1 dx =
0
1 1 x = 6= 0. = 2(k + n + 1) 0 2(k + n + 1) 2(k+n+1)
¾´ñ6cdKe¥ ,ª0,=he^f0? , ¤ ch ¤ cC¥
^ w þ ¤ þf0=f 0¦ ~c¤,p=6eð ªC¤ c¬ d996¡ª cf º= c=C,Q,=6©c=e6ª e6 hc¤c^c,? ¬ ce6þñ60¦ºk¨ª0 nf « ±þ!ª0f0pQ= cd l( ,ôT e¥ ^ h c¤ d9 ce6 cp ¬=^ª^d9eðh¨(c¤ d#ªC c±»÷Ðq?g0 3 ø¥ r Z1 1/2 h 4k+1 i p 1/2 1 x 2 = kx2k k = hx2k |x2k i = (x2k )2 dx , = 4k + 1 −1 4k + 1 −1
kx2n+1 k =
p
hx2n+1 |x2n+1 i =
Z1
(x2n+1 )2 dx
1/2
h x4k+3 1 i1/2 r 2 . = = 4k + 3 −1 4k + 3
l( ,y cp ,ª0« ¥ §¦ k n f c=c¤ Td9*Q= ·7^dº0
k = l + 1/2 n = m + 1/2 T e¥ ^ ¯e^f0? , ¤ c^c ¤ cC¥
^ ô
=6 k, m = 0, ∞ 2k
hx |x
2n+1
−1
i = hx
2l+1
|x
2(m+1)
i=
Z1
x
2(l+m+1)+1
x
2(l+m+1)+1
−1 2k
hx |x
2n+1
i = hx
2l+1
|x
2(m+1)
i=
Z1
1 x2(l+m+2) = 0, dx = 2(l + m + 2) −1
1 1 x2(l+m+2) dx = = . 2(l + m + 2) 0 2(l + m + 2)
^ cct¤c=cC^,c?,,?= 6¬=Q TK,c»~ª06f0^pchQ= ¤, =Tc±¨ ª0cp f c« ² c4¤ c¤ ^ccd9,?6 ¡¬ª f »[0, ¤ 1]c;d9l(6¡ ,ª f0T [−1, e¥ ^1] c¤ d9 67(¤,pIce^ cp ¬=^ª^d9eÅ~¨(c¤ d#ªQ c±»÷Ðq?g0 3 ø¥ 0
C
2k
kx k = kx
2l+1
k=
Z1
x
2(2l+1)
−1
kx2n+1 k = kx2(m+1) k =
Z1
h x4l+3 1 i1/2 r 2 , = = 4l + 3 −1 4k + 3
x4(m+1) dx
1/2
h x4k+5 1 i1/2 r 2 . = = 4k + 5 −1 4k + 5
ó§ e6^dPy¨ª0 f « ± {f (x)} i = 1, ∞ ÷|0 i = 1, nøT,pCT=6eð!c¤c^c,? ¬pµ ,^e¥ ~e^(^(¨ª0 f « h c,=¤ cc¤c^c,? ¬= ,? = [a, b] −1
c±ô,
dx
1/2
i
hfi (x)|fj (x)i =
Zb a
fi (x)fj (x) dx = kfi (x)k2 δij ,
÷Ðq?g0ùnø
M¥àRB
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ¤ c= ° c¤± c^^e¥c , ? ¬,ph, [a, b] e^ e6^dPy¨ª0 f « ± {f (x)} ,pCT=6eð!c¤c c¤ d90µ ÷Ðq?g0 jø kf (x)k = 1, n = 1, ∞. e¥ ªe¥ c ÷Ðq?g0 jø hT cp ^ c &c»d9c?¡V c© ^¤ ^±²fe^ e¥Cd9!¨ª0 f « ± / 1
n
n
{vn (x)}
fn (x) , kfn (x)k
vn (x) =
Q=¥
cd9c p , ¢M7^± eðhc¤c c¤ d9 ¤ c= c± ,e^cA =e^ c!÷Ðq?g0ùnø¥ ¹ªe6¬ {f (x)} 4 c¤c^c=,? ¬,p,y ¤ cd96¡ª f [a, b] e^ e6CdP¶¨ª0 f « ± .¾§§µ Q ^e6 Q?
=c ¤ cc^±~c,,? ¬ c^,c7[,ñ6 c[a,db]pCp Ce^ (eC d90b^6cVA
I0
¤,p6 cQ¡^ ^^¤ f ^c=d»cñ6¤ =cªV±he^¨ e6ª0 ^f d#« ªy ¯f0ϕ(x) [a, b] X ÷Ðq?g0; :ø C f (x). ϕ(x) = =mcñ^¨(¨( « ^ ¢!¨ª0 f « hpC e^ .? =f cñ^¨(¨( « ^ ¤,p6µ cQ¡^ ÷Ðq?g0; :ø9d9c?¡V c( ^^f cϕ(x) ,=± ^e¥ ¶ ¤,{f==ª0(x)} ¢ ^=ª0¢ ,=e6÷Ðq?g0; :ø9e^fC? , ¤ c ª0d9d9cQ¡V cQ ¡Vc c¬td9,^ f¬V(x) c¤§0b
ccAf~
»e6ª0 d9¤ d9 ¤ ªce¥ =c ~²~¤,= c^d9^¤ ^ ¤ ¤ cc±²= eA¦cCh
~d9 ccpe6 ,ª0 ¤ 0¬
²÷Ðq?g ; :ø n
∞
n n
n=1
n
n
m
hϕ(x)|fm (x)i = =
c=fªC
∞ X n=1
Cn =
∞ X n=1
Cn hfn (x)|fm (x)i =
Cn kfm (x)k2 δnm = Cm kfm (x)k2 ,
hϕ(x)|fn (x)i = kfn (x)k2
Zb
ϕ(x)fn (x) dx
a
Zb
.
÷Ðq?g0 oø
fn2 (x) dx
# 0
À÷Ðq?g0;:ø[ewf cñ^¨(¨( « ^p=d9 §c ¤ ¥
¥ , ^d9Td9² ct¨(c¤ d#ªQ =d£÷Ðq?g0 oø¥ ¤06Q
=cd& øTe^ ®¯d9ª0c¶¤ ¬c==V c^c ;eA¦ cC
eðcw0 !C 6?;,pCT=6eð¤0
cd ÷|0 =e6¤,=f Td ¡I,=dþ ¤ ¥
e6côò e^ ¬ ¤ hf0=f 0¦~ªe¥ c0¦h¤0
®¯ª0¤ ¬¯eA¦cQ
eð ♦
c¤ e6 p =^ dc f d9º^ Q ?c
Qf ¨ &ª0 ¾§f cp« µ¸ ^¤ f(x) §¦;#¹
,¤ [ Q?e^e¥
=¥
c=c^=c© c¤ñ6c^cc^,c(? c¬ ¤ cc^c©eCpc=CC d9eCcQ ¡V{f ©(x)} c eCp¬¶f0 =e^e^ ¨ª0 f « ± ;f c=c¤ T[d9c6ª ôò¬¶¤,p6 c?¡^ ¼¶¤0
»®¯ª0¤ ¬ ñ6,?c dt¬ Tp±ôC e^p=C ¾§e cpµ¸=c¤ §¦;,¾&
,ϕ(x) ¤,Qp?¶
Q=?
Qc,^[c¯,f0p 7=e^(eC¯e^¨6^ª0c f c=« C ± ϕ(x) [cdPe6p¤ c^dPp¬ c ¤^e^cf0^cc?± µ c 0 f = 6 V ¨(C f ¶7e^0 ,ªy{fñ6(x)} c^c¤,=e^e^dPp¤ =6eÅô7¤,p6
¥ ?¦; ce^=7^ §¦¶e^ ^« ,? ¬ Td ¨÷Ðq?ª0g0 úq?f ø&« cQdw
Iª0eð¤,~=cp ^^ ( cd!=7dP^p(c^dP ¤ p¥
¥ ^^e^ f0 c±7¯e^¨(f0?C , f ¤ ô c÷|e^^dTc0 û p¤ üýcø¥pC¾¥ñ6
^c d!e¥ , ª0,=§d9^e6c a
n
n
n
hf1 (x)|f2 (x)i =
Zb a
ρ(x)f1 (x)f2 (x) dx
OÊ Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í M¥àWG
e4¥ ,^ e^¢Mcceð±yh¨ ^ª0¤ 6f ¯« ñ6 ^c± e^ρ(x) f? , >¤ 0c
( , ¤ cx∈C(a,¥
^b) y= e^ce6p? ¬ T¥ ² ^¤ ^c ¤ ¥µ Ç
^e^¬d9*¤,=e^e^d9c=¤ dº ^¤ =ª0¢ Q?
QªT¤,=77fQ ^e6c¤c^c,? ¬ c^c?µ C eC[ce^ cª0¢¤ ^c cd96¤ , ^eCfª0¢ e^ e6^d#ª~¨ª0 f « ± =uD Ex D x %I L=' / 0 ² ;;" w1. ó§ e6^dP¨ª0 f « ± n ÷Ðq?g0 ø πnx o n πnx o cos , sin , n = 0, ∞, ,pCT=6eð ce^ c c±²¤ ^cl cd96¤ ,^e^f0cl ± e^ e6^d9c± ¨ª0 f « ± ,t ¤ cd96¡ª f ¹^¤ pôh÷Ðq?g0 ø§ cC
e^ e6^dPV,pCT=6eðh 6 c± =c¤,p 4 ^ 6 c± [−l,kl] Tp c e^ e6^d#ªh÷Ðq?g0 øªC
c cQ= e^Tp¬I0
sin 0 = 0 F cos 0 = 1 n ÷Ðq?g0úqQiø πnx o n πnx o 1, cos , sin , n = 1, ∞, §
¥ ce6cQ ª0¢ e^ce6==p , l ¢M¯ª0¢¯¹¤ ll = π e^ e6^dPw÷Ðq?g0úqQiø&ª0 ¤ c==¥eð ÷Ðq?g0úqq?ø 1, {cos nx}, {sin nx}, n = 1, ∞. <~ ' # =uD Ex D x(¹cfpQp¬ c q?øycd9e^6 ¡cª ,f pI¤ ^c ~c d9(6c¤ ¤ c^^eCcf0,p? ¬e^, e6[,^dP7 ¢M¨ª0c ±~f « ^ ^c±~ ÷Ðq?cpg0 úcqQi ø cV¤÷ c^c,? 0¬ ,4,M ¤ ø cpµ gø ¢4pôI[−l, cQ
l]e^ e6^d¼÷Ðq?g0úqQiø§c¤c^c,? ¬,[f0=f~, [−l, l] ==f~[−l,~,0] [0, l] [0, l] 5 ¶Q= eCp¬Vc¤c c¤ d9 ¤ c= ª0¢ e^ e6^d#ª~¨ª0 f « ± @ 9 xq?øÉóP Å
,ªhc ¤ ¥
¥ ^ ¢¯,?¦ cC
dþe^f0? , ¤ TI ¤ cC¥
^ Zl D πnx E l πnx l πnx dx = sin = 1 · cos 1 cos = 0; l l πn l −l −l
Zl D πnx E πnx l πnx l = 1 · sin 1 sin dx = − cos = 0. l l πn l −l
°4e^¢§
e¥ ¥
,ª6?c[¥
«,[f0=fô¨ª0 f « ôc¤c^c,? ¬,e^ e6^dP=d {cosl7πnx/l} ? ^=
,
,~n6=e^¥m¦ n5 n=≥1,1∞ m ≥ 1
÷Ðq?g0úq?gø ÷Ðq?g0úqQø
−l
D
πmx E πnx = sin sin l l
Zl
sin
πmx πnx sin dx = l l
−l
Zl h
1 π(n − m)x π(n + m)x i cos dx = − cos 2 l l −l l h sin π(n − m)x/l sin π(n + m)x/l i l = − = 0; 2π n−m n+m −l l Z D πmx E πnx πnx πmx cos = cos cos dx = cos l l l l =
{sin πnx/l}
−l
÷Ðq?g0úqV3 ø
M¥àRI 1
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í
Zl 1 h π(n − m)x i π(n + m)x = dx = + cos cos 2 l l −l l h sin π(n + m)x/l sin π(n − m)x/l i l + = = 0. 2π n+m n−m −l
Ð÷ q?g0úq?nø f0?pI¡[¨(
cc¤ ±ôd#ªQI ÷ÐcCq?
g0e^úq? ge6ø¥^d ÷Ðq?g04 úqV 3 6ø§ »c÷бhq?g0~úq? n^øò e¥6 ¥
,cª±h6¶[c= Qc
,¥= ¤ ¬, pc~e6c ¤c^c,? ¬ ce6¬Vª ¤ e¥ ,ª0,==(e^e^ d9¤ c= ¤ d ^ ^¤ ¬©c¤c=^c,? ¬ ce6¬©ñ60¦º cC
e^ e6^dÀd96¡[
,ª»e^cc± 9¾Kñ6cd n 6= m 5 n ≥ 1 m ≥ 1 D
πmx E πnx sin = cos l l
sin
πnx πmx cos dx = l l
−l
Zl h
π(n − m)x i 1 π(n + m)x dx = − sin sin 2 l l −l l h cos π(n + m)x/l cos π(n − m)x/l i l =− + = 0. 2π n+m n−m −l =
¹¤
Zl
d9^^d÷Ðq?g0úqQø¥
?m ,==f00c ^« n≥
,1~e^¥¦ D
n = m = 1, ∞
πnx πnx E sin = cos l l =
1 2
Zl
sin
÷Ðq?g0úqQjø
Zl
sin
πnx πnx cos dx = l l
−l
l 2πnx l 2πnx dx = − cos = 0. l 2π l −l
÷Ðq?g0úqW:ø
b#,==¤ f c[dcc¤¤,cp^Ccc,dT? ¬e^ É ¨,ª0 f « ¤ cyd96c¡e^ ª cf0 c±[¤ ,^c ccd9ô6¤ ¤ ,^ ^ce^f0Qc ±yce^e^¬[ e6
c^f0d9p Qp÷Ðq?¬ g0 úqQiø# c?µ [−l, l]e6^d9( ,= ¤ d9^¤ ,( ¤ cd96¡ª f ( l , [
^ e ¥ e ¥
c =
V c ¤ c ^ c , ? ¬
c 6 e V ^ e
¤,¥=
e^e^6d9eðc=h¤ c[dñ6e^cfd#? ,ªô ¤ ¤ cTd9T6 ¡¤ ª cfCª ¥
^ w÷Ðq?g0úq?gø µ6÷Ðq?g0úqW:=ø¥pf c=c¤ §¦[ ^^¤ ¤ c=[0, l] −l
Zl D πnx E l πnx l πnx = 1 · cos 1 cos dx = sin = 0; l l πn l 0
÷Ðq?g0úqQoø
0
Zl D πnx E πnx l πnx l = 1 · sin 1 sin dx = − cos = l l πn l 0 0
l l = − (cos πn − 1) = 1 − (−1)n = πn πn n = 2k; 0, 2l = , n = 2k − 1; k = 1, ∞. π(2k − 1)
Ð÷ q?g0úqQø c¤c^c,? ¬, 7C ce^^^dºc¶
¨cª0e6 pf0?µµ « ?c ÷Ðdq?cwg0e^ú
qQ ,e¥otø¥C d9ª ÷Ðq? g0ú^qQ÷Фq?¡[g0øPú
e¥qQ ^i ø¥¥
, ªQ¶6Pp ,e^cf0 c[e^¢4 Å e6
^ ^ dP C«,d [÷Ðq?¨f0g0=ª0úf¶qQ if ¨øI« ª0 ¶± f c« sin¤ôπ(2k c^c,?− ¬1)x/l ,!,! ¤ cd96¡ª f
OÊ Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í M¥àRL ¡V cV cpµ [0, f0pQl]p;¬t6 ^cT¤ c ^e¥c ,^ ? ¬w
c¤e6ª0¬^0e^¦ e6e^f0^?d9 , ¤ ¨§ª0¦! f « ¤ c± C÷Ðq?¥g0
ú^qQ i ø[± ,.»½4 ,¤ ?c d9c^6 ¡ ª c¶f d9c?[−l, 0] cp ^c^c,¯ ¢4cdþ ¤ cd96¡ª f [a, b] ,« ¥ f cdº ¥¡p7^dþ [−l, l] g ø = ^ e ^ e 9 d = c
¤
À d ^
^
¤ » ¬
6 0 ª £ ¢ þ
^
6 0 ª ¢
C c
^ e
6 e ^ 9 d
¤ c 9 d 6 ¡ ª f [0, l] ¤,p6
¥ ¬ c øPl( ,ô 6 c±h cQ
e^ e6^d9 n
1,
cos
πnx o , l
n = 1, ∞,
6weðh÷Ðq?g0TúqQ o ø4e¥e¥ ¥
,¬Vª6e^pf0? , c~¤ ¥T
I ¤ «,c~Cc¤¥
c^^ c, ? ¬,~=e^^d ¨ª0 f « d
,
n, m = 1, ∞
D
n 6= m
D
cos πnx/l
°e6p?µ
πnx πmx E cos cos l l
?y÷Ðq?g0úq?n=ø§ d9^^d
πmx E πnx cos = cos l l
Zl
πnx πmx cos dx = l l 0 l sin π(n + m)x/l sin π(n − m)x/l i l = + = 0. 2π n+m n−m 0 cos
øPl( ,ô ^ 6,c±h cC
e^ e6^d9 n
πnx o sin , l
÷Ðq?g0ùg=iø
n = 1, ∞,
ce6ppc cV¤,=e^e^d9c=¤ 6¬ye^f0? , ¤ TI ¤ cC¥
^
,
n 6= m
D
πnx πmx E sin sin l l
,f c=c¤ TIeª0 6cd¼÷Ðq?g0úqV3 ø&d9cQ¡V cQ= eCp¬ Zl πmx E πmx πnx πnx sin dx = sin = sin sin l l l l 0 l h sin π(n − m)x/l sin π(n + m)x/l i l = − = 0. 2π n−m n+m 0 D
Ð÷ q?g0ùg0q?ø b#=f dtc¤,pCcdTf0p¡[
pyC ( cC
e^ e6Cdº p , 6eðôe^ e6^d9c±~c¤c^c,? ¬ §¦ô¨ª0 f0µ « ±~[ ¤ cd96¡ª f [0, l] c[¡e^ ¤,=¥
cVy
,~ ¤ cd96¡ª f0 [−l, 0] øIl( ,º ^¤ ¥¦cC
©fºc¤c c¤ d9 ¤ c= c±þe^ e6^d9t÷Ðq?g0úqQiø¯T e¥ dKe^cc=6pµ e6=ª0¢M7 ¯ c¤ d9 k1k = h1|1i
1/2
=
Z l −l
dx
1/2
=
√
2l;
Zl
D πnx πnx πmx E1/2 h πnx i1/2
cos2 dx = =
= cos cos
cos l l l l −l
M0N
1 =
h 1 Zl 2
1 + cos
−l
2πnx i1/2 √ dx = l; l
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ÷Ðq?g0ùggø
Zl
i1/2 D πnx πnx πmx E1/2 h
2 πnx sin = = dx
sin
= sin sin l l l l −l
h 1 Zl 2πnx i1/2 √ dx = l. 1 − cos = 2 l
ó*ª0 6cd¼÷Ðq?g0ùggøPQ= ·7^dc¤c c¤ d9 ¤ c= ª0¢ e^ e6^d#ª~¨ª0 f « ± −l
Ð÷ q?g0ùg=ø ¹ªe6¬ô^ ^¤ ¬ 4 ^f c=c¤,p ^^¤ ¤ ªCdPp, [−l, l] ¨ª0 f « .¾§T^¤ ^d
,¶ªQ
ce6[e¥ ¥
,fª0(x) ¢M ±hc¤c^c,? ¬ T±hpC e= n ÷Ðq?g0ùYg 3ø 1 πnx o n πnx o , sin , n = 1, ∞. , cos p6 cQ¡^ I¨ª0 f « 2 f (x) c[l ñ6cd#ªôpC e6ªôl pªQ
6V d96¬V0
X ÷Ðq?g0ùgnø πnx πnx a + + b sin a cos . f (x) = 2 l l C p!¨(c¤ d#ªQ !e¥ ¥
,ª6©©÷Ðq?g0 oø¥!f0c=c¤ c±f cñ^¨(¨( « ^ Q=d9^ ^ ö,
,© 6, ?c¦±tcQ
, beð~
, c»¨( ^c ¤ 6d#ªC c=±d cC
e^ e6^d pC eCt÷Ðq?g0ùYg 3 ø¥óò=Cd9»f0cñ^¨(¨( « ^a a b 1 √ , 2l
1 n πnx o √ cos , l l
1 n πnx o √ sin , l l
n = 1, ∞.
∞
0
n
n
n=1
n
n
n
n
n
Zl
1 an = l bn =
1 l
−l Zl
f (x) cos
πnx dx, n = 0, ∞; l
f (x) sin
πnx dx, l
÷Ðq?g0ùg=jø
n = 1, ∞;
f c=c¤ TIe¥ ¥
,ª0¢Myy÷Ðq?g0 oø§eª0 6cd ÷Ðq?g0ùg=gø¥ ¤0
¹cd©¤ # ®¯0
ª0¤ ÷Ьq?Tg0¨ùgTª0n ¤,øf pew¡[« f ^ c ñ^f¨(y(x)¨(
,« ~ f ¤^c0ñ^
p¨(V=d9¨(®¯À ª0« ¤ ÷Ð q?¬^g0(ùg=~j ø^^a,cpf C cTbñ^¨(=4 ¨(6f0 eðc=« *ñ^ ¨(^¤ ¨( c^« c ª0^c d9¤ =6=c=d9¤ y,=¢M ®¯^ª0eCeðf ¤ ¬ =d −l
n
l=π
n
∞ a0 X f (x) = + (an cos nx + bn sin nx); 2 n=1 Zπ 1 f (x) cos nx dx, n = 0, ∞; an = π
bn =
1 π
−π Zπ −π
f (x) sin nx dx,
n = 1, ∞;
÷Ðq?g0ùgZ:ø
OÊ Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í [0/
e¥ w¨ª0 f «
M0 M
f (x)
4
6,p cVf cñ^¨(¨( « ^ö÷Ðq?g0ùg=jøòd9cQ¡V c ¤ ¥
e6=p¬ Zl
2 an = l
ô¤ 0
!®¯ª0¤ ¬(Q= e^T=6eðwf0=f
f (x) cos
πnx dx, l
bn = 0,
÷Ðq?g0ùg=oø
0
÷Ðq?g0ùg=ø
∞
πnx a0 X an cos + . 2 l n=1
f (x) =
/
e¥ ~¡¨ª0 f «
f (x)
^ 6,p ,c 4
Zl
2 bn = l
an = 0,
f (x) sin
πnx dx l
÷Ðq?g0 iø
0
f (x) =
∞ X
bn sin
÷Ðq?g0 q?ø
πnx , l
e^cc=6e6^, c ½4,? c^ c[ª0 ¤ c==¢MeÅh~¨(c¤ d#ªC ´÷Ðq?g0ùgZ:ø¥ cC
♦e^ ve6c=^Qd! p6C eCT÷Ðq?0g0 ùYg !3 ø¥?^ ¤ 6C c ÉCcp ¨¬ª0, pf 7« ¨ ª0 ff (x) « d9d9cc?6¡ª ¶6¤,p6ò =¬¥p¤,p¬6eÅ !cQ¡ cy^,cQM
cpc ±w¬f c c¤c7 ccp^ c ,c? ±ô¬e^ c=e6^c^d9VpC÷Ð q?eCg0hùYg ¤ 3 0ø¥
!®¯c[ª0e^¤ =¬=Q#= f c7c=ec ¤ ccyd9´ ^dº¤,= e^cpe^ d9 c=c=¤ À d ÷|0 c=y^¡Q==d99f Ç
ª ^e^c¬!e6d9 ø c^¤,= d9eðw0 ¢4e6¤,=« ^±!¯^fc¤ c±!? ^^¤ f cA
,h¤,p6 cQ¡[^ ~ ¤ cpµ Ccp ¬ c^c¤ ¥¦d9C¤ c^ct^fc¤,©¤ ^=ª6eÅ*c¤c^c,? ¬ T±*pC e #¦ c=Q c^¤fp=cd ¤ ; 6¡[ p7 ± .,= ¤ d9^¤ ô0 ce^f ce6 xOy .d9cQ¡6~ò¬~¤,p6 cQ¡[~ı ^~ ~kcy
=ª0d l( ,~ı e^~e¥ ¥
c= yeA¦ cC
d9ce6y¤ 0
ô®¯ª0¤ ¬÷Ðq?g0ùgnø¥f0=fV ¢4c±[
¤ª0^c±y¨ª0 f0µ « c,? ¬ T±~¤0
, ¤ ¥
e6p=p , ¢My0
X ÷Ðq?g0 gø πnx πnx a + + b sin a cos f (x) = = S (x) + r (x), x ∈ [−l, l], 2 l l A
X ÷Ðq?g0 ø a πnx πnx ; S (x) = + + b sin a cos n=1
∞
0
n
n
k
k
n=1
k
0
k
2
rk (x) =
n
n=1
∞ X
an cos
l
n
l
πnx πnx . + bn sin l l
÷Ðq?g0 d3 ø
¯® ª0¤ ¬=¾§¥®¯ ª0 f « ¢ SS(x)(x) rcA(x)
¶,,ppCCTT==¢M¢Môþ,¤ = e6^c c d9c6± ¤,e6 ª0 d9^d9eCcf ± d ccpe6 p p cf0d9cdõ cd²¤00
cp ¾§ c=Ccd9¤,cpd®¯p ª0e^¤ ¬¬f7 k¤,µ¸p^6c[ cQ ¡c¤^0
f0¢[÷Шq?g0ª0ù gnf ø¥« ? c= d9f6(x) dT =c^^cM ^=ª0¢þ,=e6¬ p? ==¨ª0 f0µ « ¢ f (x) 0d9*c ¤ Å
¥ ,0 ¶,¯ ¤ c=dP¥¡ª f0 [−l, l] cA
¯f0=fy^^c7 ¤,=p,=e6¬ e^cpµ e6 cQe¥ cpc©±hc[e^w¤ e¯ c^c= 7c d9d*6 ¤ ^,¤ ^cCe^
f,c0d ¦©¨ª0 f « C ± cyc c^=e^7cc=[^c6ce^¤ c d9¤ cQ¥¡V
¥ c^ª,e6h¤,,=¶ e^^¬ ± ^e¥ ( c4c^¤,= ¬Cd9^ ^ xT4= ¤,2l=c=±¯,=e6ô÷Ðq?g0ùgnø;cQ
dh ^¤ cQ
cd [−l, l] n=k+1
k
k
k
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ^dc[¨ª0 f « ¢ f (x) ¤ cC,
A
cp , ¡V¬« ¥, [c(e^ ¢¼ e¥ c e¥ c=ª0¢¼ce^¬e( ^¤ cC
cd 2l e^cc= c=·7ŵ óP f¥ (x
,c ª6±72nl) Q=d96=f(x) ¬ 0,cycI^¨e¥nª0 V4 f ¨« ª0 f « f(x) Q ?¢M
=,p( eð,w Q=^d9¤ f cC
ª c^d»e^f ¤ dcd9 6¤ ¡cCª
cpf µ
p , [−l, f (−l) ¡^ l]^d f (x) .¶6=c ff0(l)?¦ x = (1 ± 2n)lf =(x)ªQ
6h ^cC
c=C,? c± C c^c~Qp¤ªQ
^ d94cQ¨¡Vª0 cMf « C¢¯6p¡Q?p
¬ = p cCª0
¢º^ ¤ ¥pc7, ¤ cC
cp ,¡[Q^ ¢þc, §¨ª0 f « ? ¢ fc(x), ?Q?
= =¾»ª0¢þ¤ 6,^ªC [−l, l] ? ¬ p p l[ p=f c^c¶ ¤ cQ
c? ,¡[^ hc f ^cC]−l,
c=Cl],Q ce6 [−l, e6]−l, p= cl[=eð!c f=d9 x = (1 ± 2n)l
¤, ¤ pC ¤ ^Te^^f d c¯ ^¤ ¤ cC
ccp^ ,c©¡¤ ^cQ
f¨ª0(x) f « = ªC
6fy (x)&^^m ¤ ,ñ6¤ªcCd#d9ªºcye¥ c¥
, ª¢46»c
d#cªôc==¤ 6C¬ fPª¶
c» ^ ¤ 2lcpµ Z Z Z Z ÷Ðq?g0 nø f (x)dx = f (x)dx = f (x)dx = f (x)dx. ¹ò(ce¥c ¬ ¥¤
¥cp
^¥6ò C^e^ ccc= c ^¤ cc=~·7 ^T ^ ^ɤ,e¥c ? ^ ^ 00w
¤ f Tcd9ñ^d¨(^ c¨(^ ¤, « pc C^c^cd!fVcò! ^÷Ф 6q?f g0=cCùg=6
(j ø¥ ^Te^^f0^cc±yd9¨6ª0¤ f « C e^ f c0^Vc(d9e^cQd9¡[T6e¥ µ b#=f d¼c¤,pCcdT^e¥ ¨ª0 f « Q?
= ,p, [−l, l] #¤,p6 =¥=6eðþ!¤0
®¯ª0¤ ¬=cVñ6c=¶¤0
!=ªQ
6yeð¦cC
¬eð!f (x) fhC ^¯ ^¤ cC
^e^f0cd#ªh ¤ cC
cp ,¡^ ¢ ,e^¢
^ ¤ e¥ cCc
=ª0 ¢ë c ^ e ¬ ? = f ¨ 0 ª
f
«
^e^f 0¦h ¤ c« ^e^e^c ,=e6fc(x)e6¤ ^,=ch¢Mc7f00p¦ CeÅTh=[6 ¤ eð©0 cQcp¡[ ^6 C T0¦;d ¤ c eC= M0S
1
∗
∗
∗
l
x0 +2l
l
∗
−l
x0 +2l
∗
x0
−l
x0
∗
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ãdÖ]×@çØrÝ4Ð 0RäáRâuÐÏráRÙ@4Ö 3bÖÊâdçY×t Ð Ã ådâ 8WÒ OM S ê G? !ê 2 ÐÏZÖVßÛ@Ö]×rãdÖ]×ÖVÛrâuÐàV×ÐÙäàWÖRÒßådÙrØràWÒâ@äOÐbÜ 8RÙ@ÖRä?á 8?u¥Ø ×ÐVÒáàWÖVÝOÓÖOÑÙ@ÖRä?á 8àRÞçräÙ@Ør?á 8 Û@ÖÜuÖOÑÒÙrØ@ÒáRârØ3]ÖVÙuÖRÓÒáWârØræ@Ò]ä]ÏdÖ43bÖòâYçd×Ð9à ådâ8WÒ£àÖ?0VÕÒßÔáWÒ]ÖVârØrØÔâdçd×uÖVàFà ådâ8WÒRê t ×@ÙuÐÏZÖ äådÕÒ]äáRàRåWÒá ×@ârå!3OÖV߸Û@ÖO×rãYÖO× ÚWÛrârØràWÖO×@çZÕØrßϸÛ@ÖVÙrçZáRØu"áRârØ3OÖVÙ@ÖRÓÒáRârØ@æ@Ò]äÏZÖ43OÖâdçd×Ðeà ådâ8WÒRÚ ÖRäÙ@ÖVàYÐÙrÙrÞßÙuÐ äàWÖVß@äáRàYÐbãgáRârØ 3OÖVÙ@ÖRÓÒáRârØ@æ@Ò]äÏrØrf ã ådÙrÏ rrØrßgäâuÐÝ]ÙrÞÓØÛ@ÒârØ@ÖO×ÐVÓØØgØZã äåZÓÓg@ Ø 0RÖÜuÒ]Ò±ä]ÖRÖVáRàWÒáWäáRàR!å u ÕØrßÍØ@äáWÖVârØrØÛ@ÖbçZàVÜuÒÙrØrçâdçd×uÖV¥ à à ådâ 8WÒRê 5ÊÝ]àWÒ]äáRÙ@ÖrÚ@æráWÖÛrâ@ÖRäáWÒßréØ@ÓØÛ@ÒârØ@Ö]×@Øræ@Ò]äÏrØuÓ Ø ådÙrÏ rrØrçrÓØÍçZàVÜ@+ç u áWä
h ç ådÙrÏ rrØrØ MOS ê àRB a cos(ωx + ϕ), a sin(ωx + ϕ). ÄAäÜ@ØÛ@Òâ@Ò]ÓÒÙrÙuÐç ?Ö 0RÖVÝ]ÙuÐ]æuÐVÒá àRâ@Ò]ÓçÚráW> Ö G]áRh Ø ådÙrÏ rrØrØÖVÛrØ@äÞàY Ð u á Ûrâ@ÖRäáWÒßréØ@ÒÛ@ÒârØ@Ö / ×@Øræ@Ò]äÏrØ@ÒÛ@ÖàRâ@Ò]ÓÒxÙrØÛrâ@4Ö r@Ò]ä]äÞ±ê sÊâ@4Ö r@Ò]ä]äÞ±ÚÖVÛrØ@äÞàYÐVÒ]ÓÞl Ò ådÙrÏ rrØrçrÓ¥ Ø MOS ê àRB ÚOÙuÐÝ]ÞàY Ð u áWä
ç±Ûrâ@ÖRäáRÞÓØØZÜ@ Ø 3Ðâ@ÓÖVÙrØ / æ@Ò]äÏrØ@ÓØ ÏZÖÜu(Ò 0WÐÙrØrçrÓظä¦ÐVÓÛZÜ@ØráRåb×uÖVß a ÚVæuÐVäáWÖVáWÖVß ω Ø ÙuÐ]æuÐbÜ 8RÙ@ÖV¥ ß ÐÝOÖVß ϕ ÚWÐ ØZ> ã 3]âu4Ð ØrÏrØ r Û @ â R Ö ä R á Þ Ó Ø @ 3 Ð @ â Ó V Ö r Ù r Ø r Ï V Ð Ó Ø ê Å ½ ÒÝ]åb Ü 8áVÐáäåZÓÓØrâ@ÖVàYÐÙrØrçg×@àRåY9 ã 3]Ðâ@ÓÖVÙrØræ@Ò]äÏrØZãÏdÖÜu(Ò 0WÐÙrØrß äådÕÒ]äáRàWÒÙrÙrÞÓ(?Ö 0VâuÐÝOÖRÓ ÝbÐàRØ@äØrá¸ÖVá¸ä]ÖRÖVáRÙ@ÖVéÒÙrØrçØZãÍÐVÓÛrÜ@ØráRåb×ØæuÐVäáWÖVáVê ½ÐVä]ä]ÓÖVáRârØ@ÓÚ@ÙuÐÛrârØ@ÓÒâ ÚuäåZÓÓå
ƱÐ]ærÙ@Ò]Ó ä]ÖäÜ@ådæuÐçÚrÏdÖ43×Ð
3
×uÒ
OM S ê àWG? êsÖØrÝ]àWÒ]äáRÙ@ÖVßØrݱáRârØ3OÖVÙ@ÖRÓÒáRârØrØ&±ÖVâ@ÓåbÜuÒÊÙuÐßZ×uÒ]Ó
y(x) = A1 sin ω1 x + A2 sin ω2 x. A1 = A 2
y(x) = A(x) sin
A(x) = 2 cos
MOS ê àRI
MOS ê àRL
ω1 + ω 2 x, 2
ω1 − ω 2 x. 2
Ç àVÜuÒÙrØ@ÒRÚbÖVÛrØ@äÞàYÐVÒ]ÓÖRÒl±ÖVâ@ÓåbÜuÖVß MOS ê àRI ÚOÓÖbÑÙ@ÖâuÐVä]ä]ÓAÐáRârØràYÐá?8 ÏZÐϱÏZÖÜuÒ(0WÐÙrØrç äæuÐVäáWÖ áWÖVß (ω +ω )/2 ØZÜ@Ø Û@ÒârØ@Ö]×uÖRÓ T = 4π/(ω +ω ) Ø Û@Òâ@Ò]ÓÒÙrÙ@ÖVß ÐVÓÛZÜ@ØráRåb×uÖVß A(x) ê E ÓÛZÜ@Ø // 1 2 áRåb×Ð A(x) ÓÒÙrçrÒáWäçÛ@ÒârØ@ÖO×@Øræ@Ò]ä1ÏrØäæuÐVäáW1ÖVáWÖVß 2 (ω − ω )/2 ØÛ@ÒârØ@ÖO×uÖRÓ T = 4π/(ω − ω ) áWÒ] Ó 0RÖÜ8RéØ@ÓÚ@æ@Ò]Ó0Ü@ØrÑÒ æuÐVäáWÖVáRÞ ω Ø ω êÄAäÜ@Ø 1 ÖVáRÙ@2ÖVéÒÙrØ@Ò T /T Ørârâu2 Ð rrØ@ÖVÙuÐbÜ8R1Ù@ÖrÚáW2Ö â@ÒÝ]åbÜ 8báRØrâr!å u ÕÒ]Ò×@àRØrÑÒÙrØ@ÒÙ@h Ò 0Våb×uÒáò1 Ù@ÒáW2ÖÜ 8RÏZS Ö 3Ðâ@ÓÖVÙ@Øræ@Ò]äÏrØ@1ÓÚ2Ù@Ö ×ÐÑÒÛ@ÒârØuÖ]×@Øræ@Ò / äÏrØ@Ó ê 2ÐÏrØ@H Ò ådÙrÏ rrØrS Ø 0Våb×@ådá£Ø@ä]äÜuÒ×uÖVàYÐÙrÞyàâuÐÝ×uÒÜuÒRÚ Û@ÖRäàRçZÕÒÙrÙ@ÖRÓØrÙráWÒv3]âuÐbÜ@å9à ådâ8WÒRê
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PÝpÙu¿Í
M0à âuÐ rrØ@ÖVÙuÐbÜ8RÙuÐVç×@â@Ö?048@Ú@áWÖâ@ÒÝ]åbÜ8báRØrârå!u ÕÒ]Ò×@àRØrÑÒÙrØ@Ò ÖVáRÙ@ÖRäØráWäçÍÏÍáVÐÏÍÙuÐ ÄAäÜ@Ø Ý]ÞàYÐVÒ]TÓ1Þ/TÓj20VÅ Ø@ÒÙrØrçrÓÚÏdÖVáWÖVârÞÒãYÖVâ@ÖVéÖØrÝ]àWÒ]äáRÙrÞ(àÍÐÏråYäáRØrÏdÒRê ƱÐÛrârØ@ÓÒâ Úà£âuÐb×@Ø@ÖVáWÒãZÙrØ // ÏdÒ±æuÐVäáWÖVáRÞÉÛ@Òâ@Ò×ÐàYÐVÒ]ÓÞã GÜuÒÏráRâ@ÖRÓAÐ 3]ÙrØráRÙ@ÞãàWÖÜ@Ù àWÖ¸ÓÙ@Ö43OÖ¸âuÐÝÛrâ@ÒàRÞéÐ u áæuÐVäáWÖVáRÞ Ý]àRådÏdÖVàRÞã ÏZÖÜuÒ(0WÐÙrØrß ê t ×@ÙuÐÏZÖâuÐÝ]Ù@ÖRäá?8 Û@ÖVÛÐb×ÐVÒáÍàÖ?0ÜÐVäá?8 ÐÏråWäáRØræ@Ò]äÏrØZãòÏdÖ ÜuÒ(0WÐÙrØrß ê}BáWÖ¸Û@ÖVÝ]àWÖÜ@çrÒáä±Û@ÖRÓÖVÕe8?u(àRÞä]ωÖV1ÏZÖV−æuÐVωä2áWÖVáRÙ@Þã (ω + ω )/2 GÜuÒÏráRâ@ÖRÓA Ð 3]ÙrØráWÙrÞã / ÏdÖÜuÒ(0WÐÙrØrßÛ@Òâ@Ò×ÐàYÐá?8 ÙrØrÝ]ÏdÖVæuÐVäáWÖVáRÙrÞ¦Ò (ω − ω )/2 ÐÏråYäáRØr1æ@Ò]äÏrØ@2 ÒÊÏZÖÜu(Ò 0WÐÙrØrçÚ@ÐVÓÛZÜ@ØráRå / 2 ê$ƱбârØ@äRêd0 ØrÝOÖ?0VâuÐÑÒy Ù 3]âu4Ð ØrÏ£äåZÓÓÞ ×@Þ¨ÏdÖVáWÖVârÞã£ÓÒÙrç+u áWä
ç¸Û@Ö ÝbÐÏZÖVÙrå 2 cos(ω −1 ω )/2 1
2
y(x) = sin 7x + sin 9x = 2 cos x sin 8x.
t æ@ÒàRØZ×@Ù@ÖrÚZæráWÖÛrârØ Ør Ý MOS ê àWG? ÓÞÛ@ÖÜ@ådæuÐVÒ]Ó63]Ðâ@ÓÖVÙrØræ@Ò]äÏdÖWÒÏdÖÜuÒ(0WÐÙrØ@Ò±ä æuÐVäáWÖ áWÖVß ω Øåb×@àWÖRÒÙrÙ@ÖVß ωVÐ 1Ó=ÛZÜ@ωØr2áRåb×uÖVß Ú@áVê ÒRê y(x) = 2 sin ω x ê 2
/
1
9 e=3
ê MOS ê àW?G Ú@ÏdÖ43×Ð è ÐÏ&Ø&âuÐÙ@Ò]ÒRÚ ×rÜ@çÙ@Ò]ä]ÖVØrÝOÓÒârØ@ÓÞã&æuÐVäáWÖVáVÚ ÏZÖ43
×Ð ω /ω AØr1âr6= ± âuÐ A rrØ@2ÖVÙuÐbÜ8RÙ@ÖrÚ ×@àRØrÑÒÙrØ@Ò Ù@ÒÛ@ÒârØ@Ö]×@ØrærÙ@Ö¸àÛrâ@ÖVáRØràWÖVÛ@ÖÜuÖbÑÙuÖRäá?8£äÜ@ådæuÐ uÚZÏZÖ43
×Ð ω 1= ω2 Ø y(x) = (A + A ) sin ω x ÖVÛrØ@äÞàYÐVÒá&3]Ðâ@ÓÖVÙrØræ@Ò]äÏZÖRÒ£ÏdÖÜuÒ(0WÐÙrØ@ÒRê t áWÓÒáRØ@ÓÚ æráWÖ 1Ò]äÜuØg2äÏZÜÐb×@ÞàYÐá?8q31Ðâ@ÓÖVÙr2 Øræ@Ò]äÏ@1Ø@Ò ÏdÖÜu(Ò 0WÐÙrØrç MOS ê 0 N y(x) = A1 sin ωx + A2 cos ωx, áWÖâ@ÒÝ]åbÜ 8áRØrâr!å u ÕÒ]ÒÊÏdÖÜu(Ò 0WÐÙrØ@ÒáWÖbÑ^ Ò 3Ðâ@ÓÖVÙrØræ@Ò]äÏZÖRÒRêr Î 0RÒ×@Ør?á 8Wäç£* à G]áWÖRÓÓÖOÑÙ@Ö äÜuÒ×@!å u / ÕØ@Ó?Ö 0VâuÐÝOÖRÓê sÊâ@Ò×uäáVÐàRØ@Ó MOS ê 0 M A1 = A cos ϕ, A2 = A sin ϕ, áW4Ö 3
×Ð MOS ê 0 S y(x) = A(sin ωx cos ϕ + cos ωx sin ϕ) = A sin(ωx + ϕ). È ÐàRØ@äØ@ÓÖRä?á 89 MOS ê 0 S ÖVÛrâ@Ò×uÒÜ@çrÒh á 3Ðâ@ÓÖVÙrØræ@Ò]äÏZÖRÒÏdÖÜu(Ò 0WÐÙrØrçòäÐVÓÛZÜ@ØráRåb×uÖVß A ÚuæuÐVäáWÖVáWÖVß È ÙuÐ]æ@ÒÙrØrç Í Ø ä u × R à Ø 3OÖRµ Ó ÐÝ]Þ ê Ø ×uÖRäáRÐáWÖVærÙ@Ö Ûrâ@ÖRäáWÖÙuÐbãdÖ]×@çZáWäç£Ø@ Ý MOS ê 0 M w ω ϕ A ϕ ½ÐVä]ä]ÓÖVáRârØuÓÔáWÒÛ@Òâ80RÖÜuÒ]Ò±Ö?0VÕØrßäÜ@ådæuÐßF
sÒâ@ÒßZ×uÒ]Ó¨áWÒÛ@Òâ8£Ï @r ÒÜ@ÞÒ±ærØ@äÜÐZê n1 n2 Å
q A = A21 + A22 ,
ϕ = arctg
A2 . A
MOS ê 0
à
äÜ@ådæuÐ uyä]ÖVØrÝOÓÒârØ@ÓÞã æuÐVäáWÖVáVêDs åYäá?8 ω = n ω Ø ω = n ω Ú3
×uÒ ÜuÖOÑÒÙrØ@ Ò 3Ðâ@ÓÖVÙrØræ@Ò]äÏ@ØZãÍÏdÖÜuÒ(0WÐÙrØrß6 MO1S ê 0 S 1àHG]áWÖRÓÔ2äÜ@ådæuÐV2 Ò MOS ê 0R0 y(x) = A1 sin(n1 ωx + ϕ1 ) + A2 sin(n2 ωx + ϕ2 ) ÛrârØràWÖ]×@Ørá¸Ï×@àRØrÑÒÙrØuÚ@ÏdÖVáWÖVâ@ÖRÒRÚ@Ù@ Ò 0Våb×@ådærØådÑ Ò 3]Ðâ@ÓÖVÙrØræ@Ò]äÏrØ@ÓÚ@çZàVÜ@çrÒáWä
çÚuÖ]×@ÙuÐÏZÖrÚrÛ@Ò / ârØ@Ö]×@Øræ@Ò]äÏrØ@Óä Û@ÒârØuÖ]×uÖRÓ T = 2π/ω ê . Òß@äáRàRØráWÒÜ 8RÙ@ÖrÚäåZÓÓA& Ð MOS ê 0R0 Ù@Ò ØrÝOÓÒÙrØráÍäàWÖRvÒ 3]Ö Ý]ÙuÐ]æ@ÒÙrØrçÍÛrârØÛrârØ0WÐàVÜuÒÙrØrØ Ï£Ü u0RÖRÓåÝ]ÙuÐOæ@ÒÙrØ u x àWÒÜ@ØrærØrÙrÞ T = 2π/ω ê 2ÐÏÏrÐÏÍærØ@äÜuÖ çZàVÜ@çrÒáWä
çÙuÐØ@ÓÒÙ 8RéØ@ÓòÛ@ÖÜuÖOÑØráWÒÜ 8RÙrÞ¦Ó ærØ@äÜuÖRÓÚW?Ö 0ÜÐb× Ð u ÕØ@q Ó G]áRØ@Ó äàWÖVß@äáRàWÖRÓÚRáWÖ 2π/ω ÖVÙ@ÖäÜ@ådÑØrá Û@ÒârØ@ÖO×uÖRÓ äåZÓÓAÐârÙ@4Ö 3OÖÏZÖÜu(Ò 0WÐÙrØrçê èÊÖÜvÒ 0WÐÙ@Ø@ÒRÚÖVÛ@Ø@äÞàYÐVÒ]ÓÖRÉÒ ±ÖVâ@ÓåbÜuÖVy ß MOS ê 0R0 ÚbÙuÐÝ]ÞàYÐVÒáWäçäÜuÖOÑÙrÞ Ó 3Ðâ@ÓÖVÙrØræ@Ò]äÏrØuÓ ÏdÖÜu(Ò0WÐÙrØ@Ò]ÓÚÐvÒ 3O> Ö 3]âu4Ð Ø@Ï Å äÜuÖbÑÙ@ÖVh ß 3]Ðâ@ÓÖVÙrØrÏZÖVß ê Ú
1 ×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í Ê-ÞÜuÖÝbÐVÓÒæ@ÒÙ@ÖrÚbæráWÖäåZÓÓÞÛrâ@ÖRäáRÞã^3]Ðâ@ÓÖVÙrØrÏäAä]ÖVØrÝOÓÒârØ@ÓÞÓØæuÐVäáWÖVáVÐVÓØ Ú ωn = nω ÐVÓÛZÜ@ØráRåb×ÐVÓØ A ØÙuÐOæuÐbÜ8RÙrÞÓØ ÐÝbÐVÓØ ϕ M 0R0
n
n
y(x) =
k X
An sin(nωx + ϕn )
∞ X
An sin(nωx + ϕn ),
MOS ê 0 7
n=0
ÛrârØ£àYÐâ8RØrâ@ÖVàYÐÙrØrØ A Ú ω = nω Ø ϕ Ó4Ö 3]ådá ÖVÛrØ@äÞàYÐ?á 8 äOÐVÓÞÒÊâuÐÝ]Ù@ÖR?Ö 0VâuÐÝ]ÙrÞÒ±äÜuÖOÑÙrÞÒ n Û@ÒârØ@Ö]×@Øræ@Ò]äÏrØuÒ±Ûrâ@Ö4r@Ò]nä]äÞ±nê 7äàRçZÝ]Ø^ ä G]áRØ@ÓÚuÒäáWÒ]äáRàWÒÙrÙuÖrÚ@àWÖVÝ]ÙrØ@ÏÍàWÖVÛrâ@ÖRä?w@Ù@ÒÜ8RÝ]çÜ@Ø ÚrÙuÐÛrâ@ÖVáRØrà@Ú@Üu0RÖRÒÛ@ÒârØ@Ö]×@Ø / æ@Ò]äÏdÖRÒ ×@àRØrÑÒÙrØ@ÒÛrâ@Ò×uäáVÐàRØr?á 8 ÏrÐÏÍäÜuÖOÑÙr!å uË3Ðâ@ÓÖVÙrØrÏråYÚZáVê ÒRêZäåZÓÓåÛrâ@ÖRäáRÞy ã 3Ðâ@ÓÖVÙrØrÏ ê 7ÓAÐáWÒ]ÓAÐáRØræ@Ò]äÏZÖRÓ&ÛZÜÐÙ@1 Ò G]áWÖ ÖVÝ]ÙuÐOæuÐVÒá Å àWÖVÝOÓÖOÑÙ@Ö±Ü@Øä]ÖRÖVáRàWÒáWäáRàR!å u ÕØ@ÓÛ@ÖO× 0RÖVâ@ÖRÓ A Ú Ø ϕ ÝbÐâuÐÙ@Ò]Ò±ÝbÐb×ÐÙrÙr!å u Û@ÒârØ@Ö]×@Øræ@Ò]äÏr!å uºådÙrÏ rrØ uÉÛrâ@Ò×uäáVÐàRØr?á 8à àRØZ×Ò±äåZÓÓnÞ ω = nω n Ûrâ@ÖRäáRÞx ã 3Ðâ@Ón ÖVÙrØrÏ ê t ÏrÐÝOÐbÜÖRä 8@Ú æráWÖÍÒ]äÜ@ØòÛrârØràVÜuÒæ 8 0RÒ]äÏdÖVÙuÒæ@Ù@ÖÓÙ@4Ö 3OÖÛrâ@ÖRäáRÞS ã 3Ðâ@ÓÖVÙrØrÏ Ú áVê ÒRê Ò]äÜ@ØòÏdÖ / Ù@ÒærÙr!å uÉäåZÓÓq å MOS ê 0 7 ÝbÐVÓÒÙrØr?á 8 âdçd×uÖRÓ y(x) =
MOS ê 0 B
n=0
áWÖrÚRàWÖRÖ?0VÕÒ13OÖVàWÖVâYçÚRÜu0Vå!ujådÙrÏrrØu y(x) äÛ@ÒârØ@ÖO×uÖRÓ T = 2π/ω ÚWÛrârØrÙuÐb×rÜuÒÑÐÕå!u"àWÒ]äv8 / ÓAЦéØrâuÖVÏdÖRÓåÏZÜÐVä]ä> å ådÙrÏ rrØrß ÚRÓÖOÑÙ@ÖâuÐÝÜuÖOÑØrá?8±ÙuÐ Ûrâ@ÖRäáWÒßréØ@Ò´3]Ðâ@ÓÖVÙrØrÏrØ ÚRáVê ÒVêRàÊâdçd× MOS ê 0 B ê ÄAäÜ@ØÍàWÖRäÛ@ÖÜ 8RÝOÖVàYÐV?á 8WäçÍáRârØ 3OÖVÙ@ÖRÓÒáRârØræ@ÒOäÏZÖVß ±ÖVâ@ÓåbÜuÖVß9 MOS ê 0 S ØÍÖ?0RÖVÝ]ÙuÐOærØrá?8 MOS ê 0 ?G An cos ϕn = an , An sin ϕn = bn , áWÖ âYçd× Ûrâ@ÖRäáRÞh ã 3]Ðâ@ÓÖVÙrØrµ Ï MOS ê 0 B ?Ö 0VâuÐáRØráWä
ç ¥ à 0RÖÜuÒ]Ò±ÛrârØràRÞærÙ@Þß áRârØ 3OÖVÙ@ÖRÓÒáRâ@Øræ@Ò]äÏ@Ørß âYçd× ∞ X MOS ê 0 I y(x) = a cos nωx + b sin nωx, n
n
n=0
ÏdÖVáWÖVârÞß ÛrârØ ω = π/l ä]ÖVàRÛuÐb×ÐVÒáÍäàRàWÒ×uÒÙrÙrÞÓâuÐÙ@Ò]ÒØrÝ×@ârå!3]ØZã ä]ÖRÖ?0VâuÐÑÒÙrØ@ß áRârØ3OÖVÙ@Ö / ÓÒáRârØræ@Ò]äÏrØ@Ó âdçd×uÖRÓ Ã ådâ8WÒ¥ MOS ê SR7 êuèÖ4G¾ØrrØ@ÒÙráRÞÌG]áWÖ43OÖ âdçd×ÐÓÖbÑÙ@Ö ÙuÐßráRØ ÓÒáWÖO×uÖRÓ BßZÜuÒâuÐ Å Ã ådâ8WÒRÚdä]ÓÞäܸÏZÖVáWÖVâ@Ö43OÖ±ÝbÐÏZÜu æuÐVÒáWäç à±Ù@ÒÛ@ÖRäâ@Ò×uäáRàWÒÙrÙuÖRÓØ@äÛ@ÖÜ8RÝOÖVàYÐÙrØrØy±ÖVâ Óåb Ü MOS ê MOS Å MOS ê MbG 0RÒÝ àWäçdÏZ4Ö 3OÖådÛ@ÖRÓØ@ÙuÐÙrØrçòÖ?0ÍÖVâráWÖ43OÖVÙuÐbÜ8RÙ@ÖRä]áRØFådÙrÏrrØrß ê}7 â@ÒÝ]åbÜ8 // áVÐáWÒRÚ@Ò]äáWÒ]äáRàWÒÙrÙ@ÖrÚuÛ@ÖÜ@ådærØ@ÓàRÞâuÐÑÒÙrØrçF MOS ê SWG? ê sÊâ@ÖRäáRÐh ç 3Ðâ@ÓÖVÙrØrÏrÐ ä à âuÐÝÜuÖbÑÒÙrØrf Ø MOS ê 0 B ØZÜ@F Ø MOS ê 0 I ÙuÐÝ]ÞàYÐVÒáWä
ç ÖRäÙ@ÖVà ÙrÞÓ ÏdÖÜu(Ò 0WÐÙrØ@Ò]ÓÚtØZÜ@Ø ÖRäÙ@ÖVàRnÙ@ÖV=& ß 13Ðâ@ÓÖVÙrØ@ÏdÖVß Ú ådÙrÏ rrØrØ f (x) ê ÍrÐâ@ÓÖVÙrØrÏrÐä n = 2 ÙuÐÝ]Þ // àYÐVÒáWä
ç Û@ÒâràRÞ9 Ó 3Ðâ@ÓÖVÙrØræ@Ò]äÏ@Ø@Ó&?Ö 0RÒâráWÖVÙ@ÖRÓÚdØZÜ@Ø Û@ÒâràWÖV¥ ß 3Ðâ@ÓÖVÙrØ@ÏdÖVß !ê ÍrÐâ@ÓÖVÙrØrÏrØ£ä n ≥ 3 ÙuÐÝ]ÞàY Ð u áWä
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@ ç ådÙrÏ rrØrç f (x) ê
¤d90c
ºe6¹®¯~¤ ª0¤ ¤ 0¥¬
=^d®¯l¯ª0¤
¤ ª0¬^ô = V¤ c¤ e6 §d9¦»^¤ ´¤ d9d9K^¤,¤,=e^e^0d9 c=¢4¤ e6 d¼¤ ¤cªe¥¢M 7V¦©ò ¤,e^p 6^ cQ¡[©^ ª e¥ c¨ ª0±t f eA« ¦ cQ
©0 µ
¤<~0
®¯'ª0 ¤ #¬==u D xEH x(#p6 cQ¡V¬~¨ª0 f « ¢
f (x) = x
Q?
= ª0¢ë,ôc=¤ ¥Cf0
[−π, π]
.
OÊ Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í @9 x (=±0
^dþf cñ^¨(¨( « ^ ®¯ª0¤ ¬= 1 a0 = π an =
1 π
Zπ
−π Zπ
M07
1 x2 π xdx = = 0, π 2 −π x cos nx dx.
¹¤ c ^^¤ ¤ª^d´ c,=e6Q dT& cp cQ¡V ,bcA
(sin nx)/n
−π
U = x dU = dx dV = cos nx dx V =
Zπ 1 x sin nx π sin nx dx = 0, an = − π n n −π
c~e¥ ¥
c=? cc?¡V0
p¬[[e^0 ,ªô ^ 6 ce6h¨ª0 f « −π
1 bn = π
Zπ
f (x) = x
l7? ^
x sin nx dx.
ó§ cw ¤ c ^^¤ ¤ ªCd c!,=e6? dT# cp cQ¡V bcA
V = (cos nx)/n
−π
U = x dU = dx dV = sin nx dx
Zπ cos nx 1 π cos nπ 2 1 x cos nx π + dx = − −2 = (−1)n+1 . bn = π n n π n n −π
°f c ,p¥ ¬ c[Q= ·7^d
x=2
−π
∞ X (−1)n+1 n=1
n
sin nx,
x ∈] − π, π[.
÷Ðq?g0 3ø
9 e=n Tce^±¬ ¤ 0cô
^®¯^côª0¤ e6¬ª0ôd9dP÷Ðq?yg0 =3ªQ
øT6e¯~ eA¤ ¦cCc
d96¡¬ª eÅf0f! ] −^¤ π,cC
π[ ¤,^=e^e6f0 c¤ d#cª!e6 ¤,¤ =cC
cp ,¬ô¡[,Å µ / e^e¥¢À w cpe¥ ,cª0= ª0^¢K ¢ f (x) ==f ,c X (−1) ÷Ðq?g0ùn=iø f (x) = 2 sin nx, ∗
∞
n+1
∗
n=1
x ∈ R,
n
x 6= (1 + 2k)π,
k = −∞, ∞.
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í =
,f c=c¤ c±V¤,p6 pcQ ,¡ ^¢M 7p ÷Ðeðq?ôg0 3 ^ø¤ e^ cC
¤,0Qµµ = x¶¨ª0 f « (¥
¤ e=c7ncp Cc¬f0¤,c[p¡[ ^¤ cþd9¨6¡ª0 ª f f « ] −f (x) π[ ¤0
®¯ª0¤ ¬÷Ðfq?g0(x) ^e^f d¼ ¤ cQ
cp ,¡^ ^d f (x) f c=c¤π,ª0¢õ ùn=iø( ¤ ¥
e6p=p , 6©,he^^± e¥Ç c=d9c6±ô cdTe^ô#Q[c c=e^f0y ¢4e6ª0 d9^ d9 ^dc ÷Ð^q?f g0 x3=ø¥(1+÷Ðq?2k)π k = −∞, ∞ g0¢Mùn=ieðø²4 ©,wªQ f ¬ c§ «, ?h¦t ¤ ¤ ¥
cd9e6=6=¡pª ,f0*9 c , ¤ p = p=f¡wc©§ e^¥¦þc f0§?¦ b#=xf c=w (1c4 +¥
2k)π ^ !e6ª0d9d ÷Ðq?g0 3ø[ ÷Ðq?g0ùn=iø[=ªC
6ºc=³. e^ ^ c f c=(x) 6
=^Ix ¤ ~§f,e6(x) ^ hªe¥ c ±~eA¦ cC
d9ce6w¤0
®¯ª0¤ ¬= <~ ' # =uD Kx JxI®¯ª0 f « ¢ ÷Ðq?g0ùn0q?ø 0, x ∈] − π, 0[; f (x) = 1, x ∈]0, π[, ¤,p6 cQ¡V¬[¤0
®¯ª0¤ ¬( ¤ cd96¡ª f ] − π, π[ @ 9 x.ó§cð =e^ cc ¤ ¥
¥ ^ ¢÷ qQg0ùZg :ø¥ ,=±0
^d M0B
1
∗
∗
,
n≥1
1 a0 = π
Zπ
1 f (x)dx = π
−π
1 an = π bn =
1 π
Zπ
−π Zπ
dx = 1,
0
1 f (x) cos nx dx = π
Zπ
cos nx dx =
1 π
Zπ
sin nx dx = −
0
f (x) sin nx dx =
−π
bcA
Zπ
0
1 1 = − (cos nπ − 1) = [1 − (−1)n ] = πn πn ( 0, n = 2k, 2 = , n = 2k − 1, k = 1, ∞. π(2k − 1)
π 1 sin nx = 0, πn 0 π 1 cos nx = πn 0
÷Ðq?g0ùngø
∞
1 X 2 f (x) = + sin(2k − 1)x. 2 k=1 π(2k − 1)
7® c¤ ¾§d#cpªCµ¸ h^¤ ÷Ðq?§g0¦;ùng^ø¥e¥ f0w=fôce^ª0 f0cpCTª0¢¼? c¤ e^ ¬V^c= òc·7d9=6
¤ c ªCe^e^ff=ª06¢À[e^
e6( ^d#ªt^¤ ÷Ðq? g0¤ ù^gY=3 =øò« ¤, = e^ e^dPp¤ 0µ p¬f0=fc¤c^c,? ¬= T±þ¨ª0 f « c,? ¬ T±ºpC e¶ ¤ ce6¤,= e6~f ?
¤,p c ¨qQg0ª0ù n0f q?« ø (± ñ6#ccdf cpñ^C¨( e^¨(= ?« =^=¦ =õ ®¯ª0^¤ ¤ ¬ ^ªa¢M, bp ¤ pc ,^f « ¢M eð¢ pf cc^^¤¤
¤,ªp^pd9=d9ò¦¢¯,¨ ª0[−π, π]
f
«
h ÷ , ¤ = f ¨ª0 f « »÷Ðq?g0ùn0q?ø&,c¤ ¤ ^c cd96¤ ,^e^f0c^cVpC eC~÷Ðq?g0ùg=ø¥ C dº ^^f cc=³ µ l¯ e^ e^^d9± d9e666e𤠺 ¥^c= ±»¬e6 c=ª c Ée^c¨e^ ª0~ ¥f « ¬e6 ª0chd9÷Ð d9q?¤©g0 ùn0d9÷Ðq?q?cg0øh± ùn÷|¤ gøI e=e¥É =j¥U¤Q=ø^d9,.§»¶¦;, Pôcpe^ ,cC ª0
cp ^ ,^¤ª0¤ ¡ ?^cC7¤
0cC¦
] −cosπ,nx0[ cpc=sin ?
2nx c=e^60 µ x ¨ =ª0 −π/2 ]0,¥π[¦4 ¡y cp ,ªµ ¥^ ¤ ¬ cCc
? ¦!¤ c pd9 c?±
=x¢M=¶e^π/2 c ð
! f p f
f
«
, h cos nx sin 2nx d9d96¤ C±c= ce66 ¬ c¶c ^f ÷| c¶ 7 ¤ d9§¦,ø þ0¦º ¤ e6ª e6 w,=¤ª ·? c»e^ d9d96¤ ¢ c= ce^¥x =¬ ct±π/2 ¤ d9§¦ x = ±π/2 n
n
Ê©=Ë+Ï4Ñ6Þ¬× ÍÐÏ0ÎÐÒß6Úq£ÉÑðÙÞßTÙÚpÛÑ¥Ø&
PÝpÙu¿Í
M0G
9 e=j ¹cñ6cd#ª¥p ¤ c^f « ¢h¤,p6 =6=^d9c±¨ª0 f « *÷Ðq?g0ùn0q?øM,ôñ6©c¤
cp ,¡V ò¬ ¤ ,e^= 6eð÷|y0¤,¦=c= e6ª e^e^cð pe^ ¯c¯e¥ª0TdP d9 7e¥ ÷Ðq?^g0 ùn gd&ø¥øPáI^ªC^ f ¢¯cI ª0 ^0dT
6 ¬ cC
0 ^¤ c7f c e6^pdt==6·77 M^¤,eðyp¯e6ª0cd9=d9³ µ ÷Ðq?g0ùngø¯e¥ =¥=^d9T sin(2k − 1)x ,f0p¡[
cd cp ,ª0 ^¤ cC
he^ d9d96¤ ,õc= ce^0µ ¥ ¬ cw ¤ d9§¦ x = ±π/2 e^cc=6e6C cº÷|¤ e=#júa^ø( f ¤ cd9Vc^c eV¤ ce6cd k e^c¤ c= 6e6=ª ¢M7 Í p ¤ c^f « ¢ 2/π(2k − 1) d9c c=c cIª0T=¢M? e6¤ ^d# e^¬7fªC ¢ k→∞
9 e= : ¾§cpµ¸=c¤ §¦;^e¥ w¨ª0 f « ¢÷Ðq?g0ùn0q?øò¤,=e^e^dPp¤ p¬yf0=fhe6ª0d9d#ª~¥=¤ d9c ^e^f 0¦ f cp ^= ± c0
c ,cce^ c,pô¥=¤ d9c f 1 2 + sin x 2 π
Q ?
=6tf0Q ^e6^ ,c! c¥
6 !¨ª0 f « ÷Ðq?g0ùn0q?ø~÷|¤ e=:0U¤Qø¥§6=¤ d9c f f ¼c¤ ÷ ¤ 0^ f0 µ c ^¤¤ª0¢Mc[ M^4ø4 cTe6¥·7
^0 ¦ =0c ¤0¤
0f
c=¶pe¯yª0^±yT¨(=c¢M¤ d#7ª[ d9^¤,©=¨(=d9 0f I¨=ªCª0
=f d9« 2/π(2k ÷Ðq?g0ùn0q?øÉ−÷|¤ 1) e=:0úa? RCø¥ Vf0c=c¤ §¦wc,V¤,=,~Wq eg0=ªQ
6 0, π c=³.¾w¹ e^cQ =^¥f0
c[^¢4 c=^76
e6 ª0=^d9?=d9 c=öC÷Ф,q?pg0ùnpg øTe^¬Éf(cc e^f ?c¦ −π, Td~¨(c¤ d#ªQ =d¶
,(¤0
M®¯ª0¤ ¬=CQ=d96 dT c cd9 d9ch÷Ðq?g0 3jø&»÷Ðq?g 3oø§d9C cQ¡V c ¤ ^e6w67(cC
ª¶¨(c¤ d#ª¶^^c[C= e^ ^e¥ ce^ cp ¬=Cc=p¬eÅ~¨(c¤ d#ªC =d9 ±0 ^¤,
,¶¤ ^c cd96¤ Ce^f 0¦h¨ª0 f « ± ÿA°± г% ! Ñy%&! Ê Ò#$Ê, =e^e^d9c=¤ d*¨ª0 f « ¢ f (x) e( ^¤ cQ
cd 2l ~e^=Q= T±~e( ^±h¤0
®¯ª0¤ ¬ X ÷ÐqQúq?ø πnx πnx a + + b sin , a cos f (x) = ∞
0
2
n
n=1
l
n
l
M0I
1
A
1 an = l
Zl
πnt f (t) cos dt, l
1 bn = l
¾§ce^ cp ¬=^ª^d9eðh¨(c¤ d#ªQ =d9 C 0± ^¤, 0
Zl
f (t) sin
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í
πnt dt. l
0
cos
÷ÐqQùgø
∞
÷ÐqQ ø
1 πnx = einπx/l + e−inπx/l , l 2 i πnx = e−inπx/l − einπx/l . sin l 2
óP ¥
cp¥ ¬= c
i 1 a0 X h 1 + (an − ibn )einπx/l + (an + ibn )e−inπx/l . f (x) = 2 2 2 n=1
¹ce¥ ¥
^Ie^cc= c=·7^ Id9c?¡V c7Q= eCp¬Vf0=f ∞ X
f (x) =
A
cn einπx/l ,
÷ÐqQ 3 ø
n=−∞
1 1 c0 = a0 , cn = (an − ibn ), 2 2 1 c−n = (an + ibn ), c−n = c∗n , 2
eª0 6cd¼÷Ðq?g0ùg=jø
÷ÐqQùnø
Ð÷ qQ jø ®¯ª0 f « c,? ¬ T±[¤ 0
h÷ÐqQ 3 øeòf cñ^¨(¨( « ^=pdPh÷ÐqQùnø0 ~÷ÐqQ jø,pCT=¢M ¤0
cd®¯ª0¤ ¬I¨ª0 f « f (x) [f0c=dP0 6f0e^ c±h¨(c¤ d9= <~ ' #=uH x>=,xI®¯ª0 f « ¢ f cd90 ^f e^ c±h¨(c¤ d9= f (x) = x V ¤ d9^¤,©q?g0 ô¤,p6 cQ¡V¬h~¤0
º®¯ª0¤ ¬V @9 x.ó§cð =e^ cc ¤ ¥
¥ ^ ¢÷ qC jø¥0
, n = 0 ,=±0
^ d 1 cn = 2l
Zl
f (x)e−iπnx/l dx.
−l
l( ,
1 c0 = 2π
Zπ
x2 π = 0. 4π −π
^^¤ ¤ c= I c,=e6Q deª0o6 cd¨(c=¤ d9ªC C 0± ^¤,
6 −π
n 6= 0
d dx =
Zπ π 1 1 x −inx −inx −inx xe dx = e dx = − e + 2π in in −π −π −π π 1 π i 1 1 = − [e−inπ + einπ ] + 2 e−inx = cos nπ + (e−inπ − einπ ) = 2π in n n 2πn2 −π i i i = (−1)n − 2 sin nπ = (−1)n . n πn n 1 cn = 2π
Zπ
Ê ©=Ë+Ï4Ñ6Þ¬× ÍÐÏ0ÎÐÒß6Úq£ÉÑðÙÞßTÙÚpÛÑ¥Ø&
PÝpÙu¿Í b#=f dºc¤,pCc=dÉ ¤0
®¯ª0¤ ¬(f cd90 ^f e^ c±h¨(c¤ d9( d9^6V0
∞ X (−1)n i −inx e , x= n n=−∞
M0L
÷ÐqQ;:ø
x ∈] − π, π[.
p6 c?¡^ h»¨(c¤ d9t÷ÐqQ;:ød9cQ¡V c©,=± § eA¦ cC
,*w¤ 6^ªQ ¬Qppct ¤ d9^¤, q?g0 0
=¢M7^^c n6=0
ô e^ cp ¬=^ª~e^cc= c=·7^ t÷ÐqQùnø¥ bcA
a0 = 0,
an = 0,
2 bn = (−1)n+1 , n
1 i 2 i cn = (an − ibn ) = − (−1)n+1 = (−1)n , 2 2 n n
1 c0 = a0 = 0, 2
c−n = c∗n ,
cÉe^cc=6e6=ª^MC,Q ^ dhf cñ^¨(¨( « ^c Q,=±0
^ §¦I ^ ce^¤ ¥
e6^ Édh§µ e¥( ^= ¤ ^c=d ^(^¤,f ?c d9c0! ÷Ð^qQf e^ jcø¥±~ ¨(c¤ d9¤0
®¯ª0¤ ¬÷ÐqQ; :ø9 ^^f c[ cp ,ª0 ¬¤,p6 cpµ ¡C ^ [¶¤ ^c cd96¤ ,^e^f0c±©¨(c¤ d9~÷Ðq?g0 3ø¥ ^e¥ ce^ cp ¬=Ccp¬eð¨(c¤ d#ªQ pdP ±0 ^¤,h÷ÐqQ ø¥0l¯^± e66 ¬ c y÷ÐqQ; :ø§ d9^^d x=i
X ∞
−∞ (−1)n −inx X (−1)n −inx e + e , n n n=−1
x ∈] − π, π[.
¹cd9^ tct=c¤ c±*e6ª0d9d9!C,=f² 0
^f eCe6ª0d9d9 ¤ c= ²,» ¤ c= c cp cQ¡V T± ÷ n → −nø¥, cp ,ª0 dþT¤,p¡[^ n=1
x=i
X ∞ n=1
= −2
∞ ∞ X (−1)n −inx (−1)n −inx X (−1)n inx =i e − e (e − einx ) = n n n n=1 n=1
∞ X (−1)n
e^c,?
=¢M76IeV÷Ðq?g0 3ø¥ <~ ' # =uH Ex D xI®¯ª0 f « ¢ n=1
n
sin nx = 2
∞ X (−1)n+1 n=1
f (x) =
n
sin nx,
x ∈] − π, π[,
0, −l < x < 0; 1, 0 < x < l
¤,p6 cQ¡V¬[¤0
®¯ª0¤ ¬(f cd90 ^f e^ c±h¨(c¤ d9= @ 9 x.ó§cð =e^ cw÷ÐqQ jø¥0
, n = 0 y
,
1 c0 = 2l
Zl
1 f (x) dx = 2l
1 cn = 2l
Zl
−l
f (x)e
dx =
Zl
e−inπx/l dx =
1 2
0
−l
n 6= 0
Zl
−inπx/l
1 dx = 2l
0
1 i =− (e−inπ − 1) = [(−1)n − 1] = 2iπn 2πn 0, n = 2k k = 1, ∞, i = k = −∞, ∞. − π(2k − 1)
1
MO7RN
bcA
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í
∞ i X 1 1 ei(2k−1)πx/l . f (x) = − 2 π k=−∞ 2k − 1
eA¦cQm(
,=~fI¤ 6^¤ ªC¥
¬?§p
,pª 7cVCd© ¤ ¤ d9 ^d9¤,^ô¤ =q?g0¤, 3 p06
=c?¢M¡7^ ^ ^cI÷ÐqQ oø ¤ a0 = 1,
an = 0, b2k = 0,
b2k−1 =
ô e^ cp ¬=^ª~e^cc= c=·7^ t÷ÐqQùnø¥ bcA
1 1 c 0 = a0 = , 2 2
2 , π(2k − 1)
l=π
÷ÐqQ oø d9cQ¡V c cp ,ª0 ¬
k = −∞, ∞,
1 c2k = (a2k − ib2k ) = 0, 2
i 1 , c2k−1 = (a2k−1 − ib2k−1 ) = − 2 π(2k − 1)
c−(2k−1) = c∗2k−1 ,
cÉe^cc=6e6=ª^MC,Q ^ dhf cñ^¨(¨( « ^c Q,=±0
^ §¦I ^ ce^¤ ¥
e6^ Édh§µ e¥ ^ ^d ^^¤,? c!÷ÐqQ jø¥ ¾f0Q ^e6 ¤ c^¤ f ,= ¤ c= cp ,ª0 dºf cd90 ^f e^ c±ô¨(c¤ d9 ÷ÐqQ oø#
^±0µ e6¥ ¬ª0¢ ¨(c¤ d#ª÷Ðq?g0ùngø¥ l( ,~ñ6c^c[Q= ·7^d¼÷ÐqQ oø§[0
∞ −∞ X 1 i X 1 1 i(2k−1)x i(2k−1)x e + e . f (x) = − 2 π k=1 2k − 1 2k − 1 k=0
÷ÐqQ ø
¹cd9^ C,=f~ 0
^f eCe6ª0d9d9 ¤ c= »÷ k → −løòc[=c¤ c±~e6ª0d9d9= cp ,ª0 d /÷
∞ ∞ X 1 i X 1 1 i(2k−1)x −i(2l+1)x f (x) = − . e − e 2 π k=1 2k − 1 2l + 1 l=0
e¥ ! ¤ ¬¶cy dP= =;c¶e^c¶eA
^cd² 0
^f eC¶e6ª0d9d9 ¤ c= !,V¥
«ª ø§=c¤ª0¢ e6ª0d9d#ª¶dPcQ¡V c7Q= eCp¬Vf0=f l =m−1 ∞ X l=0
∞ X 1 1 e−i(2l+1)x = e−i(2m−1)x , 2l + 1 2m − 1 m=1
c¤,p6 cQ¡^ V÷ÐqQ ø§ ¤ d96V0
∞ ∞ X 1 1 i X 1 i(2k−1)x −i(2l+1)x f (x) = − e − e = 2 π k=1 2k − 1 2l + 1 l=0
e^c,?
=¢M7 ±~eV÷Ðq?g0ùngø¥
∞ 1 2X 1 sin(2k − 1)x, = + 2 π k=1 2k − 1
Ê ¼ Ë}PÎÐÒ Ñ¥ØQÒß6Ú[× Í Þ.Þß4Ü¥ßÅÙÞÑ¥ÒÖÓ=ÍÅÎÐÏ Ñ6ÜCÑ(ß6Òß6×Ö@®6ß ÿ À " ! %&%&! t!%& ± ! µ!´d
!! µ!
OM 7ZM ed
¹^¤ ^±0
^d^ ^¤ ¬yf~cp ^
6=? ¬ cd#ª~^ª0 6 ¢À c¥
^ ~¤ ^c cd96¤ ¥µ e^¨f ª0c ^f ch« ¤ 0c
,w? ®¯¬ ª0c¤ ^¬c4!¤÷Ð0q?
g0ùg?n,ø¥?A ? e^^e¥dw e&¥
¤,c==e^=e^ d9 c=y¤ eA^¦ cC
d9^c^ce6tµ¸,ñ6=e6c^ cw ¤0c
±7e6ª0f0d9=d9f©t0 I¢4¤ c^0c µ ^c cd96¤ ^e^f0c^cV cp cdP kµ¸^c c¤0
f0h÷Ðq?g0 ø¥ k Qh ''¥ =rJ>x =,n x Ó:]6ZN,\0 D rº*b] Y 4@ ,CB D>(`=@Y6
Zl
f (t)
−l
sin[(2k + 1)π(t − x)/(2l)] dt, 2 sin[π(t − x)/(2l)]
÷Ðq?g0ùUg=jø¥ cp , ª0 +- d
xɹcC
e6== »²÷Ðq?g0 ø7 T±0
²f cñ^¨(¨( « ^c ®¯ª0¤ ¬ Zl
1 Sk (x) = 2l
f (t)dt +
k Z X 1 n=1
−l
l
l
−l
πnx πnt dt cos + f (t) cos l l
Zl 1 πnt πnx + f (t) sin . dt sin l l l
÷ÐqV3 ùgø
l( ,ôf c ^ c±he6ª0d9d9 c ^¤,=« ~e6ª0d9d9 ¤ c= ~~ ^^¤ ¤ c= hd9cQ¡V c[ c?µ 9d ^¬hd9^e6p=d9 / e¥ © ¤ ©ñ6cd ª0 ^e6¬wC^e6 c¤ ^c cd96¤ Ce^f0cVe^cc= cpµ ·7^ ÷ q3 ø cos(α ± β) = cos α cos β ∓ sin α sin β, c,=e6 ª0¢ e6ª0d9d#ª S (x) ÷ q3 ùgø§d9c?¡V c7Q= eCp¬V[0
−l
k
1 Sk (x) = l
Zl
−l
k πnt 1 X πnx πnt πnx cos dt f (t) = + cos + sin sin 2 n=1 l l l l
1 = l
Zl
k 1 X πn (t − x) dt. f (t) + cos 2 n=1 l
ó&ª0d9d#ª[I ce¥ ¥
^dt ^^¤,? T e¥ C d»e¥ ¥
,ª0¢M dtc¤,pCcdT0°c=C,Q d ~ce^ cp ¬=^ª^d9eðh¨(c¤ d#ªQ c± ±0 6¤,,cA
x)/l = α −l
k
1 1 X cos nα = + cos α + · · · + cos kα = + 2 n=1 2 = =
1 eiα + e−iα eikα + e−ikα + + ··· + = 2 2 2
1 −ikα e + e−i(k−1)α + · · · + e−iα + · · · + eikα . 2
÷ÐqV3 3 ø π(t −
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ¹ c7d ¨(ec¤ d#ôªQ C,e6=ª0d9d9^d9,p ¥^ ^^cd d96e¤ , ?^¦eCcQf
c ±~d ¤ c^¤ ^e^e^ ~e ^¤ Tdº0 ^ cd e ce¥ ¥
µ 1
MO7RS
−ikα
ikα
iα
k
1 eikα eiα − e−ikα 1 X cos nα = + . 2 n=1 2 eiα − 1
kd9 cQ¡V [ e¥ ¥ ¬VôC,=d9^,p¥ ¬V,
e−iα/2
cp ,ª0 d
k
1 ei(k+1/2)α − e−i(k+1/2)α 1 X sin(k + 1/2)α cos nα = + = . iα/2 −iα/2 2 n=1 2 e −e 2 sin α/2
l( ,¶ªQ
ce6[[ ^^¤? y÷ÐqV3 úq?ø§ ¤ c¥
^dºQ=d9^ª
÷ÐqV3 jø
ôce^ cp ¬=^ª6dPeðh¨(c¤ d#ªC c±»÷Ðq?g0 nø¥cA
t−x=τ
Zl−x
1 Sk = 2l =
f (τ + x)
−l−x Zl
1 2l
f (τ + x)
÷ÐqV3 ùnø
sin[(2k + 1)πτ /(2l)] dτ = sin[πτ /(2l)]
÷ÐqV3 ;:=ø
sin[(2k + 1)πτ /(2l)] dτ. sin[πτ /(2l)]
ó cd9c=7¬¢÷ÐqV3 ùnø kµ¸,=e6 ª0¢ e6ª0d9d#ª~¤0
y®¯ª0¤ ¬¶÷ÐqV3 3 øòd9cQ¡V c ¤ ¥
e6p=¬y 0
( ¹¤ ^^¤,¥?
^d*w÷ÐcCqV
3 úq?cyø¥, =b#e6=f0 =cdIcC,¤,?p C^c dº 7ª ^¤^¡[^¤,
?^ l¯
c¤ f0p¦ Q =¶ c÷ÐqV3 úq?øT ¤ f (t) ≡ 1 ¾*e^0♦ ,ª©÷ÐVq 3 3 ø§©÷ÐVq 3 ; :ø§ d9^^d −l
1 l
Zl
Zl k sin[(2k + 1)πτ /(2l)] 1 X 1 πn + (t − x) dt = dτ. cos 2 n=1 l 2l sin[πτ /(2l)]
?^^¤,? !7 ^c±~,=e6hñ6c^c¤,=^ e6¤,=^ô¥
« = ce^f cp ¬fª −l
1 l
−l
Zl X k
Zl Zl k πnt πnt πn 1X πnx πnx cos sin cos (t − x)dt = cos dt + sin dt = 0. l l l l l l n=1 n=1
óP ¥
cp¥ ¬= c −l
−l
1 2l
Zl
−l
sin[(2k + 1)πτ /(2l)] dτ = 1. sin[πτ /(2l)]
k Tp~ 6 ce6¬V cC
T^^¤? ¬= c±w¨ª0 f « ,?¦cC
d −l
Zl
sin[(2k + 1)πτ /(2l)] dτ = l. sin[πτ /(2l)]
÷ÐqV3 oø
?p=f µ¸,=e6 ª0¢²e6ª0d9d#ª7¤ 0
(®¯ª0¤ ¬Td9º ¤ Å
e6==0 0
ò ^^¤? 4l¯0µ ¤ 0¦ V÷ÐqV3 k;:ø¥e^cC
^¤¡p76^c e¥ c [f0Q ^e6I,=¤,=d96¤,0l7? ^(,pdº ¤ ¥
e6c 0
k
Ê ¼ Ë}PÎÐÒ Ñ¥ØQÒß6Ú[× Í Þ.Þß4Ü¥ßÅÙÞÑ¥ÒÖÓ=ÍÅÎÐÏ Ñ6ÜCÑ(ß6Òß6×Ö@®6ß MO7Qà l¯ e^ e¥¤ 0¥
¦ c4pQ=¬ºf0 ¢4c,¥p
6^ eŶ !¯ñ6ccdT^0cº cI
^ ,^y¤,? ^^c7 ¤ 4 d9kcQ¡→67∞òT°¬7e^ c¤ c^ ¥
c^e6¶¬º ¤ ¥
¥^ ^¬¤, ?T ± ^¤ ¥¦cC
k → ∞ cC
hC,=f cdþ ^^¤,? ce^f cp ¬fªô cC
T^^¤,? ¬ c¯T¤,p¡^
¤ =6 k → ∞ ¯ d9^6ô ¤ ¥
¥ i(=e6 T±§¦cC
7e^c=6
==·7^^ceð!Qp¤ªC
C Qh ''¥ :=ap]^> ºT Z=rDJ>[Ex N,D D,¶ Z[Y @N= @ Yº:y+R 'DY6'_ >QZ[LQ >A@ >'´`=@>/( 0Y¥B³ } L=Zt_ Y +G& Z[ ·?> £x ò
] N[O D,_ N,uN f (t) [a, b]
lim
Zb
÷ÐqV3 ø
f (t)eiλt dt = 0.
U +-
x ó§cA =e^ cV¨(c¤ d#ªC C 0± ^¤,
ce6=pc cV cf0pQp¬ ,c λ→∞
a
Ð÷ qV3 úqQiø =e^e^d9c=¤ de^,?,? ~ ^¤ T±» 6^¤? hñ6c±»¨(c=¤ d9ªC =¹¤ ¥
=¤ ¥ ¬ chQ=d9¥µ e6 dTc§= cw^e¥
,*!¨ ^ª0[ f f« ªe^ c¢ fcp(t)µ¸ ^ ¤ cC^
¤ É c±¬ ¤ cp c^hC¡[cC
^ e6c±f cd#ªþ;¤ ^côce^ =¤, = ¢ ¥
4 e6ª c7e6¥¬ µ ÷ÐqV3 úqQiø.ñ¥ ^d9^==¤ Tdc¤,pCcd!ò^f=6(T¨(c¤ d#ªQ f^(t)^¤ ¤,c= c,=e6Q d lim
λ→∞
Zb
f (t) sin λt dt = lim
λ→∞
Zb
f (t) cos λt dt = 0.
a
a
0
lim
λ→∞
Zb a
1 f (t) sin λt dt = lim f (a) cos λa − λ→∞ λ
−f (b) cos λb +
Zb
0
f (t) cos λt dt = 0.
¾§c=C¤,pp e^¬f¶Q?
= cd#ªyªe¥ c ¢ ^d9d9,= cd9 dþC6e6 ª0¢ c« ^ fª Z cos λt − cos λt 2 ÷ÐVq 3 úqq?ø sin λt dt = ≤ λ λ , ¤ cCcp ¬ cdþf c ^ cdþ ¤ cd96¡ª f c [tf0=,d9t ] l7? ^(¤,pCc¬^dº ¤ cd96¡ª cf [a, b] t ÷ÐVq 3 úq?gø a = t < t < · · · < t = b, d9^e6c cp cQ¡^ If0cpc¤ §¦hc ¤ ¥
Å d ¡[= = ^ e p c M ¢
c
^ ^
¤
¤ ª C P d , ^ e c ¥ e ^
c * d ^ e 9 d T ¥ e p ¥ e h ¨ 0 ª
f
« f (t) [a, b] / dPc? ¬i µ¸ccdº C¤ d9c^d9 6^¡ ª ¯f ,I^ñ6(c,d= d9¤ ^c d9¬=6·7¡^ª (Cf , ? ^ ^ ¤ I6cc=C ,bQ c A
dº ^¤ 6 m ,[6IdP=f e^0µ 4 ∆ ÷ÐVq 3 úqQø ∆ ≥ f (t) − m . ¾ 0
e^c c=6e6 eV¤,pC ^ ^d÷ÐVq 3 úq?gø eA¦cQ
T± ^^¤,? ÷ÐVq 3 úqQiø ¤ ¥
e6p= d a
t2
2
1
t1
1
2
i
1
2
n
i
i
i
Zb a
i
t n−1 Zi+1 X f (t) sin λt dt = f (t) sin λt dt = i=1 t i
t t n−1 Zi+1 n−1 Zi+1 X X = [f (t) − mi ] sin λt dt + mi sin λt dt, i=1 t i
i=1 t i
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í c=fªC
eª0 6cdc« ^ cf»÷ÐqV3 úqq?ø&»÷ÐqV3 úqQø&d9c?¡^dº cp ,ª0 ¬Ve6ª0d9dP=¤ ª0¢ c« ^ fª MO7 0
1
t Z b X Zti+1 n−1 Zi+1 X n−1 f (f ) sin λt dt ≤ mi sin λt dt ≤ |[f (t) − mi ] sin λt| dt + i=1 t i
a
≤
n−1 X
∆i
i=1
Zti+1
dt +
n−1 X i=1
i=1 t i
|mi |
Zti+1
n−1 n−1 X X |mi | . sin λt dt ≤ ∆i (ti+1 − ti ) + 2λ i=1 i=1
÷ÐqV3 úqV3 ø
b ^ ^¤ ¬[d9c?¡V c7ª c ¬¦ =¤,=f^¤h¤,pC ^ »÷ÐqV3 úq?gø¥ c¾» e^0 ,ª7 ^^¤ ,¤ª^d9ce6¨ª0 f « f (t) ¤,pC ^ Te^^ð
d9cQ¡V cT¤,p¬p=f ti
ti
n−1 X
ε ∆i (ti+1 − ti ) ≤ , 2
A
ε 4 ¤ cCc? ¬,ph cp cQ¡V¥ ¬,p~e^f cp ¬ª0^cC
cVdP? p~¥ ,¹¤ dPp ,c[=¤,= d9 6dP=¤ =,[e^cVc¢¼ ¤ cc ^¥¤
¥^
¬ ,cd9¯cQ¤,¡VpC c[ ^ T I¤,pcC
¬[ c=pC=,f ? dT cc c ¤ ¥
¥ 0 cy¥ m λ i=1
i
n−1
4X |mi |. λ≥ ε i=1
bcA
n−1 X |mi |
ô¤ 6^ªC ¬Q=p
i=1
2λ
≤
ε 2
ε ε Z b f (t) sin λt dt ≤ + = ε, 2 2
c=fªC
~e¥ ¥
,ª6÷ÐqV3 úqQiø¥ b^ ^¤ ¬ ?^e¥ 7¨ª0 f « =e^cp ¢M c4 ^^¤ ,¤ª^dP4, [a, b] 4 ^e^ce6=^ ,cd e^6d9eðT~e¥ c= 0f0c(
ce6=p0c cc6¤,f=(x) ¾² ¤ c=¬eÅ ¶ ¤ c¥dþ
e¥ ,cpª0 ,cQ¡= ^ ^dTe^^Ac7
ce^d9ccQc=¡V± c[c¤, pf0Cc±V p¬ ,,0 µ ¤ ce6cd9t6¡ce^ª cf c x±©&=c=ca ^f ,c=±»¢Mx7c= f~bª
= cd#ªºb c¤ A¥
h cp eA¦cQcC¡[
^ T[a, ±¢¯b]§ ¾§T^^¤,^¤ ? ¼^d ÷ÐqV
3 ,úqQiø¥c e^¤ c¥A
¥= e^^ 0c µ ¤,pC ^ ¢ö÷ÐVq 3 úq?gø¥ ¤ ¥
e6p= xd=a0
a
Zb
f (t) sin λt dt =
Zx1
f (t) sin λt dt +
Zb
f (t) sin λt dt.
÷ÐqV3 úq?nø
¹^¤ T±~ ^^¤,? h÷ÐqV3 úq?nø& 6Q= e^ d9cVc= λ d9^6Vc« ^ fª x1
a
a
Zx1 Zx1 f (t) sin λt dt ≤ |f (t)|, dt.
÷ÐqV3 úqQjø
&¾ c¤ c±[ ^^¤,? y, ¤ cd96¡ª f0 p , 6eðe^ce6^ Td©[ cª ¡§
cf0pQ=0µ cd#ªªC
cp 6c¤ 6º÷ q3 úqQiø¥6Q=[x ,e^b] d9ce6¬!c« ^ f ÷ÐqV3 úqQjøc= λ »=e^cp ¢M,p a
a
1
Ê ¼ Ë}PÎÐÒ Ñ¥ØQÒß6Ú[× Í Þ.Þß4Ü¥ßÅÙÞÑ¥ÒÖÓ=ÍÅÎÐÏ Ñ6ÜCÑ(ß6Òß6×Ö@®6ß ^^¤ ¤ª^d9ce6¬V¨ª0 f « f (t) c=Ccp , ¢MT¤,p¬ x p=f c=
MO7R7
1
Zx1 Zx1 f (t) sin λt dt ≤ |f (t)| dt ≤ ε,
Ae¥
,ª0,ε= cpª0 c?6¡[¡V
=¥^ d9¬eÅ,~?[e^e^ f0¤,cp= ¬w¥
ª0 ^ cQ
c e6chdP? ^p¤ tc^¥c e^ c c= , c=P·7óP^ ¥
hcwp÷ÐqV¥3 ú¬qQ iø¥c 9!ñ6cd ½4,? c^ c
cf0pCT=6eÅh=c¤ c(e^cc= c=·7^ = ¹¤ d9^ ^ h ^d9d9ÄVq 3 ùgf¨(c¤ d#ªQ =d£÷Ðq?g0ùgpjø[¼÷ÐqQùnø7
,f0cñ^¨(¨( « ^c= ®¯ª0¤ ¬( ¤ cC
Vf~e¥ ¥
,ª0¢M7^d9ª¶ª ^¤ ¡[
^ ¢¯ %&N,D, Z[ Y @'N= @ =r
Y J(>x >Q=F¶ OID,'_ N,N,N! ·?]6¥x Z4Ö¯@Y> Ì O,O9Z[ N ] N,!N_¶Y6D,D ZW× E ºÔ`=^@ @ N Y©÷ÐqQùnø!GPN¼Z3 ÷ÐYCq?3 g0ùg=jøw:=ap]^> ºTZD> n→∞ a
lim
n→∞
Zl
f (x)e
iπnx/l
a
dx = lim
n→∞
Zl
πnx dx = lim f (x) cos n→∞ l
Zl
f (x) sin
πnx dx = 0. l
°e^ c,pV ^d9dP¥=¤ d9c ^e^f0c^cV=,? Q7 c=Ccp , 6[p=f¡e^¨(c¤ d#ªQ ¤ cp¬ e¥ ¥
,ª0¢M7^Iª ^¤¡[
^ = %&Z[ >Q\0 _ Y ' =rJEx Dɶ ) $)*+ ;+0 $·?x9RA>Q?]6Z!@u,QF >Q`=@Y6
x °c=C,Q ÷ÐVq 3 úWq :ø f (x + τ ) = F (x, τ ), 2l sin[πτ /(2l)] ÷ÐVq 3 úqQoø (2k + 1)π =λ 2l ôT¤,=[ ¤ cCcp ¬ c e¥ c 0 < δ < l ,¤,pCc¬^dº ^^¤,? þ÷ÐVq 3 ; :ø§,7¤ −l
−l
−l
Sk (x) =
Zl
F (x, τ ) sin xτ dτ =
−l
+
Zδ
Z−δ
F (x, τ ) sin λτ dτ +
−l
F (x, τ ) sin λτ dτ +
Zl
F (x, τ ) sin λτ dτ.
÷ÐqV3 úqQø
¹ ¤ p=f cdë¤,pC ^ *c f0 τ = 0 e^cQ
^¤¡Veð*cp ¬f ctc? =e6² ^^¤ ¤,cpµ == e^cp !¢M= ccI¤ c ^c~^ ^¤ ^¤6ª¤^?dP ¯(¶ ¤ ñ6ccôd96C¡,Qª f0??.¦ c .e^cA =e^ c»÷ÐqV(3 f úqWc=:ø¥c;¤ ¨§ª0¦ f « F (x, τ ) [−l, −δ] [δ, l] sin(πτ /2l) c¤,p=6eðh[ªC ¬ cA
/ e¥ »÷ÐVq 3 úqQø ^¤ ^±f© ¤ ¥
¥ ,ª» ¤ k → ∞ cwhe^0 ,ª*÷ÐVq 3 úqQoø λ → ∞ δ
−δ
lim Sk (x) = lim
k→∞
λ→∞
Z−δ
−l
F (x, τ ) sin λτ dτ +
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í Z Z ÷ÐVq 3 ùg=iø + lim F (x, τ ) sin λτ dτ + lim F (x, τ ) sin λτ dτ. 1
MO7RB
δ
l
λ→∞
λ→∞
¶ó§cAc ð=
e^ cyce^ c c±ô ^d9d9=; ^¤ T±ww ce¥ ¥
±! ¤ ¥
¥ ¼!÷ÐqV3 ùg=iøT¤,= ªQ ¢¯ −δ
δ
lim Sk (x) = lim
k→∞
λ→∞
Zδ
F (x, τ ) sin λτ dτ =
−δ
1 k→∞ 2l
= lim
Zδ
f (x + τ )
sin[(2k + 1)πτ /2l] dτ. sin(πτ /2l)
÷ÐqV3 ùg0q?ø
b#=f dºc¤,pCcdTe6ª 7^e^c= I ¤ ¥
¥
, kµ¸±ô,=e6 c±he6ª0d9d9 ¤0
V®¯ªµ ¤ ¬cV yS ñ6(x) c=¶~ ^^c[^^C¤,,?Q © ^p ¦ cQ
,¯ cp c·7e6¬y¬=C¢ ,Qc ^¤ ¥
!¥ ,¨ ª0¢M f eÅ« h ¤ f¥
(x)¥ cc=d ^,=^¢M^¤,7? !÷ÐCqVd93 ù^g0 q?¥ø¥µ ¶¢*
=c¤ f06pª0Cd9T^==6(Ve^ ¤,¤ =cd9¥6
¡ ª cf e6Éc=¬ x−δ
c x+δ =c(e^0 ,ª7 ¤ cCcp ¬ c±[dP? ce6 ^ c
¤ ^ 9 d δ ?eA¦cQ
,4 ¤ « ,( cf0? Q=« 0d9cQ¡V cª ^¤¡[
p¬ 0c¯^e¥
M¨ª0 f0µ ♦ « f (x) f (x) e^c,?
=¢MI( ^f c=c¤ c± δµ¸cf ¤ ^e6 ce6Vc f x =cIe^cc=6e6=ªµ ¢Mf c7c 07 ñ6 cd²c¨ª0 ¤,f =« eð¦cC
,d²¤eÅ0
®¯ª0c¤ c¬7eA¥¦
,cCª
,ôe^6eð ?w y¤ ñ6 c^±hdf~c cCf0
7 ce^±~cô^¤·7c±~^ ¡ c¶(e6cCª0
d9 d9,?=µ « ° e^^c^c ô c¤ e¥0
¬cñ6~,~c®¯^ª0ce^¤ ^ª ¬d {a^¤¤ ¡[c,d9
b6^¡ }ª ~ f e^{ace6,cb}6 d97cc 6¤ ôª !¥7
c¥ ,f0cp dTQ^pd9§¬¦eðc7»
e^cc7¥e^^
¤^^·7 f ^^ ¤,d [c!¨f ¤,cª0pñ^ 6¨(f ¨(« 00± µµ f T(x) [−l, l] d9 f (x) ÿ \ & 'Â#$Ê,} $¦ lØY@ !- * ($Êedz m(=f[d9ª ¡òªe6== c0 ¨ª0 f « =Q?
= ,p, ¤ cd96¡ª f [−l, l] [ ¥µ ¤ cC
^e^f ! ¤ cQ
cp ,¡^ ,ph,Ve^¢ e¥f (x) c=ª0¢Àce^¬ d9c?¡6yò¬y¤,p6 cQ¡^,VV¤ 0
®c¯=ª0ª¤ e¥¬ cp÷Ð q?^g0,ùgIne6ø¥ ª m(7=^e6fVe¥ c¥
,=ª 6 [^dt ¤ ¤ ¥
¥ « y ,÷ (q3 ùg0cq?f0ø#? (e^Qf0=cp« ¬( ª0^eAcC¦
cC
c¯ d9dPc?e6 c¬±^c^f c7¤ I^e6c cf e6 x ñ6c±c f ¹cñ6cd#ª¯ªe¥ ceð¦cC
d9ce6¤ 0
(®¯ª0¤ ¬=cc=7ò^cc¤ =c ¤ ¥
¥ ,0µ ¢M ^ceðC©¦cCª
e¥ d9c§¦ d9!»
eAc=¦e6cQ=
p cd9 c e6§t¦ ª¤ e¥ ¥
c¥ t±÷Ð^Vq ^3 côùg0eðq?¦ø¥cC
¹ d9ce^cf e6cp ¬
fcôªe^c0=¦ pc©¤¨( c7¤ d#ªe6ªCp = ¤ ccp f¥ µ ,d9 ¤,=e^e^d9c=¤ d
¯ ,[,=e6 §¦ôªe¥ c ± ,[c==7^dþe¥ ,ª0,=( ò^f0=¢M7¦
®¯¤ª0ª0¤ M¬=É°M
¤
ª0 6cIV(cCp¦=f p0¦ôTªe¥= ¢Mc7 0±h¦yeA¤,¦pcQ6
d9 c e6T4hf0ñ6 =ce^^e^cV*¤0¨
ª0¯
f « =6 V± 0e¥ ¤,¥p
,6 ª0=¢M¥=^pd9 T4I¤ 0
%& ' =uP >x =,x ò
] NhO D,_ N,uN f (x) `Y¸@N>QQ\0DR>
:= (Y¥³B Z _ NwY _ [−l, l]GZ[>ÉY^Y.@ ,y< Ô^ @ Y&] R>QQF!_ Ì Z[Z[>Q>Q\0F_ YTO @ W: D,Å_ @N,EPN,RpN[:yRM`Z[Y¸@>QR=\0W> _¥R: >[@ÛY >QÜ Ú ?]6ZN [f (x + 0) + f (x − 0)]/2 U -+
x(l( , c=7 ce6 6 cQ¡[^ =ªC
^d eA¦ cC
¬ þ ¤ ¥
¥ e6÷Ъ0Vq 3 d9ùd9g0q?(ø¥¤ 0¦
c=?þ®¯ñ6ª0¤ c©¬ye^c÷Ðf q?¤,g0pùgn0ø& ¨c©ª0 ëf «
c f0pfQ(x)p¥[ ¬e60
=(c § c^^¤,¤,?? [dPl¯eðþ ¤ f 0¦ k µ¸y±º÷Ð,Vq =3e6; :ø c± −δ
k
1
2
n
1
n 1
n
n 2
2
1 Sk (x) = 2l
Zl
−l
f (x + τ )
sin[(2k + 1)πτ /2l] dτ. sin(πτ /2l)
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PÝpÙu¿Í&ÏÝ ÎÅÑpÓCÒ Ñ Ý Ü ×ßÅÛ0ÏÖZq£PÝÒÏ[ÖÕ =e^e^d9c=¤ d ¤ ¥
¥ !ñ6c^cV¤,=^ e6 ¤ 1 lim Sk (x) = lim k→∞ k→∞ 2l
Zl
k→∞
f (x + τ )
MO7WG
sin[(2k + 1)πτ /2l] dτ. sin(πτ /2l)
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^dþ[−l,e^ l]cd9¤,pcC6cp¬¥ ^dþ¬,ª0¯¢
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[0, l]
÷Ðq?n0úqQø
[0, l]
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p ,
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p c ¸ µ
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¤ ^
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,³. p , ^d9§¦»ft ¤ cCcC
c± f (x)ϕ(τ)h d9^ τ c!→¶+0e6ª 7^e6c= ¤,=ce6c¤ c 6±w ¤ c^cQ
c± f (x + 0) ϕ(τ ) =
π f (x + τ ) − f (x + 0) . 2l sin(πτ /2l)
0
0
π f (x + τ ) − f (x + 0) = τ →+0 2l sin(πτ /2l) π f (x + τ ) − f (x + 0) = f 0 (x + 0). = lim τ →+0 2l τ lim
=e^e^d9c=¤ d6 ^¤ ¬y ^^¤,? 1 2l
Zl 0
πτ i ϕ(τ ) sin (2k + 1) dτ. 2l
Zl
h πτ i ϕ(τ ) sin (2k + 1) dτ = 0. 2l
h
÷Ðq?n0 ø
^e¹f cce^^f cVcp =,¬f? ª¶¨Qª0, f ¤ « d9 ^ϕ(τ ^ ) IªQ
^c(?f~ 6 c^¤ ^ ¤,6?V ,ªª»e¥ ÷Ðq?cn0 øPdþ
=c6e^ c c±y ^d9d9 ¥=¤ d9c ¥µ 1 lim k→∞ 2l
°4e^¢§
ce^ cp ¬pCc==·7 e^¬w÷Ðq?n0ùgø¥ cp ,ª0 dþ¤,=^ e6c 0
1 lim k→∞ 2l
Zl
f (x + τ )
Zl
f (x + 0)
sin[(2k + 1)πτ /2l] dτ − sin(πτ /2l)
0
1 − lim k→∞ 2l
0 1 lim k→∞ 2l
sin[(2k + 1)πτ /2l] dτ = 0 sin(πτ /2l)
0
Zl 0
f (x + 0) sin[(2k + 1)πτ /2l] dτ = f (x + τ ) sin(πτ /2l) 2l
Zl 0
sin[(2k + 1)πτ /2l] dτ, sin(πτ /2l)
1
MO7RI
f c=c¤ c(eª0 6cd¼÷ÐqV3 oø§d9cQ¡V cQ= eCp¬ 1 lim k→∞ 2l
Zl
f (x + τ )
f (x + 0) sin[(2k + 1)πτ /2l] dτ = . sin(πτ /2l) 2
½4,? c^ c,?¦cQ
d
,ôc=¤ 6Cf 0
1 lim k→∞ 2l
Z0
f (x + τ )
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PÝpÙu¿Í
[−l, 0]
÷Ðq?n0 3 ø
f (x − 0) sin[(2k + 1)πτ /2l] dτ = . sin(πτ /2l) 2
÷Ðq?n0ùnø
ó&ª0d9d9 ¤ c= V÷ qQn0 3 ø§»÷Ðq?n0ùnøP
=6 −l
1 lim Sk (x) = lim k→∞ k→∞ 2l
Zl
f (x + τ )
sin[(2k + 1)πτ /2l] dτ = sin(πτ /2l)
−l
f (x + 0) + f (x − 0) . = 2
÷Ðq?n0 jø
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,T ^c¤e6¡[
=f6V(xe^ +¤,0)=¥=
f (xc−e6¬0)=^c¤ f^(x)d9 c [f (x + 0) +áIf^(x ^f c−ª0)]/2 e6p= c=f¬ (x)c(¨ª0 f « f (x) ¤ d9^¤ q?g0 4ªC
cp 6c¤ 6(ªe¥ c d ^cc ¤ f ^Id9 ²^ q?¤ n0^ú¤ q&T¯ cpc ,e6ª0 .^ T[±cp =f0d!?¤¦~0¤,
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e6==p ,
=66ñ6C=,ª¯Q¨ ^ª0 f « ¢þ4fp¡[
c± x = ±nπ /
f (x + 0) + f (x − 0) −π + π = = 0. 2 2
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c¨c ª0¤ ¥f
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f»6t¨ª0 e^¢§f «
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¤ ?
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p=^6dPpc=¨6=ª04 f ,« Mtc ¤ Vce§ªC
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pd9¡6I¬ ^ ¤ ¤ c^¤ CTcC
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« t dT¨ ª0 f « Q?
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1
0,
x=0
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PÝpÙu¿Í f2 (x) =
MO7RL
÷Ðq?n0 oø
x cos(l/x), x 6= 0; 0, x = 0,
ôf f0c=l¯^c± ¤ e6Tdº^6c ¤ ¬^ dPc ~ò q?^n0e¥úq( cQ ¡V( c»¤ ªd9e6p^= cdP ¬ òct¨ª0 f « ¤ T c±t,wc=¤ 6Cf [−l, l] ey ^¤ ? =d9d9c c=c ce6 ]f−(x)l, 0[,? ,]0,,¥l[eÅ* ¤ ^ ¤ ^¥d µ I,?ª0 f0 p Q =ô w§¦yc f0 ^¤ ? ?¦yc,( p , 6eðy ^ ¤ ^¤ T c¯
¨(¨(^¤ ^ « ¤ªCd9c± ,°M
,=f c ^c^¤,= ^ c±º ¤ cCcQ
c± ô c=Ccp , 6c= ^e6þ^ #,cf
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?^
¤0f ¡[0p¦þ7¨ ±ºª0 f c« f± ª ¦ ic©f0=9eC =^6c=eðCd9¨cQ¡Vª0 f c« ¤, p Cf(x) © ¬ e¥ c[ ^¤ ? cd9c c=c ce6 [e^0 ,xª¶= ^^0c^p=f¡[ ¥ ¬=^~c= ^e6hf~f0 =e^e6ª fªe^cóë ª0cp µ¸6A c?dë
f 0ò¦~·7¨^ª0e^f0 pf Q« = ± c^c 6¤ ^±0
^dëf ¤ C,=f0=d´eA¦ cC
d9ce6²¤0
t®¯ª0¤ ¬ ¨ª0 f « f (x) [cp ^·7 ¤ cf 0¦; ^dþ^c¤ ^d9Vq?n0úq ¤ ¥
c? c?¡^ 0¦; ÿ ~± xyµ !Ñß# * %& + !Ò#$Ê, b^ ^¤ ¬d9^e6ce^ cd9c¥p¥ ¬ c±h¨ª0 f « t÷Ðq?n0ùgø§Å
^dþ¨ª0 f « ¢ ÷ÐqQjúq?ø ψ(τ ) = f (x + τ ) + f (x − τ ) − [f (x + 0) + f (x − 0)], ªC
cp 6c¤ ¢M¯ª¢¼ªe¥ c ¢ 1
2
lim ψ(τ ) = 0.
%&`Y¸ @N >Q =`=@¶ U>Q
Y^]N!`=@N!DY6_ >QZ[>A@>
f ∗ (x) = 0<δ
f (x − 0) + f (x + 0) , 2 Y V@P
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÷ÐqQjùgø
|ψ(τ )| dτ τ
b< OD,_N,N,N G§>C`=@Y6
sin[(2k + 1)πτ /2l] sin(πτ /2l)
^^¤,? º÷ÐqV3 ;:ø§d9cQ¡V c[ ¤ ¥
e6p=¬y[0
1 Sk (x) = 2l
Zl 0
[f (x + τ ) + f (x − τ )]
sin[(2k + 1)πτ /2l] dτ. sin(πτ /2l)
÷ÐqQj ø
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ^?ªC~ ¬?÷ÐpqQpj ¯øÉ cpT ,ª0 ^ dÀd ÷ÐqV3 oø¥ cQ
¥ ^ T±, l wª0d9 cQ¡^ T±,y e¥ c f (x) ¾ ¤ ¥µ 1
MOBRN
∗
Zl sin[(2k + 1)πτ /2l] 1 [f (x + τ ) + f (x − τ )] Sk (x)−f ∗ (x) = dτ − 2l sin(πτ /2l) 0
1 − l
Zl
f ∗ (x)
sin[(2k + 1)πτ /2l] dτ = sin(πτ /2l)
0
=
Zl
1 {[f (x + τ ) + f (x − τ )] − [f (x + 0) + f (x − 0)]}× 2l 0
1 sin[(2k + 1)πτ /2l] dτ = × sin(πτ /2l) 2l
Zl
ψ(τ )
sin[(2k + 1)πτ /2l] dτ. sin(πτ /2l)
÷ÐqQj 3ø
e6ª 7ó§^e6cA c=e^ = c ¯¤ « ^ ^¤,ª? c f? Q=« ?òe6ª 7^e6cp ^^¤,? ÷ÐqQjùgø e¥ ¥
,ª6 Z ÷ÐqQjùn=ø 1 ψ(τ ) dτ. 2l τ pC ce6¬!÷ÐqQj 3 ø& ^¤ ^ ·^d0
0
l
0
1 Sk (x) − f ∗ (x) = 2l
ôT e¥ d^( ¤ ¥
¥ ! ¤
Zl 0
k→∞
ψ(τ ) τ πτ sin(2k + 1) dτ τ sin(πτ /2l) 2l
lim [Sk (x) − f ∗ (x)] =
k→∞
1 k→∞ 2l
= lim
Zl
ψ(τ ) τ πτ sin(2k + 1) dτ. τ sin(πτ /2l) 2l
¾*e^0 ,ª~=e^cp ¢M c±~eð¦cC
d9ce6w ^^¤,? c=¨ª0 f « 0
ψ(τ ) τ τ sin(πτ /2l)
¤ ¥
¥ ![ ¤,=c±ô,=e6t÷ÐqQj jø§¤,=^ôªQ ¢¯ e6=? cò¬ 0
lim [Sk (x) − f ∗ (x)] = 0
k→∞
lim Sk (x) = f ∗ (x) =
k→∞ f (x) = f (x + 0) + f (x − 0) 2
f (x + 0) + f (x − 0) = 2
[c f0?¦~ ^ ¤ 6¤ T ce6 , c f0?¦~¤,pC¤ T ,
÷ÐqQj jø
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PÝpÙu¿Í MOBZM cô¤ ^c=? ce^¬
cfpQp¬ cp °4cQ¡Vd96¬[ e6dTª #7^e6ch
c ,»= e6 ª I7 ^ce6¤ c=Cc ¬p ^ ^¤,?^ ^c¤,V? ~0
l¯ ÷ÐqQjùgø4
ce6ppc c ¤ ¥
µ Zδ
|f (x ± τ ) − f (x ± 0)| dτ, τ
δ > 0.
0Á°M¥
¬?
d~^¤,w9÷|ªe^e¥dT ,c ¤ ±I0 0cQ¦¡e6^ª 7 ^ e6=c^e6=c e6Id9^cQ ¡ É6ɯ ò¬ ^Q^T¤,?, =e6¢! cû g?e6üýø .Cªe¥ c #áI ·7 «, 4 f (x ± τ ) − f (x ± 0) ÷ÐqQj; :ø , 0 < α ≤ 1, ≤ Lτ τ A
L α 4 cp cQ¡V¥ ¬ TI ce6cQ T= 9 k ¥ e c
w Ð ÷ Q q j ; :ø4 ¤ d9^ ¥ ¬ cwf¤ 0
,ª®¯ª0¤ ¬ ¤ 0µ c(,pCTp¬ªe¥ c ^dáI ·7 «, 4 Á0¥ ¬p
^¤,eA¦ cC
d9ce6 ¤
®¯ª0¤ ¬([c f x eð k9^e¥e¥ Vc(wc ÷ÐqQf0j ; :xø#=
¤ p ¡αÉ^=e¥ 1V=ñ6ªQ
,c¯ª ¯cQ =f0=I¥
¤,cpd9C¤ cITTI cp ^¤ c^¬pc µ ¤ cC
,e6ª 7^e6=ª0¢Myf0c ^ TIcC
ce6c¤ c I ¤ ¥
¥ 9 e=o 0
α−1
f 0 (x ± 0) = lim
f (x ± τ ) − f (x ± 0) = tg α± , ±τ
¦c=Qh~¤,pC T=f0=f~,¤ e=o c7ñ6c7ªe¥ c= ¤ ¥
e6p=p , 6e^cc±y ¶c c=f=fVªeÅ c fªe^c c±ôA ?
µ f ce6cf ¤ ^e6 ce6c f x É°4e^¢§
0
c Tc^c¤ ^d9q?n0úq ¤ ¥
e6p=p , 6 e^cc±ô,=e6 T±he¥ ,ª0,=±~^c¤ ^d9ÀqQjúq c¶ fe^¥(x)¦wcQ ?f0
?=¦; . §f0¦» ¢4¤,=, p^w¶e^cc c=fpª µ c=·7¹^¤ Qd9 =d9^ þ¥
d¼÷Ðcq?d9n0 cw;¤ :øMCTº, =÷Ðcpq?f» n0l¯ o ø¥ 6 l(»eðt ,ft!ªe¥¨¨ ª0cª0 f f « V« áI d ff·7(x)(x) «, 4 Á0¥ ¬p
^¤,º÷ÐqQj; :ø4
, α = 1 x C,=Q 0?¶ªe¥ c l¯ »÷ÐqQjùgø¥ ce^f cp ¬fª τ →+0
1
2
2
l |f (x) − f (0) = x cos − 0 ≤ |x|. x
C Q?
cM=c= C ,cQ^,c[= 6?¤ Qcd96cM¡¨ª ª0 ff « f (x) ¤ ¥
e6p= dPMe^c d~¤0
cdw®¯ª0¤ ¬PcTe^¥¦c f?¦ ic4f0=eC=6eð¨ª0 f « QcÉ
,¯ ^§ ^^¤,? 7l¯ ?c= ceÅ7 ± eð7fIc f ffª (x) , , ¤ = A e ¦ C c
Å e
c ^ e f p c ¬ x=0 2
1
Zδ
|f (τ ) + f (−τ ) − 2f (0)| dτ = 2 τ
Zδ
dτ = τ ln(τ /2l)
0
0
=
Zδ
τ δ d[ln(τ /2l)] = 2 lim ln ln = ∞. γ→+0 ln(τ /2l) 2l γ
C ¨eA¦ª0cQc
f c=« d9C ,c¯Qe¥,M=w6?¤c0
f c cyx ®¯=¤ ª00¤ C?¬, ==c= fVñ6l¯c d# ªI¶d9 »4 ª¤ e6pcC=
,cp= ,p¡V dw=¤,6=e^eAe^¦d9cCc=
¤ d9^ ce6 &¬[
¤ ¤0ª0
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4 ¤ e^c= d#d¼ª~Q=·7 ¯ ¤ § c¤ ¦¶f C±t^, =f0cf0e^ c¤ = ¥e^
eyce6 f ª¤,e^Cc = ¢M c[7cpeMµ¸0d9¦ceð¤ ! ^c=,cc =¦0 =§¤, =¦td9f¨¶^ª0^¤! Mf
« C d9 ¨± ¥µµ l( § ,¦~¶¨ñ6ª0c ^f c[« , =± dºf c=c=c¤ ¤^ª0=¢ª6d9eð ô 6¤ dP dP¥
^¦0d=¤,=6f(
^c=¤ fp^Qª0p¢4¥ p¬~e6cC
w c[÷
c(fpe^Cpc± ¥e6 ¬[e6d9ccV c=e^dTc,0 µ ¤,p6
Ql¯¥ ¬Qp?µ¸¨ª0 f « ôl¯ ¤,=f0 ¢hû g?üýø¥ Qh ''¥ OD,_N,Nu =T;>x =,x òR] N²Y6Z[D,]:©`=>W@ > @ (:=D,Y¥³B N,\Y6ZD,D_0>QY FG&(Z[>QD> >QZ[>QD,D>?]6ZN²>QZ x = 0
Zδ
lim
λ→∞
f (x)
sin λx π dx = f (+0). x 2
÷ÐqW:0úq?ø
0
%& ' =T;x>=¶ U - + t·?xò]NOD,_N,N@ f (x) `Y¸@N>QQ\0D>R>(>QD>QZ[>QD,D,:R NW> @ :=D,N,\Y6D,D,:RR`=@>](Y¥B³Z_ Y [−l, l]G&Z[>Y6YÉ@,<6Ô^@ Y¯RV_:pB
U +-
xTl( ,² ¤ cCcp ¬ cµ¸±»±*,=ce6 f xc± e6ª0[−l, d9d9´l] ¤0c
e^! cp®¯ ¬=ª0^¤ ª¬^yd9¨(eðc¤ d#¤ ªQ ,0ª µ « cd cf0? Q=«
=¢M7 d
, k ÷ÐVq 3 ùg0q?ø¥f0c=c¤ª0¢¼e^0 ,ª¶ 6 ce6hd9 c?¡V¥ ,~ ¤ ¥
e6p= d[0
1 lim Sk (x) = lim k→∞ k→∞ 2l
Zδ
[f (x + τ ) + f (x − τ )]
sin[(2k + 1)πτ /2l] dτ. sin(πτ /2l)
÷ÐqW:0ùg=ø
¹ ce^f cp ¬fªy¥ , δ ¤ cCcp ¬, T^¤ ^dº^p=f c± c ¤ cd96¡C ª f [x − § © ¤ cd96¡ª f0=d9»d9c c=c ce6»¨ª0 f « cwd9cQ¡V c δ, eA÷Ð
qWx]:0¥ ùgpø&~¬V[x, ¤ xe^c0+ ,¥ªôδ]
^ªdþe¥ c ^±~dþe¥^ c¥¤
,^ª0d9¢M7;^I=e^cQe^¡[d9c=
^¤ e6 d*^ ^c¤ ¯T ±w¤ ^ c¤,^p^C¤,c? f= (x) = ¤,=c±~,=e6 0
1 lim k→∞ 2l
Zδ
f (x + τ )
sin[(2k + 1)πτ /2l] dτ = sin(πτ /2l)
0
1 = lim k→∞ 2l
Zδ
f (x + τ )
sin[(2k + 1)πτ /2l] dτ + πτ /2l
0
1 + lim k→∞ 2l
Zδ
1 1 − f (x + τ ) sin(πτ /2l) πτ /2l
(2k + 1)π τ dτ. sin 2l
l( ,ô ^¤ c^c ^^¤,? h÷ÐqW:0 ø¥ ^e¥ h^^c[Q= eCp¬0
0
1 lim k→∞ π
Zδ 0
f (x + τ )
sin[(2k + 1)πτ /2l] dτ, τ
÷ÐqW:0 ø
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PÝpÙu¿Í « d9cQ¡Vf c7(x+ce^τ )cp d9¬=cC cc=pc ¬,eðy, b^d9cAd9
c ±qW:0úq ce^f cp ¬fªy,7 ¤ cd96¡ª f0 1 lim k→∞ π
Zδ
f (x + τ )
MOBQà
[x, x + δ]
¨ª0 f0µ
1 sin[(2k + 1)πτ /2l] dτ = f (x + 0). τ 2
ce^f m cpc¯ =¬fcª¶¤ c¤ d#cªV C¥
^^^ ¤, ? ,ªy ¤,=c±y,=e6÷ÐqW:0ùgø9d9cQ¡V c¯ ¤ d9^ ¬7 ^d9d#ª~qV3 ùg0 ÷ÐWq :0 3 ø πτ /2l − sin(πτ /2l) 1 1 = f (x + τ ) f (x + τ ) − sin(πτ /2l) πτ /2l (πτ /2l) sin(πτ /2l) p , 6eð=e^cp ¢M cy ^^¤ ,¤ª^d9c±, [0, δ] ¨ª0 f « ^± 6¤ ^d9^ c± τ ,l¯^± e60µ ¥ ¬ c 0
πτ /2l − sin(πτ /2l) = 0. τ →0 (πτ /2l) sin(πτ /2l) lim
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n
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0
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Zl
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l bx dx = (b − a); 2
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−l
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n
V =
πnx bx cos dx . l
l πnx sin πn l
÷ÐqQoùgø
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Z0 l πnx 0 a lx πnx sin dx + an = sin − l πn l −l πn l −l
Zl
b lx πnx πnx l l + sin sin dx = − l πn l 0 πn l 0 bl al πnx 0 πnx l = cos cos + = (πn)2 l −l (πn)2 l 0 πnx i l(a − b) l(a − b) h 1 − cos [1 − (−1)n ] = = = (πn)2 l (πn)2 0, n = 2k, k = 0, ∞; 2l(a − b) = , n = 2k + 1. 2 π (2k + 1)2
÷ÐqQo ø
l( ,ôf0cñ^¨(¨( « 6c b n
1 bn = l
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πnx ax sin dx + l
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πnx bx sin dx , l
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−l
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U = x,
dV = sin
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V =−
l πnx cos πn l
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b lx πnx l l + − cos + l πn l 0 πn
Zl 0
πnx cos dx = l
÷ÐqQo 3 ø
(a + b)l (a + b)l =− cos πn = (−1)n+1 . πn πn
b#=f dºc¤,pCc=dÉ ¤0
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f (x) =
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cp ,¡^ ∗
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∞
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2
k=0
l=π b=1
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∞
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2
n+1
2
n=1
k=0
b=l=1
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c?=¡V6 cTe(, =cp± ,ª0I ^e6 ª0 d9Td9d*©¤, ^=f c=^(c[¤ § ¤ ¦I eðd9¦cC^
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, x = l ,?¦cQ
d ∞
n+1
n=1
∞
l l(b − a) 2l(a − b) X 1 cos π(2k + 1), (b − a) = + 2 2 4 π (2k + 1)2 k=0
c=fªC
¤,p6
¥ ,
l(b − a) 6= 0
ôª0Q c
cos π(2k + 1) = −1
, cp ,ª0 d
∞
1 π2 X . = 8 (2k + 1)2 k=0
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l¯^± e6¥ ¬ c c ^0
c ,c
∞ X k=1
1 , (2k)2
67
=ª¦ô e¥ c§¦~¤ 0
c
∞ X 1 . k2 k=1
∞ ∞ ∞ X X X 1 1 1 = + 2 2 k (2k) (2k + 1)2 k=1 k=1 k=0 ∞ X k=1
÷ÐqQoúqQiø
∞ ∞ ∞ X 1X 1 1 X 1 1 1 = = + . (2k)2 4 k=1 k 2 4 k=1 (2k)2 k=0 (2k + 1)2
÷ÐqQoúqq?ø
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3
0
∞ X k=1
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
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π2 1 = . (2k)2 24
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=6 k=1
m¤ cd9c^c cp p6pôh÷ÐqQo ø f =2
c=fªC
X ∞ k=0
π 2
∞ X π2 π2 π2 1 = + = . k2 24 8 6 k=1
x = π/2
÷ÐqQoúq?gø ÷ÐqQoúqQø
, d9^^d
∞ X (−1)n+1 πn π sin = = =2 2 n 2 n=1
∞ ∞ X (−1)(2k+1)+1 π(2k + 1) X (−1)2k+1 (−1)k sin + sin πk = 2 , 2k + 1 2 2k 2k + 1 k=1 k=0
Ð÷ qQoúqV3 ø = ^¨(c¤ d#ªC ÷ÐqQoúqV3 øV§ cp ,ª0 ^ cþ¤,p6 cQ¡[^ arctg x º ¤0
b^±0 c¤, ÷ÐqqùnZ:ø¥ <~ ' # =uX Ex D x(#p6 cQ¡V¬[¤0
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= ,p¨ª0 f « ©ªQ
cp 6c¤ 6wªe¥ c= d l¯ ¤ 0¦ V ce^f cp ¬fª c, ^ 6,p ,c a = 0 ∞
π X (−1)k = . 4 2k + 1 k=0
n
1 bn = l
Zl
sin ax sin
πnx dx. l
e^¦e6=,=,0=
±0=
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cpµ ¾§cQc¡Ve^ cp ¬=Ccc =a=·76= πn/l −l
1 bn = 2l
Zl h
−l
=
πn πn i cos a − x − cos a + x dx = l l
1 h sin(a − πn/l)x
sin(a + πn/l)x i l = 2l a − πn/l a + πn/l −l 1 sin(a − πn/l)l sin(a + πn/l)l − = = l a − πn/l a + πn/l (−1)n 1 1 = . sin al − l a − πn/l a + πn/l −
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f (x) =
MOBRL
(−1)n 2πn sin al l2 (a2 − n2 )
∞ X (−1)n 2πn sin al
l2 [a2
−
(πn/l)2 ]
sin
πnx . l
d9^^d b = 0
, k 6= n b = 1 ? =¤ 0
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, µ¸ ^¤ cC
^e^f 0¦7¨ª0 f « ±~÷ ø d9^^d ♦ 2π l = pi X ÷ÐqQoúq?nø b sin nx; f (x) = n=1
k
n
∞
n
n=1 n
bn =
(−1) 2n sin aπ π(a2 − n2 )
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cp 6c¤ 6¯ªe¥ c dl¯ ¤ 0¦ =0¹ce^f cp ¬fª¨ª0 f « [ pµ , 6eðh 6 c± ,c b = 0 n
1 a0 = l
Zl
2 cos ax dx = l
an =
,¹?cp¦ cCcQ
¡V d
1 l
cos ax dx =
0
−l
Zl
Zl
cos ax cos
πnx 2 dx = l l
Zl
l 2 2 sin ax = sin al, al al 0
cos ax cos
÷ÐqQoúqQjø
πnx dx. l
e e^ cp ¬=Cc= ^dC^e6 §¦©¤ ^c cd96¤ Ce^f 0¦t¨(c¤ d#ªC 0
−l
a 6= πn/l
Zl πn πn i 1 h cos a + an = x + cos a − x dx = l l l 0 1 h sin(a + πn/l)x sin(a − πn/l)x i l + = = l a + πn/l a − πn/l 0 1 h sin(al + πn) sin(al − πn) i (−1)n 2a = = sin al. + l a + πn/l a − πn/l l a2 − (πn/l)2
÷ÐqQoúqW:ø
b#=f dºc¤,pCcdT
∞ X 1 πnx (−1)n 2a sin al cos ax = sin al + cos . 2 2 al l[a − (πn/l) ] l n=1
¹¤ d9^^d 0 a = 1 a e^ce6caV=Iπn/l eC=d9c±~¨ª0 f a« = cos(πnx/l) 0
n
k
= 0
,ºe^¥¦
n 6= k
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²®¯ª0¤ ¬
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I§e^
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^^f d cpµ 1
MbGVN
∞
sin al X (−1)n 2a sin al + 1= al l[a2 − (πn/l)2 ] n=1
0
÷ÐqQoúqQø
∞
cp cQ¡V
x=l
1 (−1)n 1 2a X = + , sin al al l n=1 a2 − (πn/l)2
cp ,ª0 d
∞
2a sin al sin al X cos al = + 2 al l[a − (πn/l)2 ] n=1
0
Ð÷ qQoùg=iø °c=C,? al = z .!÷ÐqQoúqQø¥P÷ÐqQoùg=iø4d9cQ¡V cô,=±»¤,p6 cQ¡^ ¨ª0 f « ± ♦ ,[ ¤ ce6T
¤ c 1/ sin z ctg z 1 1 X 1 1 X (−1) 2z 1 ÷ÐqQoùg0q?ø = + = + + (−1) , ∞
cos al 1 2a X 1 . = ctg al = + 2 sin al al l n=1 a − (πn/l)2
∞
sin z
z
∞
n
n
z 2 − (πn)2
z
z − πn ∞ ∞ 1 X 2z 1 X 1 1 ctg z = + . = + + z n=1 z 2 − (πn)2 z n=1 z − πn z − πn
z − πn
Ð÷ qQoùggø f ®7ccd9¤ 0d# ªC^ f ´e^ ÷ÐcqQ^ocVùg0 q?^ø¥¤ .^÷Ðd9qQo^ ùg gcø&^d9cw û g?ü cp ,ª0 d
¤ª0^ de^ ce^c=cdþ ¤ ô^ª0 ^ w¨ª0 f « ± <~ ' # =uX Kx Jx(¹ªe6¬( ^¤ cC
^e^fp[eò ^¤ cQ
cd ¨ª0 f « d9^6(e^c d9 f ®¯cª0ñ^¤ ¨(¬¨( « ^p=e6d9dPº¥7®¯^ª0 ¤ ,¬cV±h,¥ 7 Q ?
= aª0¢ b¥ ¾§ T ¤,pª C2l¬!¨ ª0^ ¤ f 6« y 0f¦(x)f0cñ^¨(¨( « ^ A B x f (x + x ) « @ 9 x¹ce^f ccp ¬e^f0ªw ,ª¶e^d9c6 7¤ ^¥
¥ V^= ¤ 6tª0d9÷ qQ^g0ùpg=ôjø§ ~,e=ª0¤ ª 6·c==d 6h÷Ð q?g0^¤ ncCø&
, ? ¦ cQc
e6 d ©¨ª0 f0µ n=1
n=1
n
n
n
n
0
0
f (x + x0 ) Zl
1 A0 = l
1 f (x + x0 )dx = l
−l
An =
1 l
Zl
l+x Z 0
1 f (x)dx = l
−l+x0
f (x + x0 ) cos
−l
Zl
÷ÐqQoùg=ø
f (x)dx = a0 ,
−l
πnx 1 dx = l l
l+x Z 0
f (x) cos
πn(x − x0 ) dx = l
−l+x0
=
1 l
l+x Z 0
h πnx0 πnx πnx0 i πnx cos + sin sin f (x) cos dx = l l l l
−l+x0 l+x Z 0
πnx0 h 1 = cos l l
−l+x0
πnx i πnx0 h 1 f (x) cos dx + sin l l l
l+x Z 0
−l+x0
f (x) sin
πnx i dx = l
Ê ¡=ËZ&Ù0Ö ÞÍ Ù0ÔtÙ,ß®ð× Ñ¾(ÍÐÒÖÚq£PÝÒÏ[ÖÕØÙÚpÛµ
PÝpÙu¿Í πnx0 πnx0 + bn sin . l l
= an cos
½4,? c^ c 1 Bn = l
Zl
f (x + x0 ) sin
πnx0 πnx0 πnx dx = bn cos − an sin . l l l
bM GdM ÷ÐqQoùgY 3 ø
÷ÐqQoùgnø
<~ ' #=uX xEP x(#p6 cQ¡V¬¨ª0 f « ¢ !¤0
*®¯ª0¤ ¬¶,w ¤ cd9¥µ f (x) = (l − x)/2 ¡ª f [−l, l] @ 9 x 6 ªQ
^d eA¦cC
¬yI¤,p6 cQ¡^ »÷ÐqQo ø −l
x=2
,²f0c=c
¤ ,c~^c ¨aª0 f =« 0
x0 = −l
n
∞ X (−1)n+1
n
n=1
,=±0
^d
bn = 2(−1)n+1 /n x−l
A0 = a0 = 0,
óP ¥
cp¥ ¬= c 0
sin
πnx , l
−l < x < l,
ɹcp =¥p²¨(c¤ d#ªQ ?¦ ÷ÐqQoùg=ø 4 ÷ÐqQoùgnø
An = an cos πn − bn sin πn = 0,
Bn = bn cos πn + an sin πn = bn (−1)n = −2/n. x − l = −2
∞ X 1 πnx sin , n l n=1
÷ÐqQoùg=jø
−l < x < l
Ð÷ qQo0ùgZ:ø ª0 ¤ ¤ m(ce6ª e6=fpd9þ¬~ ^¤,ª T=¡ ¨( [ e¥ f00=^
dÀ¥ 6f0c e^ñ^§ ¨(d9¦þ¨(d9º 6« ¤ ^^ »6c=c hò ®¯c¦þe^ª0¨¤ ª0¬6¯ f ¬=« = cwf ± 0cP¦we^e^ ¨cc=Oy ª0 f 6« e6,± Q,^~ ? c?ôc ¡V&f0 ccyc=c C¤¤
cp¥ ,
, p6cp? µ e6cQ¡V¬V ¬ ¤ c« ¥c~
,ª0 ¤¢4ªôp0¦~
cT cp e¥ ^ ¥ ¬ ,p»e^ d9d96¤ te^ ce^c,~67V ^e^f0cp ¬f chª0 ¤ cpµ ®¯ª0 f « ¢ f (x) =ªQ
^d©,pCTp¬I¨ª0 f « ^±Veò
c± c±[e^ d9d96¤ C± ^e¥ Vc, e^ pd9 ,d9 6e^¬¤ ^6±w 0c ±þ~0c= ºc e^^ 6¥ ¬c ± cV& c¤ ? d9?c
± =6©,! cp, ,0ª0 ~^¤ c= cC
ce^ [0,¥ l]¬
cVc cpc f ~e(¥ f ¬c cc¤0± µ x = l/2
,pp=d9 (l/2, 0) ô¨=e^ª0e^ d9f c=« ¤ ±h döe
^ce^± f0 cpc ±ô¬f e^c d9 dP¤ ¥ d9¤ ^,¤ ^c± ò0 ¢4e6¤ ¤ª¢M7¦²ce^c^ ce6¤,p6 cQ¡[ŵ <~ ' # =uX Ex Q x(¹cfpQp¬ c^e¥ ~ 6,p~ ^¤ cC
^e^fp~¨ª0 f « , cpµ ,ª0 q?^ø§¤ c=cC
ce^[0,l]¥ c¬=p cy?
¤= 6d9c
± c cp ¥ ¬cV c^±w(¤,e^ pd96 d9cQ6¡¤ ^ ^±( d9^6y0
f (x) x = l/2 X ÷ÐqQoùg=oø a π2nx f (x) = + , a cos 2 l A
Z ÷ÐqQoùg=ø 4 π2nx dx, n = 0, ∞; a = f (x) cos ∞
πnx l−x X 1 = sin , 2 n l n=1
∞
0
2n
n=1
l/2
2n
l
l
0
−l < x < l.
MbGVS
gø§c= ce^¥ ¬= cVc f
(l/2, 0)
f (x) =
A
1
c
∞ X
a2n+1 cos
n=0
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ÷ÐqQo iø
π(2n + 1)x , l
Ð÷ qQo q?ø
c @ 9cp ¬[x§ªq?e¥ø ¾¼c ñ6^cd d e¥ ,ª0,=ªe¥ c [ 6 ce6©¨ª0 f « f (−x) = f (x) e¥ ÷ÐqQ¥o
, ª6g ø f (l − x) = f (x), a2n+1
4 = l
Zl/2 π(2n + 1)x f (x) cos dx, l
n = 0, ∞.
0
9 e=.qV3 p , ¢M7 ÷| d9eð©T¤,p¡^¤ e=^d qV3 e^ U¤Qd9ø¥d9 i66¤ tce6^¤,¬¶=¨¨(ª0 f0f ~« ¨ ª0 f « ©c=Cc=,Q ,c=e^6?;¥ ¬ c cw ¤ d9c± x f (x) b =0
,=ôl/2 T e¥ x^ =w−l/2 f cñ^¨(¨( « ^c a d9^^d n
n
2 an = l
Zl
l/2
Zl Z 2h π2nx π2nx π2nx i dx = dx + f (x) cos dx . f (x) cos f (x) cos l l l l
¾§c[=c¤ cdþ ^^¤,? ¯ ¤ c=¥
^dþQ=d9^ª¶ ^¤ ^d9^ ò¦ 0
0
2h an = l 2h = l
l/2
x=l−y
cA
Zl/2 Z0 πnx πn(l − y) i f (x) cos dx − f (l − y) cos dy = l l 0
l/2
Zl/2
πnx f (x) cos dx + l
Zl/2 πn(l − y) i dy . f (l − y) cos l
¾§ce^ cp ¬=Cc==·7 e^¬¤,=^ e6cdí÷ÐqQo gø¥Teª0 6cdõe^cc= c=·7^ d9cQ¡^dQ= eCp¬ (−1) cos α 0
0
n
2h an = l
cos(α + πn) =
Zl/2 Zl/2 πnx πny i f (x) cos dx + f (y)(−1)n cos dy = l l 0
0
2 = l
Zl/2 πnx [1 + (−1)n ]f (x) cos dx = l 0
0, n = (2k + 1), k = 0, ∞; Zl/2 4 2πkx = dx, n = 2k, f (x) cos l l 0
Ê ¡=ËZ&Ù0Ö ÞÍ Ù0ÔtÙ,ß®ð× Ñ¾(ÍÐÒÖÚq£PÝÒÏ[ÖÕØÙÚpÛµ
PÝpÙu¿Í MbGCà ce^cc=6e6=ª6÷ÐqQoùg=ø¥ gø.¾ñ6cd!e¥ ,ª0,=&ªe¥ c § 6 ce6¨ª0 f « f (−x) = f (x) e¥ ¥
,ª64
c cp ¬ ªe¥ c ^d ÷ÐqQo ø f (l − x) = −f (x), p , ¢M7 d9eð T¤,p¡^ ^dõe^ d9d96¤ ¼^¤,=¨( f0þ¨ª0 f « c= ce^6 ¬ cc f û Q,(−l/2, ;¤ e=#Vq ,3 ¯úaÅ
ü . ,~m(=f!T ! ¶e¥ ^¤ ¥
h§f0
,cª ñ^¨(^¨(d e¥« , ª06,==c. 6 cd9e¥^^¬~d ¨ª0 f « (l/2, 0) 0) = c C , = 6 ? c f (x) b =0 a n
2 an = l
Zl
n
l/2
Zl Z 2h πnx i πnx πnx dx = dx + f (x) cos dx . f (x) cos f (x) cos l l l l
^^¤,? Q=d9^ c± 6¤ ^d9^ , ?ò¦ cC¦
x d = l − y ;¤,=^0µ 6e¾§ce^c dcp ÷Ь=qQCoc =ø&=·7~ e^e^c¬ôc= cyc=·7=^c ¤ c^d²d cos(α + πn) = (−1) cos α 0
0
l/2
n
2h an = l
Z0 Zl/2 πny i πnx n dx − (−1) f (y) cos dy = f (x) cos l l 0
l/2
2 = l
Zl/2 πnx dx = [1 − (−1)n ]f (x) cos l 0
0, n = 2k, k = 0, ∞; Zl/2 π(2k + 1)x 4 = f (x) cos dx, n = 2k + 1, l l
ce^cc=6e6=ª6÷ÐqQo q?ø¥ cC=u
X Ux T;x(¹ccfpp Q?p
=¬ 6#
cc cp^ e¥ º ¥^ ¬6 c,±wpþe^ d9^d9¤ 6 cC¤
^ ±w^e^c=f0p þce^¨ª0¥ f ¬« c <~cp ,ª0 '^ ¤ # q?ø§ ¤ dPc=± [0,x =l] l/2 c^(¤,p6 cQ¡[^ ( d9^6V0
0
f (x) =
A
∞ X
b2n+1 sin
n=0
b2n+1
gø&c f
c (l/2, 0)
4 = l
Zl/2 (2n + 1)πx dx, f (x) sin l
n = 0, ∞;
,
÷ÐqQo d3 ø ÷ÐqQo nø
0
f (x) =
A
∞ X n=0
b2n
(2n + 1)πx , l
f (x)
4 = l
b2n sin
2nπx , l
Zl/2 2nπx f (x) sin dx, l 0
n = 0, ∞.
÷ÐqQo jø ÷ÐqQo r:ø
MbG 0
1
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í
9 e=.q?n @9 xP¹¤ d9^¤ ´^¤,=¨( f cw¨ª0 f « ± e[
c± c±©e^ d9dP¥¤ ,^± #e^cc=6pµ e6=c ª0¢M7^±[ ª0 fT==cQd
Vq?¨(øc[¤ gd#ø¥ªCp ¤ ÷ÐqQ o ¥d
3 ^ø¥ ,º÷ÐqQo,4j¤ øf (x) e =cp q?n0 =ce6^¬ ¢²6 cc=e6c¬(¤¨ 6ª0¯ f « ò [cQ
Vc=C¨(,cQ¤ ,d#=6ªC? ÷ÐqQoùg=aoø¥=.÷ÐqQ0o iø§eª0 6cdþc^c ,c n
sin(α + πn) = (−1)n sin α
<~ ' # =uX Ex X x(¹cfpQp¬ cô^e¥ ô ¤ cd96¡ª f l] ¨ª0 f « f (x) ªC
cp ¥µ c¤ 6ªe¥ c ¢ f (l+x) = f (x) Qc a = b = 0[−l, p^e¥ ¯¡&c,ɪC
cp 6c¤ 6 ªe¥ c ¢ f (l + x) = −f (x) c a = b = 0 n = 0, ∞ @ 9 y=,? c^ c¤ 6·7^ ¢ ¤ ¥
&
,ª070¦h ¤ d9^¤ c <~ ' # =uX Ex ` x(¾² ¤ cd96¡ª f [−π, π] ¨ª0 f « 2n+1
2n
2n+1
2n
1. f1 (x) = ex , 2. f2 (x) = ecos x cos(sin x), 3. f3 = ecos x sin(sin x)
¤,p6 cQ¡V¬[¤0
®¯ª0¤ ¬= @ 9 xq¹c[¨(c¤ d#ªC =d÷Ðq?g0ùg=jø§,?¦cC
d Zπ
1 a0 = π 1 an = π bn =
c=fªC
1 π
−π Zπ −π Zπ
ex dx =
eπ − e−π 2 = sh π; π π
sh π cos nx − n sin nx x π e = 2(−1)n ; e cos nx dx = 2 π(1 + n ) π(1 + n2 ) −π x
ex sin nx dx =
−π
sin nx − n cos nx x π n−1 n sh π e , = 2(−1) π(1 + n2 ) π(1 + n2 ) −π
Ð÷ qQo oø f0 ?g0
f0¾=dTñ6,f cc=dþce¥ ,¤ ª0§,¦~=d9cQ ¡Ve^ cpc[ ¬=CCc6¡[= p ¬¨(e¥ c¥¤
,d#ª0ªQ¢M »7÷ qQ g0d*ùg=cjø&¤, p¤ C cdTcC
ó§ce6f¶p=^¤ cdd9c=e6ª06
d9f d# ª dº§µ ∞
ex =
i 2 sh π h 1 X (−1)n (cos nx − n sin nx) . + π 2 n=1 1 + n2
f2 (x) + if3 (x) = ecos x [cos(sin x) + i sin(sin x)]
ôce^ cp ¬=^ª6dPeðh¨(c¤ d#ªC c± C 0± ^¤,
ix
f2 (x) + if3 (x) = ecos x+i sin x = ee .
4Ê §pË+¢ß®ð× Ñ¾IÍÐÒÖ Í§ØÙÚpÛµ
PÝpÙu¿Í^£PÝÒÏ[Ö ÖDâu®6ßÅÛ,ß6ÒÒ Ñ¥ÕÒß ¬ Ñ¥×ÝZ¬ Í Ù0Ö Ñ ÛÍ ¹cp ,ª0 ^ ª0¢ ¨ª0 f « ¢ [ ¤ cd96¡ª f [−π, π] ¤,?6 c?¡V dº[¤ 0
hb^±0 c¤, f2 (x) + if3 (x) =
∞ X (eix )n
n!
=
∞ X einx
n!
=
∞ X cos nx
n!
+i
∞ X sin nx
n!
MbGV7
.
¹¤ ¤,= 7
^± e66 ¬ T¯~d9 d9T(,=e6h cp ,ª0 ^ c^cV¤,=^ e6,=±0
^d X cos nx ÷ÐqQo ø , f (x) = e cos(sin x) = n=0
n=0
n=0
n=0
∞
cos x
2
n!
Ð÷ qQo 3iø =e^e^d9c=¤ ^ Étò·7© ¤ dP6¤ í,=A ,0
c0 ¢4e6¤ ¤ª¢M²67»cC
ªce^cpµ ^ ♦ ce6¬y¤0
¶®¯ª0¤ ¬=cQ
d*=,? ^e^f, d²T¤,p¡^ ^d* ¤ ¥
e6==p ,¬V ¢4T fªe^c cpµ¸ ^ ¤ ^¤ T ÉI¨ª0 f « cC?,=e6=ª0¢ cp ^^,=6V¤,=c==ª¶e( d9 ÿ ©± ò!µ A ³ . #$Ê,z($ÊIdxã
f3 (x) = ecos x sin(sin x) =
n=0 ∞ X n=0
µ!- !z ! $Ê
sin nx . n!
« =Q^?¤
¥
= f c!c±e6¤ ^¤ ,c=d966eÅ¡»ª Q?f0
?[0,,~l]¤,=p?6 = cQ,¡[^ cp ,»ª0~ ^¤ ¤ cC^
c= c6 d9ªQ6
^¤ d©, ^¤ e^¥f,
±cp ¤=0¥
p¨¬ ª0 f0c µ [ñ6¹cdþce^f cp¤ c¬d9f6ª~¡ª Cf0^Ie6c ,¼7ªC
¤,c=p 06 c¤,¤ p 66 cQ¡[^^c ¤ ^hd9(Vl¯¤0
¤ 0®¯¦ ª µ = ¤ ¬I¨ª0 f « Q?
= c±h,V ¤ cd96¡ª f l] c[
ce6=?µ qQcj ø¥Q b#cM=
f0ccc#
¤ c¥c
¥¤ ¥
¥¬( ¨^ ª0 =f =« 0 ¢7 ¤ cC¤
ccpd9 ,6¡[¡^ª [−l, c=f ?d9[−l, | ÷
¤
= e 0] c?¡V cM ¤ cpµ CcC
¬ ¤ cCcp ¬ Tdºc¤,pCcdT cp=f ,c ce^^d ¤ cd~d96l¯¡ ª ¤ 0f ¦ [−l, == d9l]¥ ¨4ª0 òf « cpw ^p§= f^¡d!¯f ªCc
c^p 6cò c ¤0e¥ c4yªce¥ ^cpf µ 9 e=.qQj ¤,pC¤ Tp6 [c? ¡^ ¤ tc»^c[¤¤ 0cQ
®¯~ª0¤ ñ^¬f we6¨¤ ª0^d# f ª« d9 c¢¯Vò,c [0,¤ ¥
l]¥ ^ ª0¢Ä, § , ¤ = ^ e ^ e P d p
¤
= ^ d [−l, l] ¤,p6 cQ¡^ ·7¬V, [0, l] =fô,p^ Tcp ,ª0=^,d9=¢MT(eÅ ¯6 6 T,(p¯¶I ^ ^6 6 É,p(¯ ¨¤ cCª0
cpf ,« ¡[ ŵ ¹(Q ¤ =¤ ôI cpf 6 c=^ cc¤ ,d§=e6 ¦¯¤ c[,cCT
cpe6 ,¤ ¤ ¡c^d9^,6 =¡ ¢M~ª eŨf0ôª0 p [−l, f « l] f (x) , ¤ cd96¡ª cf [−l, 0] d9 cp ,ª0 d ¤0
X ÷ÐqQúq?ø a nπx f (x) = + , a cos 2 l A
Z ÷ÐqQùgø 2 nπx a = f (x) cos dx, n = 0, ∞. l l ¹¤ ô ^ 6 cd* ¤ cC
cp ,¡^ ~ cp ,ª0 d¤0
X ÷ÐqQ ø nπx f (x) = , b sin l A
Z ÷ÐqQ 3 ø nπx 2 dx, n = 1, ∞. f (x) sin b = ∞
0
n
n=1
l
n
0
∞
n
n=1
l
n
l
l
0
1 ×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í e6ª0d9d9¤0
cp cf0ce^ ªeC=d¢yp cVe^ ªeC=d¢
,
MbGVB
Cc
¾§c±ª ¤ cw± ¡¤ còd9¨6ª0¡ ª f « f [0,f (x)l] e^c,?
=¢É? c ¤ cd96¡ª f [−l, 0] c [¤,p6 p=ff0=fe6ª0d9dP¶¤ 0
p côf ce^ ªeC=d¢he^cc=6e6=ª6w 6 cd#ª¶e6ª0d9dPô¤0
p c e^ cf ªeC=d¢ 4 ÷|e^ dT^, ¤ 6 e=cd#qQªhjø¥ ¤ cC
cp ,¡^ ¢ ¨ª0 f « w¯ ¤ cd96¡ª f0 [0, l] V ¤ cd96¡ªµ [−l, Ç =d90]6 dTCcM 6 cP ¤ cC
cp ,¡[^ 9 ¤ ¥
c¥ ¬ C=? ce^f cp ¬fª ¤ I ^ 6pµ ♦ d9ccQdt¡6[c=Cªd9¦0cQªQ¡V
,·7 c¯ ¬VcQ eAp¦ cQ
^ d9Mce6
c¬V cp¤ 0
V®¯¥ ª0¬¤ ¬c=±¶ c f ¶¤,pC¤ T¶÷|e^dT¤ e=;qQjø¥ f0c=c¤,p e6p=p♦ , #6pV6 ,cQ=¡e6^ T ±wôe¥ ,¤ª00,
²=±~®¯¤,ª0=¤ e^¬e^d9ôc=¨¤ ª0 ^ f « c ^ cy#¤,Q?p
6 =cQ ¡ c^±º , f ¤ ccAd9
6 ¡lª =f π [0, π] # ¤ ¥
µ =uª` eC>x ==,dTx( #p6 cQ¡V¬¨ª0 f « ¢ f (x) = 1 Q?
= ª0¢ [ ¤ cd96¡ª f0 [0, 1] ¤<~0
w 'c[ e^#
@9 xl( ,V¤,p6 cQ¡[6 y¨ª0 f « y¯¤0
ô c¯e^ ªeC=d^Mª ¡V c7 ¤ cC
cp ,¡V¬¯ [−1, « ¢¼0] ^ ¤ ^ cC6
T^de^f wc,¤,pCce^d ¢¼÷| e^dT e¥ ¤ c =e=ª0 ¢ Wq :c=ø¥e^ ¬ ¯ Qp^dþ ¤ cC
cp ,¡V¬[ cp ,ª0 ^ ª0¢¼¨ª0 f0µ mcñ^¨(¨( « ^ ¤ 0
T e¥ d c[¨(c¤ d#ªQ =d
Ç
^e^¬[ª ¡V c cp cQ¡V¬
a0 = an = 0, Zl 2 nπx bn = dx. f (x) sin l l 0
l = 1 f (x) = 1
bn = 2
Z1
bcð
sin nπx dx = −
0
1 2 cos nπx = nπ 0
2 2 = − [cos nπ − cos 0] = − [(−1)n − 1] = nπ nπ ( 0, n = 2k, 4 = k = 1, ∞, , n = 2k − 1 π(2k − 1)
? = b1 =
4 , π
b2 = 0,
b3 =
4 , 3π
#0
®¯ª0¤ ¬
,¶
= c±~¨ª0 f « ~ d9^6V0
f (x) =
b4 = 0,
b5 =
4 , 5π
...
i 4 h sin πx sin 3πx sin 5πx sin(2n − 1)πx + + + ··· + + ... = π 1 3 5 2n − 1 ∞ 4X 1 = sin π(2k − 1)x. π k=1 2k − 1
<~ ' # =u` xED x(#p6 cQ¡V¬»¨ª0 f « ¢ ^¤ ? [0, π]
f (x) = x
©¤ 0
®¯ª0¤ ¬h c»f ce^ ªeC=d´,
Ê4§pË+¢ß®ð× Ñ¾IÍÐÒÖ Í§ØÙÚpÛµ
PÝpÙu¿Í^£PÝÒÏ[Ö ÖDâu®6ßÅÛ,ß6ÒÒ Ñ¥ÕÒß ¬ Ñ¥×ÝZ¬ Í Ù0Ö Ñ ÛÍ
MbGRG
9 e=qW: 9 e=.qQo @9 x#q?(ªe¥ c ±~Q?
Q he¥ ¥
,ª6?cVª ¡V c[eð
Å p¬y 6 cI ¤ cC
cp ,¡[ŵ ¤0
7Q®¯Qª0
¤ =¬ V c÷|±e^dT¨,ª0¤ f e=.« qQo ø f (x) = x , [−π, 0] VQp^d ¤,p6 cQ¡V¬¶ñ6=ª!¨ª0 f « ¢K ∞
2 an = π
a0 X an cos nx, + 2 n=1
Zπ
f (x) cos nx dx.
g0,¹ce¥
cc ¤ ¥
¥ ^ h 6 Td*c¤,pCcdº cp ,ª0 d , −x , [−π, 0], f (x) = x = |x|, x ∈ [−π, π]. [0, π] ,¾§T e¥ df0cñ^¨(¨( « ^¼®¯ª0¤ ¬ 0
∗
2 a0 = π
Zπ
xdx =
0
2 an = π
Zπ
b#=f dºc¤,pCc=dÉ 0
x2 π = π, π 0
Zπ 2 2 x sin nx π 2 (cos nπ − 1). x cos nx dx = sin nx dx = − π n πn πn2 0 0
^e¥ n 6 c , ^e¥ n ^ 6 c . 4 a = , − 3 óP ¥
cp¥ ¬ c e^f0cd9c(¤,pπn 6 cQ¡^ ( d9^6y0
n
(
0,
2
∞
π X 2 f (x) = |x| = + (cos nπ − 1) cos nx = 2 n=1 πn2 ∗
=
π 4 4 − cos x − 2 cos 3x + . . . , 2 π 3π
°f c ,p¥ ¬ c[Q= ·7^d
∞ π 4 X cos(2n − 1)x f (x) = x = − , 2 π n=1 (2n − 1)2
x ∈ [−π, π].
x ∈ [0, π].
c=V¤ 6^ªC ¬?ppye^c,?
=6Ve÷ÐqQo;:ø§ ¤ l = π b = 1 <~ ' #=u` xEH xI®¯ª0 f « ¢ ¤,p6 cQ¡V¬~~¤0
®¯ª0¤ ¬=ø4 c~f ce^ ªeC=dT ø§ c[e^ ªeC=dþf ¤,p §¦ô
,ª0Åf (x) = x C
2
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í c=¤ 6Ccf [−π, 0] [Qp^d*f (x) cp ,ª0eT ^ ¤ cª0d9¢À6¡¨ª ª0 f0f « [0, ¢ π] ¤ ¤ cC
cQ
cp ,cp ,¡V¡V dd ^6¤ cCT
d© c^e^¤,f ºpCc÷|e¯d ,¥ µ ¤ cC
cd 2π ø.,Ée^¢º e¥ c=ª0¢ºce^¬ ?bcA
É cp ,ª0 d! ^ ¤ ^¤ Tª0¢fªe^c cpµ¸A ?
fª0¢ ¨ª0 f « ¢ f (x) f (x) = f (x) ¤ |x| ≤ π l( ,ôf0cñ^¨(¨( « 6cV d9^^d 1
MbGVI @9 xø9®¯ª0 f « ¢ ∗
∗
2 a0 = π
bn = 0,
Zπ
x2 dx =
2π 2 , 3
0
an =
bcA
2 π
Zπ
x2 cos nx dx = (−1)n
4 . n2
0
f ∗ (x) =
∞ X π2 (−1)n cos nx +4 , 2 3 n n=1
x ∈ R.
,¹ cp ,¤ ª0 e=^. qQ TU¤=±I ¤0¤
¯ oeA^¦ dcC
, eð¯f( ^¤ cC
e^ ^¤,e^=f0c=¥d9
ª ¤ cQc
cp ,¡^ ¢ 0≤x≤π ∞ X (−1)n cos nx π2 , +4 x = 3 n2 n=1 2
?I cp ,ª0 ^ c^c¤,p6 c?¡^ ~ ¤
x=0
∞ X (−1)n+1
n2
n=1
c¤, pøTCc®¯d ª0 f « ¢
x2
eÅ ¥
,ª6 =
f ∗ (x)
x ∈ [0, π].
π2 . 12
9 e=.qQ e( ¤ cd96¡ª f0 [0, π] ¤ cQ
cp ,¡V dº,c=¤ 6Ccf f (x) =
CCc¤,p¡9^ cd#ª
[−π, 0]
^ 6 Td
−x2 , [−π, 0], x2 , [0, π].
Ç p^d! cp ,ª0 ^ ª0¢¨ª0 f « ¢ ¤ cC
cp ,¡V d 2πµ¸ ^¤ cC
^e^f ,4e^¢º e¥ c=ª0¢ce^¬ ¾§T e¥ df cñ^¨(¨( « ^ ¹cp ,ª0 d
2 bn = π
an = 0,
Zπ
x2 sin nx dx = −
4 2π + 3 [(−1)n − 1]. n πn
0
∞ n o X 2π 4 n f (x) = − + 3 [(−1) − 1] sin nx = n πn n=1 ∗
= −2π
∞ X sin nx n=1
n
∞ 8 X sin(2n − 1)x − , π n=1 n3
x ∈ R,
x 6= ±(1 − 2n)π.
+²?Ë+¢ß®ð× Ñ¾IÍÐÒÖ Í§ØÙÚpÛµ
PÝpÙu¿ÍPÒ߬=ÙÑ¥Ö@®ðØÑ¥×r¿QÒ Ñ6ÞÑRðÙÍ®ðÏ Í C c=©¤0
eA¦cC
eÅþfº ^¤ cC
^e^f cd#ªº ¤ cC
cp ,¡[^ ¢
MbGVL
¤,p¡9^ cd#ªt,f ¤ e=9qQúa? ,¶e^^± e¥ cc±ce^ .fªQ ¢´¶c f?¦¤,pCf¤ T(x) #x=Cc±(1 + 2n)π ¤ 0 ≤ x < π =x ∗
2
w´³±
ò!µ AÂ
¶($Êed á#$Ê, !.µ ,% +µ
e6^dP¹¤ c¤ C ^cpc ¬c d9c6d#¤ ª c= C¤ e^¥f C0f¦hª ¨[a,ª0 af +« 2l]±
2l e^cc=6e6=ª^[c¤c^c,? ¬,pVe^0µ n
o πx nπx nπx πx , cos , . . . , sin , cos ,... , 1, sin l l l l
p[= f~¤ f0¥
=f~¥ ?¦~c=^^¤,? !
c c=V ¤ c,^¤,=¥
^^~ ôªC ¢ ¢4§¦¶
=ª¦ô¨ª0 f « ±hñ6c±he^ e6^d9=^T± a
a + 2l
a+2l Z
1 · sin
πx dx = 0, l
a a+2l Z
πx dx = 0, l a ....................., a+2l Z nπx nπx sin cos dx = 0. l l 1 · cos
¹¤ ôñ6cd
a
a+2l Z
a+2l Z
nπx dx = sin2 l
cos2
nπx dx = l. l
¹ cñ6cd#ªº»e¥ ,ª0,=p§f0cA
©¨ª0 f « & d9^¢M?* ^¤ cC
2l PQ?
=,©, ,¤ wc♦ñ¥Ccd cp c=¬ ¤ c6d Cf c=9¤ 6c?C ,f ª0 [a, d a e¥+ ¥2l]
,ª0¢M
^ ô ¤,p2l6f ©(x) c¤ w6!¤ ª e¥^ cc cd9 6d ¤ l¯ ,¤ ^e^0f ¦ , ± cQ¡[ªC
^ c p 6f(x) ¤0
a
a
∞
A
a0 X nπx nπx , f (x) = + + bn sin an cos 2 l l n=1 1 a0 = l an =
bn =
1 l 1 l
a+2l Z
f (x)dx,
a a+2l Z
f (x) cos
nπx dx, l
f (x) sin
nπx dx. l
a a+2l Z a
MOIRN
A
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ¹cp cQ¡V a = 0 cp ,ª0 d¤,p6 cQ¡^ (¨ª0 f « f (x) Q?
= c±~, [0, 2l] 1
÷ g=iúq?ø
∞ nπx a0 X nπx + + bn sin f (x) = , an cos 2 l l n=1
1 a0 = l
Z2l
f (x)dx,
1 l
Z2l
f (x) cos
nπx dx, l
1 l
Z2l
f (x) sin
nπx dx. l
0
an =
÷ g=iùg=ø
0
bn =
0
¹cp cQ¡V #e¥ ¥
cp¥ ¬ c cp ,ª0 d ¤,p6 cQ¡[^ ¨ª0 f « f (x)a +Q?
2l= = cb±~, ¤ cCcp ¬ cdþc=l¤ =6C(bf −[a,a)/2 b] ÷ g=i ø
∞
A
a0 X 2nπx 2nπx f (x) = , + + bn sin an cos 2 b − a b − a n=1
2 a0 = b−a
Zb
f (x)dx,
2 an = b−a
Zb
f (x) cos
2nπx dx, b−a
2 bn = b−a
Zb
f (x) sin
2nπx dx. b−a
a
a
a
<~ ' (Dh.x>=,x(#p6 cQ¡V¬ô¶¤ ^c= cd96¤ Ce^f ±©¤ 0
¨ª0 f « ¢
Q?
= ª0¢¼,[c=¤ 6Cf [2, 6] 8x − 12 @9 x.Ç
^e^¬ a = 2 b = 6 b − a = 4 l = (b − a)/2 = 2
÷ g=i 3ø
f (x) = −x2 +
∞ a0 X 2nπx 2nπx f (x) = −x + 8x − 12 = = + + bn sin an cos 2 4 4 n=1 ∞ 2nπx 2nπx a0 X . + + bn sin an cos = 2 4 4 n=1 2
Ê?Ë}|ÑO¬=ÙÚZ¾(ÍÐÒÒÔ9Õ[ÙÚpÛµ
PÝpÙu¿Í ó§cA =e^ cw÷ g=i 3 ø¥ T e¥ df cñ^¨(¨( « ^ 2 a0 = b−a
Zb
2 f (x)dx = 4
2 an = b−a
Zb
2nπx 2 f (x) cos dx = b−a 4
Z6
2 b−a
Zb
2 2nπx dx = b−a 4
Z6
bn =
a
a
Z6
MOIZM
®¯ª0¤ ¬
(−x2 + 8x − 12)dx =
16 , 3
2
f (x) sin
?e^f cd9c(¤,p6 cQ¡[^ ( d9^6V0
a
(−x2 + 8x − 12) cos
nπx 16 (−1)n+1 , dx = 2 2 π n2
2
(−x2 + 8x − 12) sin
nπx dx = 0. 2
2
∞
cos(nπx/2) 8 16 X (−1)n+1 . f (x) = + 2 3 π n=1 n2 ∗
÷ g=iùnø
¹ cp ,ª0 ^ T±¤0
®¯ª0¤ ¬»eA¦cC
eÅf ^¤ cC
^e^f0cd#ª ¤ cC
cp ,¡^ ¢ f (x) ¨ª0 f « f (x) = −x + 8x − 12 e
,¤ cd96¡ª f0 [2,e^ ¤,6]=¥
Cc ¤,pc¡9 ^ ¤ ¥
ce6d#==ª~p, ^ ¤ e= g=i ¤ ^ d ∗
2
x ∈ [2, 6]
∞ 2nπx a0 X 2nπx an cos −x +8x−12 = + , +bn sin 2 n=1 4 4
9 ep;g=i
áIcp^ ,^¡f c^ ¤ ôc=¨^¤ ª0 f ¬ « c¤,p6 cQ¡^ (÷ g=i,ù nc[ø.e^e( ¤,¤ =cd9¥
6 ¡ ª cf0I
, ^¤ cC
^e^f c^c( ¤ cpµ 2
wTÿ
f1∗ (x) = −x2 + 4
x ∈ [2, 6].
[−2, 2]
) ÊÂ.'z´Ê #$Ê,
°4d96 dTc¤ ^c cd96¤ ,^e^f, ±h¤ 0
!®¯ª0¤ ¬ ∞
a0 X + an cos nx + bn sin nx = f (x) 2 n=1
÷ g0qúq?ø
d9cQ¡V c¤,=e^e^dPp¤ p¬yf0=f¶
6± e6¥ ¬ª0¢ ,=e6¬e6^ ^ c^cy¤0
l¯^± e6¥ ¬ c
∞ ∞ X a0 X inx + (an − ibn )e = C0 + Cn einx . 2 n=1 n=1
÷ g0qùgø
(an − ibn )einx = (an − ibn )(cos nx + i sin nx) = = (an cos nx + bn sin nx) + i(an sin nx − bn cos nx).
~ dPpô,=e6¬V¨(c¤ dP? ¬ c[ ¤ ¥
e6p=p , 6eðw¤0
cd ∞ X n=1
(−bn cos nx + an sin nx) = ψ(x).
÷ g0q ø
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ,pCT#=6
heð÷hg0qe^c ø.¤,p¡[CT^ =c6±ôeÅe fe^c(x) ¤ ¨¡ª0^ f « T d^± e§ ¤ 0
cd®¯ª0¤ ¬(÷ g0qúq?ø¥=4¨ª0 f « ψ(x) =e^e^d9c=¤ d*=cp ^( cQ
¤ c c[¤0
÷ g0q ø 1
MOIRS
ψ(x) =
∞ X n=1
einx − e−inx einx + e−inx + an = −bn 2 2
∞ 1X {(−ian − bn )einx + (ian − bn )e−inx } = = 2 n=1 ∞
iX =− {(an − ibn )einx − (an + ibn )e−inx }. 2 n=1
1 Cn = (an − ibn ), 2
bcA
ψ(x) = −i
A
X ∞
Cn e
inx
n=1
−
1 C−n = Cn∗ = (an + ibn ). 2
∞ X n=1
(
sgn n =
¾§T¤,p¡^
C−n e
−inx
= −i
∞ X
sgn nCn einx ,
n=−∞
1 x > 0, 0 x = 0, −1 x < 0.
÷ g0q 3 ø ,pCT=6eðhf cd90 ^f0e^ c±~¨(c¤ d9c±~¤0
e^c ¤ ¡^ c^cfô¤ 0
,ª~®¯ª0¤ ¬V÷ g0qúq?ø¥ ^^¤,? ¹¤ ^c¤,pCc= ^dÁ 0 ¬^¤=¯¨ª0 f « f (x) ,pCT=6eðô ^e^ce¥^ ,T±ô 0µ
ψ(x) = −i
∞ X
sgn nCn einx
n=−∞
÷ g0q=ùnø e¥ ¶ ^^¤,? h~÷ g0qùnø9eð¦cC
eÅô7e^d9Te¥ A = c^c7C,Q ^ c7c¤,p Tdº ¤ ¥µ cC/ ,¤,Q p^C cp ^d!Á.0 ¬^¤p4,pCT=6eð7 ^e^ce6^ ɱ7 ^^¤,? Me^d9Te¥ òA = c^c Z ÷ g0q= jø 1 ψ(t) f (x) = dt. π t−x c¤¤ l(ª0d9¢ ,c^ d9ch2π© µ¸c ¤ ^ ¤,¤ d9p ^cC d
c ^ ch6^e^
f ¤ c0^f0¦cpQ¤,¨ppª0C¥ c f ¬« =e6 ± ±» QÁ.?0
=¬ ^§¤¦pô,
*6 w¤ ce¥ d9¥6
,¡ª0ª ¢Mf p©[−π, ^c¤ π]^dP4e^=f0Cc?¬ µ %& @,
Z∞
−∞
f (t) dt. t−x
∞
−∞
M
1 ψ(x) = 2π
Zπ
f (t) ctg
÷ g0q=;:ø
x−t dt. 2
−π
1 f (x) = 2π
Zπ
−π
1 t−x dt + ψ(t) ctg 2 2π
Zπ
−π
f (t)dt.
÷ g0q oø
Ê?Ë}|ÑO¬=ÙÚZ¾(ÍÐÒÒÔ9Õ[ÙÚpÛµ
PÝpÙu¿Í O
MOIQà
+ "9
x ¹ªe6¬
f (x) =
∞ X
an cos nx,
ψ(x) =
an sin nx.
÷ g q ø
n=1
n=1
bcA
∞ X
1 ψ(x) = 2π
Zπ
f (t) ctg
x−t dt, 2
Zπ
ψ(t) ctg
t−x dt. 2
÷ g0qúqQiø
−π
1 f (x) = 2π
−π
l¯ ^± e6¥óP ¬ ¥
c c fp(x) ¬ ψ(x) ,c?¤ ,c,(¢Ée¥ eÅ=¶6p,^=d9e¥cI Thdº÷ g0e¥q , oª0ø§,c=^d¤,p÷ g0=q6úq?ø9eðh[÷ g0qªQ øP¬ ¤ b = ¥
c = 0 a =0 ¹^¤ ¥¦cC
c=¨ª0 f « f ψ(x) ~c= ψ(x) f f (x) Q?
==^d9T±he^cc= c=·7¥µ ^¤ pd9º ÷ g0q; :øɺ÷ g0q oø¥,pCTf(x) =6eð ¤ d9Td²!c¤,p Td ¤ ^c¤,pCc= ^dÁ.0 ¬pµ n
0
<~ ' (D=,x>=,xI®¯ª0 f « ¢
¤,p6 cQ¡V¬¤0
w®¯ª0¤ ¬,7 6¤ ? f¬(x) = sin(x/2) Ç =
C e p ¬ ¤ 0 !
¯ ® 0 ª
¤ = ^ e c
¤ ¡ ^
T ±h^d#ª,~,=±~^^ce6ª0d9d#ª ] − π, π[ @9 x ¹ce^f0cp ¬fª¶¨ª0 f « 2 bn = π
Zπ
f (x)
sin
0
^ 6,p ,c
an = 0
x 2 (−1)n+1 n sin nx dx = 2 π n2 − 1/4
÷|e^dT, ¤ d9^¤qQoùgø¥óP ¥
cp¥ ¬ c ¨(c¤ d#ªQ ~÷ÐqQoúq?nø§ ¤ ∞ 2 X (−1)n+1 n x f (x) = sin nx = sin , 2 π n=1 n − 1/4 2
a = 1/2
¤ d96V0
|x| < π.
ó§c ¤¡^ T±h¤0
®¯ª0¤ ¬=,e^cð =e^ cw÷ g0q ø¥Q= ·76eÅhe¥ ¥
,ª0¢M7 d*c¤,pCcdT ∞ X
∞ 2 X (−1)n n cos nx . ψ(x) = (−bn ) cos nx = 2 − 1/4 π n n=1 n=1
ó§cA =e^ c!^c¤ ^d9wg0qúq¨ª0 f «
f (x)
ψ(x)
e^=Q= ö ¤ ^c=¤,pCc= ^dÀÁ.0 ¬pµ
MOI 0
1
^¤p Zπ
1 ψ(x) = 2π
t x−t 1 sin ctg dt = 2 2 2π
−π
=−
Z2π
sin
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í
t x−t ctg dt = 2 2
0
1 2π
Z2π
sin
x−t x−t−x ctg dt = 2 2
0
=−
1 2π
Z2π 0
sin
x−t x−t x x−t x ctg cos − cos sin dt = 2 2 2 2 2
Z2π
Z2π x x−t x cos2 (x − t)/2 x − t cos cos dt + 2 sin d cos = 2 2 2 1 − cos2 (x − t)/2 2 0 0 2π 2π x x − t x x − t 1 1 + cos(x − t)/2 1 1 = + ln − cos = cos sin + sin π 2 2 0 π 2 2 2 1 − cos(x − t)/2 0 x 1 xh xi 1 xh x = cos − ß − sin −π − sin + sin cos π 2 2 2 π 2 2 i x 1 cos(x/2 − π) − 1 cos(x/2) + 1 − cos + ln . 2 2π cos(x/2 − π) + 1 cos(x/2) − 1 1 =− 2π
¾§ce^ cp ¬=Cc==·7 e^¬[¤ ^c cd96¤ Ce^f d9we^cc= c=·7^ d9 cp ,ª0 d 0
x − π = − sin , 2 2 x 2 x 1 − cos = 2 cos , 2 4 sin
x
x 1n x x x x cos(x/4) o − 2 cos sin + 2 sin cos + 2 sin ln ψ(x) = π 2 2 2 2 2 sin(x/4) ψ(x) =
ww
x
x − π = − cos , 2 2 x 2 x 1 + cos = 2 sin , 2 4
cos
x 1n x x o 2 x sin x − sin x + 2 sin ln ctg = sin ln ctg . π 2 4 π 2 4
"1_ 'z~Ê #$Ê, +-% !'y +-%&'ä($Êed z
e6ª 7^e6¾ ^ fc7ª0¤ ª0e^ ¤ c= =66± eŶc±e4? ¥^
^^ ¤ ^dtp^=f0c ^¤ c70 cc?¡^ 00¦f=¤ fV6·7^^f0 0©
cd9 c7c^ 0¤ ¦²ce6Q?¤,
=?0 µ e6c ~,[d9=^^ e^f0^c ^cp 6cCd9 ^T¤ d Tc±wfpe¥C ,òª0,==6± eÅ©f ccA
c[=ñ¥7 ^^ d9 ^[==d9¤,w=e^ ¤ ¤ cce6e6¤,¤,== e6^, V ñ6p ,c ^ch¢M eðc? µ ¨ª0 f « .yeC=d9cyñ6cô ¤ ce6¤,= e6c 4 ¨ª0 f « c,? ¬ TdT.(? ^d²e7c ¤ ¥
¥ ^ ñ¥ ^d9^¾§6c7V^e6 ¤ ¥^
cpc= C=¥Q= ^,d9pc©^cV¨¨ª0ª0 f f « « c,? ¬ ,cp^CcT =¤ 6ce6eÅ©¤,=f e6?
¤,p c~ ^^¤ ¤ª¥µ d9c±ô,[c=¤ 6Cf [a, b] e(^e^cd ρ(x) > 0 v(x) x ∈ [a, b] ,^eÅ he6ª 7^e6=ª6y ^^¤,? Zb a
ρ(x)v 2 (x)dx.
÷ gg0úq?ø
RË}«QÑR«¾VIÍÐÒÒÔ9ÕÙÚpÛf
PÝpÙu¿Í
MOIR7
=f 0¦¯f ?
¤,p c ^^¤ 0, ¤ôª Cd9¤ c=§e6¦c , [a,,Çb]
^¨e^ª0¬ f « ±7=ªQ
d9^d! cQc¡[cp¥µµ C~, Q,cQ¡p^¬ e6Lc([a,e^¥b],¦I=ρ(x), = L ([a, b], ρ(x)) L D⊂R4 e6c[C,Q ^ ±h¨ª0 f « D) v(x) ~ cQ¡[^e6=c L ,=A ,0
c¦ =¤,=f^¤ Cª^eðhe¥ ¥
,ª0¢M d9hª ^¤¡[
^ d9 O
=,x e¥ Éc0¦² ^± ,,p ¤ f c^d9d ,=« v ,(x) ∈ L ([a, b], ρ(x)) / p=fv¡(x), (
¤ ?
6 V ¡ V 9 d
Q c [ ¡ 6 6 e = ª λ v (x) + λ v (x) L ([a, b], ρ(x)) 2
2
2
2
1
1 1
2
2
2 2
2
Z b ρ(x)v1 (x)v2 (x)dx ≤ a
÷ gg0ùgø e6^¤,c=d^¿ e6==¤ c~«,÷ g g0ùgø#,pCT=6eð¶ ^¤,=^ e6cdºmc=·7 4u6 ª0 f ce^f c^c70 y ^¤,=^0µ l¯eC?^¦y± e6dPp^dP¥p ¬ c C§e^f c^¤ ^cVp=º,?, =e6Q~¬©÷|ª e^dTú^¤,¡[=
¤ ^ d9º^¤
cû qVf03ppüýCø¥T I=
6 ,yeð
c©f0e6ppQ=p0
¥ =¬¤e6 òh¦÷ gfg0ª0ù¤0gøµ ce^ cp ¬=^ª^d9eðwe^ ^¤,dT=¥
cy ¤ c ~ ¢4§¦h67^e6C §¦ λ
,~ ^± c±wf0c=dP= ,=« v (x) + λv (x) ≤
hZ b
ρ(x)v12 (x)dx
ρ(x)v22 (x)dx
a
a
1
i1/2 hZb
i1/2
.
2
Zb a
2 ρ(x) |v1 (x)| + λ|v2 (x)| dx −
+2λ
Zb
ρ(x)|v1 (x)v2 (x)|dx + λ2
Zb
Zb
ρ(x)v12 (x)dx +
a
ρ(x)v22 (x)dx ≥ 0.
Cª
^d ¤,=e^e^dPp¤ p¬ cp ,ª0 ^ c¶T¤,p¡^ yf0=ftf ?
¤,p cy ^¤,=^ e6cc=pµ ª 6 e¥c e^c ¥ ¢ ¬ c7
^± e6¥ ¬ c^c λ bcð
¯^^c7
e^f ¤ d9 ,=
cp ,¡^¶ªC
cp 6c¤¬ a
a
hZ b
ρ(x)v1 (x)v2 (x)dx
i2
−
Zb
ρ(x)v12 (x)dx
Zb
ρ(x)v22 (x)dx ≤ 0,
If c=c¤ c^c~e¥ ¥
,ª6y ^¤,=^ e6c!÷ gg0ùgø¥ e^f0? ,¹ ¤ ¤ cC
ccp( ,¡[ ¤ pc(=C,?¥
^c ^ ¢t= e ^± c±¯? ^^¤ c± Cc ¤ ¥
¥ dh,Td9 cQ¡^e6 L ([a, b], ρ(x)) ó§f0? , ¤ TdÀ ¤ cC¥
^ ^d
=ª¦º¨ª0 f « ± =ªC
^dÀ,pCTp¬ e¥ c hv (x)|v (x)i c ¤ ¥
¥ , ^d9cI¤,=^ e6cd v (x) v (x) Z ÷ gg0 ø hv |v i = hv (x)|v (x)i = ρ(x)v (x)v (x)dx. ó§f0? , ¤ c( ¤ cC¥
^ ¯cp ?
=6c ^0
Td9he^c± e6=d9 ÷ gg0 3 ø h[λ v (x) + λ v (x)]|v (x)i = a
a
a
2
1
1
2
2
b
1
2
1
2
ρ
1
2
a
1 1
2 2
3
ρ
= λ1 hv1 (x)|v3 (x)iρ + λ2 hv2 (x)|v3 (x)iρ = = λ1 hv3 (x)|v1 (x)iρ + λ2 hv3 (x)|v2 (x)iρ = = hv3 (x)|[λ1 v1 (x) + λ2 v2 (x)]iρ .
MOIRB
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ó² ccd9¤ c=d97c±ô¬¢¼¨e^ª0f0 ?f ,« ¤ c^c =¤ ªQc
^Cdþ¥,
p^C Thpc ¬¤ ¥
e¥¥ c d c¤ d#ª¶¨ª0 f « v(x) v(x) 1
v u b uZ q u kvk = kv(x)k = hv(x)|v(x)iρ = t ρ(x)v 2 (x)dx ≥ 0.
÷ gg0ùnø
^¤,=^ e6chmc=·7 ª0 f ce^f c^c~y^¤ d9 ,?¦e^f? , ¤ c^cô ¤ cC¥
^ c¤ d9 d9cQ¡V cQp eCp¬V4g[6 0
÷ gg0 jø |hv (x)|v (x)i| ≤ kv (x)k kv (x)k. ?^I e6÷ gg0cy jø~ ò f ^cf0=e^6f0¯c^^c e^¬dP cp 6C cɯ
ce6ppc c¯c= ^0
cÉ ^¤,=^ e6c 4 ^¤,?µ ÷ gg0; :ø kv (x) + v (x)k ≤ kv (x)k + kv (x)k. c¤b#c^=cft,?¡ ô¬ f0ò=fºdP º^!e¥ ~ 0^¦ô± e^fc?± , ? ¤ ^ ^c(¤ =¤ Pc¨ª0C f ¥
« ^ Iv ¤,(x)= cvªC(x) ¢¯,=?ªC =
^dÀ,pCTp¬ Z ÷ gg0 oø hv (x)|v (x)i = ρ(x)v (x)v (x)dx = 0. Ç =d96 dT?cc¤c^c,? ¬ ce6¬¨ª0 f « ± &? =cp ¨¬ª0f c4 f c=« 4 0 ¦ ª0 f Q« = e^ρ(x) c^c(0
cIc=¯ ¤ cd96¡ª f0 [a, b] c=¯v^e6(x)cc ±v¨(x) c0¤ ~cc^c¤,c? ^c¬, ?ò [¬ ,TyIcC
e cQc
d² c ±ô¤ cd9^e^6c¡ª c±~f ¨= ª0d9 f c6« ª ~^±h=ò (¬ôc¤ 7cc^c¤,c? ^c¬, ? ¬e4
¤ª0,^c±
¤ª0^cd mc ^ ,pI0 ¯^e^f c ^ ,p7e^ e¥CdPÉ ^± c4 6Q= e^ d9§¦7¨ª0 f « ± {v (x)} ,pCT=6eðhc¤c^c,? ¬ c± ,^e¥
, k 6= l. ÷ gg0 ø hv (x)|v (x)i = 0 <~ ' ( D D >x =,x(¹cfpQp¬ cVe^ e6^dP[¤ ^c cd96¤, ^eCf 0¦h¨ª0 f « ± ÷ gg0úqQiø {1, cos nx, sin nx}, n = 1, ∞ c¤c^c,? ¬,[,c=¤ 6Cf [−π, π] e(^e^cd ρ(x) = 1 @ 9 x.°¤c^c,? ¬ ce6¬V¤,p6 §¦h¨ª0 f « ±~e¥ ¥
,ª6yI ^^¤,? c a
1
2
1
1
2
2
1
2
1
2
b
1
2
ρ
1
2
a
1
2
k
k
Zπ
−π
Zπ
1 · cos nx dx =
Zπ
1 · sin nx dx = 0,
−π
sin2 nx π cos nx sin nx dx = = 0, 2n −π
ó² cd9c=7¬¢e^cc= c=·7^ ± −π
l
n = 1, ∞.
1 sin α cos β = [sin(α + β) − sin(α − β)], 2 1 sin α sin β = [cos(α − β) − cos(α + β)], 2 1 cos α cos β = [cos(α − β) − cos(α + β)] 2
RË}«QÑR«¾VIÍÐÒÒÔ9ÕÙÚpÛf
PÝpÙu¿Í
6¤ªC
,cª0¥
¬eð cV ¤ Zπ
MOIWG
n 6= m n, m = 1, ∞
sin nx cos mx dx =
Zπ
sin nx sin mx dx =
−π
−π
=
Zπ
cos nx cos mx dx = 0.
(=±0
^dº ^^¤,? c=f ?
¤,pc¨ª0 f « ±he^ e6^d9 −π
Zπ
ó² cd9c=7¬¢e^cc= c=·7^ ±
12 dx = 2π.
−π
1 cos2 α = (1 + cos 2α), 2
cp ,ª0 d
Zπ
2
cos nx dx =
Zπ
1 sin2 α = (1 − cos 2α) 2
sin2 nx dx = π,
÷ gg0úqq?ø
n ≥ 1,
cô¤ ^c=? ce^¬ cfpQp¬ ¾ , ¤ p 6
Å ó§ ^« ,? ¬ T©¨ª0 f « ¢¼÷|e^dTIû ?üýø¶d9£ cC
¤ c cº¤,=e^e^d9c=¤ d c¤c^c,? ¬ ò¯e^ e6^d9e^ 6« ,? ¬ §¦h¨ª0 f « ± ¨ª0 f « 6 ^e^e^¥ , {J (α x)} k = 1, ∞ c¤c=^c,? ¬ ,c=¤ 6Cf [0, 1] e^e^cd • ÷|¥
^e^¬ α 4 kµ¸T±¯ cp cQ¡V¥ ¬ T±7f c¤ ^ ¬4¨ª0 f « 6 ^e^e^¥ , J (α) = 0 5 ρ(x) = x cp cd9áI6¡[=0
¤, P (x) n = 0, ∞ c¤c^c,? ¬ þ,c=¤ 6Cf [−1, 1] eT^e^cd • ρ(x) = 1 5 cp cd9 C ¤ d9= H (x) n = 0, ∞ c¤c^c,? ¬ À,¶ ^¤ ? ] − ∞, ∞[ e • ^e^cd ρ(x) = e 5 cp cd9tá¯=^^¤ ¤, L (x) n = 0, ∞ c¤c^c,? ¬ º,4 ^¤ ? [0, ∞[ eò^e^cd • ÷ α > −1ø 5 ρ(x) = x e ¨ª0 f « C ¤ d9p U (x) n = 0, ∞ c=¤c^c,? ¬ , ^¤ ? ] − ∞, ∞[ e • ^e^cd ¨• ^ª0e^ cf d « ρ(x) á¯==^165¤ ÷ ¤, I ø¥ (x) n = 0, ∞ c¤c^c,? ¬ õ,© ^¤ ? [0, ∞[ e ρ(x) = 1d9ö α ¤,>=e^−1 m
¤ c 9 d ¶ c ^
c 9 e^÷|e^ dTe6&^û dTpüý9ø¥cô¤,hp^¤,cp6=
¥ § ¦t?e^ke^c¤,d9=c=e6 ¤ ^^ dÀ »T d9^dPf ºpc=¨^cdPª0¤ pTf h « c^ e^e^ d9f0ctc±t QT?¨(
ôQC e^ cf ± ¿ e6¢ô=ä ª0¤ dPe^c ¤ e64 c^^dTáIc,?cª0 ¤,¬0 p C§ ,cp¦ µ = ±» ÷|§e^¦dTe^ûc3püýø¥e6 ^ Td9þ¨ª0 f « d9þe^ d9d96¤ ò¦º0
^¤º ^^¤,? ¬ §¦ºª0¤,= ¥µ °¤c^c,? ¬,pôe^ e6^dP[¨ª0 f « ± {v (x)} ,pCT=6eð~c¤c c¤ d9 ¤ c=0µ c± ^áIe¥ ¢4 =ª0kv¢kc¤=c1^ c,? ¬=ª0¢¼e^ e6^d#ª¶¨ª0 f « ± d9c?¡V c7 ¤ 6¤,p¬[c¤0µ {v (x)} c ♦c¤ d9 ¤ c= ª0¢ {u (x)} , cp cQ¡V −π
−π
ν
ν k
ν k
ν
n
n
−x2
α n
−x2
α
n
α+n,n
k
∞ k=1
k
k
k
uk (x) =
vk (x) . kvk k
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í <~ ' (D D xED xI°¤c c¤ d9 ¤ cp¬É¤ ^c cd96¤, ^eCfª0¢ºe^ e6^d#ª(¨ª0 f « ±y÷ gg0úqQiø¥ @9 x.ó*ª0 6cde^cc= c=·7^ ÷ gg0úq=q?ø§ cp ,ª0 d 1
MOIRI
1 1 1 √ , √ cos nx, √ sin nx , π π 2π
n = 1, ∞.
^c,? ¬ ¯ª0e6¢ª 7e^ ^e6e6^=d#ª6ª¯ô¨c=ª0Q f « ± c±w{vc=y(x)} ?=ª0ªQ
f ^« d! ,pCTpc¬4¤ cpc ^c ,c?± p¬^ e¥ c± e^L^dþ([a,¨°ª0¤b], f cρ(x)) Q ª , w ¨ f (x) « d v (x) ñ6c±he^ e6^d9( p=f♦¹¡cp(®¯ e^ª0 ª0f ¢ p« p c¢¯¤¬y?c¤,ªC^=c ,^dT?ª0 ¢t¬ª0ªQ¢ ¢e^ ,e6 ^[a,d#ª¶b]¨e^ª0¢§ f
,« ªQ f ± ¤ cd99f0c ^ cd9^cQcÉ¡V c¯e¥ ¤,T=e^e^cdP ^pf ¤ Q= ªQ
p^d ¬ f0=f~c c¤¢4cp^~c,¨?ª0 ¬f « T±hh^¯e^f ñ6c c^^ cV c¤ d9c^e6¤ ¤,T=±w e6pCV d9e(cQ[{v¡[ 6¤ (x)} ce6ò¤,=¬V e6¤ ¥
e6Lp=([a, p ^b],,Vρ(x)) ¤,p6 cQ¡[p=Åf µ ^d X ÷ gg0úq?gø f (x) = C v (x), 4f c=c¤ cd! ce6cQ T C ^¤,=¢M¤ cp ¬ pf cc¤
,pt ¢¨ª0 f « f (x) c= ce^¥ ¬ c pC eC {v (x)} O
D x(mcñ^¨(¨( « ^ C ©¤,p6 cQ¡^ ¼÷ gg0úq?gø7c ¤ ¥
¥ , ¢MeÅ*e^cc= cpµ ·7^ ^d ÷ gg0úqQø hf (x)|v (x)i , n = 1, ∞. C = p6 c?¡^ I÷ gg0úq?gø¥pf0cpc¤ kvcd(x)k f cñ^¨(¨( « ^ c ¤ ¥
¥ , ¢Með¨(c¤ d#ªQ c± ÷ÉgeCg0=úd9qQ¯ø¥Cf0,cpñ^C¨(T¨( =« 6 eð^¤ 0
cd!®¯f ª0c¤ ñ^¬¨(§¨( cM« c ¤^cp^=cd9,? ®¯¬ª0 ¤ c¬±7§e^C¨ e6ª0 ^f d9« § ¨ ª0 f « c=± {vce^(x)} ¥ ¬
c f (x) pC eC {v (x)} C 4 U -+
x;kd9 cQ¡V Ve^f0? , ¤ cy ¤,==ª0¢ ô ^=ª0¢À,=e6÷ gg0úq?gøT,¨ª0 f0µ « ¢ v (x) ~ce^ cp ¬=Cc==·7 e^¬e^cc= c=·76 d9t÷ gg0ùnø¥÷ gg0 oø¥ cp ,ª0 d ∞ k=1
k
2
k
k
∞ k=1
2
∞
k k
k=1
k
k
k
n
n
2
n
k
k
k
∞ k=1
∞ k=1
k
n
hf (x)|vn (x)i = =
∞ X
∞ X k=1
Ck hvk (x)|vn (x)i =
Ck kvn (x)k2 δkn = Cn kvn (x)k2 .
b#=f dºc¤,pCc=dºª ^¤¡[
^ (
cf0pQ= c O
H x(#0
k=1
eA¦cQ
eð ¤ ^de^ ¤,=¥
c ∞ X
∞ X k=1
Ck2 kvk (x)k2
Ck2 kvk (x)k2 ≤ kf (x)k2 .
ó§cc= c=·7^ y÷ gg0úq?nø§,pCT=6eðh ^¤,=^ e6cd 6 ^e^e^¥ , k=1
÷ gg0úqV3 ø ÷ gg0úq?nø
RË}«QÑR«¾VIÍÐÒÒÔ9ÕÙÚpÛf
PÝpÙu¿Í MOIRL U +-
x.l , e^e¥ ¥
c= eA¦ cC
d9ce6©¤0
~®¯ª0¤ ¬~÷ g=g0úq?gøM¤,=e^e^d9c=pµ
¤ dþÅ ª
n n
2 Zb h i2 X X
= ρ(x) f (x) − f (x) − C v (x) C v (x) dx =
k k k k k=1
=
Zb
k=1
a
2
ρ(x)f (x)dx − 2
a
+
n X
Ck Cj
k,j=1
n X
Zb a
2
= kf (x)k −
Ck
k=1
Zb
ρ(x)f (x)vk (x)dx+
a
ρ(x)vk (x)vj (x)dx =
n X
÷ gg0úqQjø
Ck2 kvk (x)k2 ≥ 0.
?V÷ gg0úqQjø§e¥ ¥
,ª0¢MyeA¦cQ
d9ce6¬V¤0
~÷ g=g0úqV3 ø§~ ^¤,=^ e6c!÷ gg0úq?nø¥ k=1
(l ,c=¤c c¤ d9 ¤ c= §¦e^ e6^d´÷ ku (x)k = 1ø4 ^¤,=^ e6c 6 ^e^e^¥ , ¤ 0µ d960
♦
k
∞ X
Ck2 ≤ kf (x)k2 .
° ¤ c ^ c , ? ¬ 0 ª ¢ ^ e
6 e ^ # d º ª ¨ 0 ª
f
«
± , p C T = M ¢ Q p P d
f ª = c ± & ^ ¥ e {v (x)}
,~ ¢4c±h¨ª0 f « f (x) ∈ L ([a, b], ρ(x)) ^¤,=^ e6c 6 ^e^e^¥ ,º÷ gg0úq?nøTc¤,p?µ 6eðh[¤,=^ e6c X ÷ gg0úWq :ø C kv (x)k = kf (x)k . =^ e6c÷ gg0úWq :ø;,pCT=¢M¤,=^ e6cd!¹(=¤ e^^? , Q0 ¯¨(c¤ d#ªC c±IQ=d9f ª ce6 ®7c¤ d#ªQ Q=d9f ª ce6w c=Ccp , 6[Q= eCp¬!÷ gg0úqQjø§[0
X ÷ gg0úqQoø
C v (x) = 0. lim f (x) − k=1
∞ k=1
k
2
∞
2 k
2
k
2
k=1
n
k k
n→∞
¤° 0,
[c=®¯¤,ª0p¤ ¡¬=6e^V¤ ¥
c= ô^f ¨I=?f
?¤,p c¶ p¤ ôw Qc=^d9¤ ^6 ·77 ¨ce6ª0 ¬[f « e6 ¤ ^d9f (x) ^ ¸ µ , = 6 e
c ! ± 6 e 0 ª 9 d 9 d c ± n ¤ ô9 °4^c=d9^¤,6= dT0µ ^ cd¼c=C¤,=e6p= n ? =#¤ 0
÷ gg0úq?gø peA¦ cC
eÅ¢ft¨eÅôª0 f¶f « ªQ ¢f (x) cC,
=,6=eðf hc 0c=eAc¦cQeð¦
cC
d9 cd9e6che6¤¬[0¤
[
[c®¯ª0T¤ ¬ =cdc ÷|¤ ¥c=
¥c , ^^ dP pc¶d&e^ø cc= c=·7^ Cd ÷ gg0úqQoø¥ c=Q 0µ k=1
n X Ck vk (x) = 0 lim f (x) −
n→∞
^e d9 T e¥c d ¯hû == f~¤ ¥,
pe6CpT=p ,=^ dP6¶p~e^ceA¦ccC
±~ d9¤ c e6d9^¬y¤w eAc[¦ cC
c ¤ d9d9c(e6[w ¤ Vce^e6¤ ¤,¥
= ^e6d*0 he^¤ ¥
^f ?
ü ¤, ?µ L ([a, b], ρ(x)) ¹ c ¥ e ¥
c p ¥ ¬
c 6 e ¬ {v (x)} v (x) ∈ L ([a, b], ρ(x)) k ,pCTp¬¶eð¦cC
,7^± eÅf!¨ª0 f « f (x) L ([a, b], ρ(x)) ¶e^¤ ¥
^d= û 1,0 ∞&e^¤ =¥ªQ
^¥d µ f ?
¤,p cdT,0 ~ c c¤ d9([ ¤ ce6¤,= e6 L ([a, b], ρ(x))ü ,^e¥ ÷ gg úqQø lim kf (x) − v (x)k = 0. k=1
2
k
∞ k=1
k
2
2
2
k→∞
k
MOLRN
1
#0
∞ X
vk (x),
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í
vk (x) ∈ L2 ([a, b], ρ(x))
=ªC
^dö,pCTp¬eA¦ cC
,7 d9eðf²¨ª0 f « f (x) L ([a, b], ρ(x)) te^¤ ¥
^d û c e6ª0cd9¤ d9dþMeA¯¦ cC
¤ ce6eðw¤,=[ e^e6¤ ¥
L^d([a,f~¨b],ª0 ρ(x)) f « ü f^e¥(x) y ce¥ ¥
cp¥ ¬ ce6¬6^c nµ¸,=e6 §¦ ?¶eA¦ cC
d9ce6ºwe^¤ ¥
^d ¶e¥ ¥
,ª6eð¦cC
d9ce6¬wcT cd¼e^d9Te¥ =# cpµ ♦ e^f cp ¬fªô ^f c=c¤ §¦c f0?¦c=¤ 6Cf0 [a, b] ¨ª0 f « f (x) ¤0
P v (x) d9c6ª
cc e6p^p cc ± øòcIeA¦ e^0cC
¬d9 cIce6c=C ; Ç ,=pd9¬6eð[ dþ
¤fôª0Écc=d#ªy
¤¡ª0=6 cc(I ^ c=c=Ccd9 cQ^¡V cc(±~ e𤠦[cC
c Td9 ce6 c~±w=÷| =f0cpµµ ¡e¥ =¥.
,ª0c¢Mc7=7 d*[^ c¤ cd9¤^ ¤ .c dT[ e¥ ¥
,ª6weð¦cC
d9ce6¬w~e^¤ ¥
^dTch ¤ c0 ¢4e6¤, ¤ª^d <~ ' ( D D Ex H x ?e^e¥ ¥
cp¬V ce¥ ¥
cp¥ ¬ ce6¬ k=1
2
2
∞
k
k=1
ϕn (x) =
√
,eA¦ cC
d9ce6¬ @ 9 x° ^0
c §c e^ ¤,=¥
c
2nxe−nx
2 /2
ϕn (0) = 0
lim ϕn (x) = lim
√
n = 1, ∞,
,
2nx
enx2 /2
x ∈ [0, 1]
þ
,þ ¢4c^ct¨( f0e^ ¤ cp c^c
x ∈]0, 1]
√ 2x (−2) = lim √ 2 nx2 /2 = 0. n→∞ 2 n x e
Ç
^e^¬[d9 ce^ cp ¬=Cc? e^¬ ¤,=0 cdáIc =? , óP ¥
cp¥ ¬ c n→∞
n→∞
lim ϕn (x) = 0,
x ∈ [0, 1].
?e^e¥ ¥
,ª^d^ ^¤ ¬hc ¤ ce7c¶eð¦cC
d9ce6©¶e^¤ ¥
^d n→∞
fô¨ª0 f «
ce¥ ¥
cp¥ ¬ ce6
,#=e^e^d9c=¤ d ^^¤,? ϕ(x) = 0 Jn =
Z1
[ϕn (x) − 0]2 dx =
If c=c¤ c^ce¥ ¥
,ª6pc 0
Z1
{ϕn (x)}
2
2nx e−nx dx = 1 − e−n ,
0
lim Jn = 1
p¥ ¬ c e6¬ {ϕ (x)} ôeA¦ cC
eð c c¤ d9ô ¤ ce6¤,= e6 L ([0, 1]) f ¨ºª0 cf e¥« Å
cϕ(x) =0 § ¾ ò
^ e
T ¦ = « ~[¤0
®¯ª0¤ ¬¤,==fe^¨(^¤7c¤ eA¦d#cCªC
d9¤cªe6Cd7e¥É e^¥¤
,¥ª0
¢M 7^dwC(Iª c=C^¤ ¤,¡[?
^p e^=¬M f(¤,p6 c?¡^ ¢¨ª0 f0µ O
JxI Ç p= ,d9 f 6ª eðhpp ºcp c ¤c±ôc=^c,? ¬,pe^ e6C dP¨ª0 f « ± {v (x)} v (x) ∈ n→∞
n
L2 ([a, b], ρ(x))
2
k
L2 ([a, b], ρ(x))
∞ k=1
k
l¯^± e66 ¬ c # ¤ ¥
cp cQ¡V dTc!e^ e6^dP V p , 6eð cp c± ? =#e6ª 7^e6=ª^©¨ª0 f « ϕ(x) c¤c^c,? ¬,pt{ve^^d (x)}v (x) e^ e6Cd9 {v (x)} #c ∞ k=1
k
k
k
RË}«QÑR«¾VIÍÐÒÒÔ9ÕÙÚpÛf
PÝpÙu¿Í
MOLZM
dPc=A6
0e^
^f0cñ^¨(¨( « 6¼®¯ª0¤ ¬¤,= ªC ¢ô¨(c¤ d#ªC 7Q=d9f ª ce6h ¤ 0µ 2
kϕ(x)k =
Zb
÷ gg0ùg=iø
ρ(x)ϕ2 (x)dx = 0.
¤ ¯c¯
,V dPe^¥=¦ = c ρ(x) > 0d d9c/ cQ¥e ¤, ¡py^Cdc¨dTª0Q = ¨f « ª0eC p f « ¬ϕϕ (x).(x)c¤ ^= c^¤ 0c^,¤ ?T e^¬c,,c=7p,6 [a,e^e6^d b]^ c c ϕ(x) # b =
f
^e6¬~=0cQ¡[
^e6^ xT∈±»[a,ªQb] ¬ ¶ñ6c v (x) ôc=C,e¥? , =6 ¨cpª0 f c=« =ªô e^ e6^d9 {v (x)} ÷ gg0ù/ g=iø.e¥ ¥
,ª6pcc,4=ϕªQ
(x) 6(¤,=,,4ªQ [a,¢b]e^¢ò
,ªf=ªf e^¤ cc d9 cpòµ¸f0 c^ ^¤ ^ ¤ cT6c(, e¥ c=c C
¥ ¬p µ §c4¦I ¤ c ^f c?±7 p¤, ,= ¢M^P7
0c¦ ^eðc7^c¤,¤ =^ «,ce6=d9 7= d9t^e^¤ ?p =4^d ^ ªQ¤ ^^¤ dTT=M e¥c e6¥
cb#p=f¥ ª0¬¢º ¨c =ª0¯ Éf «
=¢¯0µ cdºe¥ ,ª0,=(Q=d9f ª pphe^ e6^dP {v (x)} p , 6eÅ~ cp c±
= 6 ð e ¶ ¥ e ¥ ,
0 ª M ¢
d 4 ° 6 , ¯ c
¤ c 4 e ¯ c A e ¦ C c
9 d c 6 e ô ¤ 0
¯ ® 0 ª
¤ ¬ ¨ 0 ª
f
«
f (x) ª ^¤¡[
^ ^dT O pQ=d9f ª =pd9IcQe^¡[ 6e6y^dPòÉ ¬yL¥
([a, b], ρ(x)) c ¢4 pw ¨P ª0x f / « e¥ f{v(x)(x)} ¤ c4 e6c¤¤,=c ^e6c,? L¬,([a,
6 e C 0 µ Tdþc¤,pCcdº¤,p6 cQ¡^,[[¤0
®¯ª0¤ ¬=eA¦cQ
,7 ± b],eðhρ(x)) f~ ^±~[e^¤ ¥
^dT l¯^± e66 ¬ c Vc¤c^c,? ¬ ce6©e^ e6^d9´ò^f0=6wc=Cd9cQ¡V ce6¬~¤,p6 cpµ ¡« ^ !¨ ª0I f Q« = cC
¶c7¤ ¶0
»cp ®¯ ª0c=¤ p¬ = Q=d9f ª 0f cc=e6c¤,pe¥ ¥7
,ªe^6hc¢ ^^ccô eA^¦¤ cQ¥
¬ d9cce6^¬ôe^ f^ ¨ ª0 =6f0 µ ¥
fe6(x)^ ce6¬Vñ6c^cV¤,p6 cQ¡^{v (x)} <~ ' ( D D Kx Jx(¹cfpQp¬ cVe^¤ ¥
^f ?
¤,p c¯c=f0 c ^
X ÷ gg0ùg0q?ø
f (x) − a v (x)
=ªC
6t,= d9^ ¬=·7 dT&^e¥ ¤ cCcp ¬ Th676e6^ Éwf cñ^¨(¨( « ^ ¤ 0µ e^ e6¬¯^d9¤,= Td9Vf cñ^¨(¨( « ^p=dþ®¯ª0¤ ¬ C ¨ª0 f « f (x) L cIc¤c^c,a? ¬ c± {v (x)} ∈ L @ 9 x #=e^e^d9c=¤ d¥ ª a
2
2
k
k
2
∞ k=1
k
k
∞ k=1
2
2
∞ k=1
k
n
k k
k=1
k
k
k
2
2
n n n
2
2
X X X
Ck vk (x) − (ak − Ck )vk (x) , ak vk (x) = f (x) −
f (x) −
f c=c¤ª0¢ e( cd9c=7¬=¢¼cc=C,Q ^ ±
k=1
k=1
f˜(x) = f (x) −
d9cQ¡V c7Q= eCp¬V[0
n X k=1
Ck vk (x),
k=1
a ˜ k = ak − C k
n n
2
2
X X
˜
a ˜k vk (x) = ak vk (x) = f (x) −
f (x) −
=
k=1
f˜(x) −
n X k=1
k=1
n X ˜ a ˜k vk (x) = a ˜k vk (x) f (x) − k=1
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í X X ÷ gg0ùggø = kf˜(x)k − 2 a ˜ hv (x)|f˜(x)i + a ˜ kv (x)k . ¹¤ [c[ dP= =,c 1
MOLRS
n
n
2
k
2 k
k
k=1
k
2
k=1
n D E X ˜ hvk (x)|f (x)i = vk (x) f (x) − Cl vl (x) = l=1
y÷ gg0ùggø&,?¦cQ
d 0
2
= Ck kvk (x)k − Ck kvk (x)k2 = 0,
n n
2 X X
2 ˜ ak vk (x) = kf (x)k + a ˜k kvk (x)k2
f (x) − k=1
k=1
n
2 X
ak vk (x) =
f (x) − k=1
n n
2 X X
= f (x) − Ck vk (x) + (ak − Ck )2 kvk (x)k2 ,
c=fªC
~e¥ ¥
,ª6?cw÷ gg0ùg0q?ø§ ¤ d96Vd9 dP? ¬ cC,Q ^ I ¤ a = C ^c ♦cd9¾§6T¤ c ¤h^e^=f0ce6±h¤,e^= fe6 ^cd9±wëc÷Фq?g0cùYg^c3 ,ø ? ¬ c±he^ e6^d9 {f (x)} 0
(ce^ c c±ô¤ 0µ k=1
k=1
k
k
n
n
cos
πnx 1 πnx o , , sin l 2 l
c=C¤,p=¥V,=e(fô¤ ^c cd96¤ ,^e^f0cd#ªô¤ 0
,ª~®¯ª0¤ ¬ ÷ gg0ùg=ø ¾§e^7^c¤ ^d9 !TcC
e^¨(c¤ d#ªC ¤ c= T7
,=e6¤,=f §¦¤0
c~®¯ª0¤ ¬= c¨(e6cp¤ =d#¢MªC eð»e^d9 cQ¤,¡V= ¥cò
ª c T d9¬47eP
,cd9c=¤ 7 ¬^¢tc cd9 6c=^¤ cM, 0^
e^òf,0¦¤ ¤^0c
ccd9÷6gg0¤,ù g= ø¥^eCf mcc±¯ e^f ¤ e66C d9T ô¥
^ c^c¤,p6
=¹c (¤ 0
®¯ª0¤ ¬d¢Ve^f0? , ¤ c^c ¤ cC¥
^ b#=f ¤ ^c cd96¤ , ^e^f, ±h cp cd*®¯ª0¤ ¬ nµ¸^c[ c¤0
f X ÷ gg0ùYg 3 ø a πnx πnx S (x) = + + b sin a cos 2 l l d9 d9C ¤ª6¶e6¤ ¥
^f ?
¤,p c¯c=f0 c ^ f (x) =
∞ a0 X πnx πnx . + + bn sin an cos 2 l l n=1
k
0
k
n
n
n=1
A
Zl
[f (x) − Sk (x)]2 dx = min
Pk (x)
Zl
[f (x) − Pk (x)]2 dx,
−l
−l
k
πnx πnx α0 X + + βn sin αn cos Pk (x) = 2 l l n=1
RË}«QÑR«¾VIÍÐÒÒÔ9ÕÙÚpÛf
PÝpÙu¿Í
MOLQà
¤ c^cp ¬÷ gg0Tù±g=øI¤ eA ¦^cCc
c=dPeź¥!¤ e^ ¤ C¥e^
f ,^± d f cp cd k¤,0=
^ ÷ g g0¶ùgYQ3 pø¥dPf m(=ª f¼c=e6e¥ ¥
,ª÷ 6o úqV3 ø eª0 6cd*e^cc= c=·7^ ±÷Ðq?g0úq?gø 4 ÷Ðq?g0ùggø&ôcf^(x) c c,=e6¬Vf0cñ^¨(¨( « ^c ccpµ C,Q,=6eðh ^¤ 6 a ,=e6¬ 4 ^¤ 6 b ,[ñ6cdþe¥ ,ª0,=Id9c?¡V c7Q= eCp¬ c
4 ÷ oùgø¥¤ 0
n
n
n
1 l
Zl
f 2 (x) dx =
Zl
a2 X 2 (an + b2n ), f (x) dx ≥ 0 + 2 n=1
÷ gg0ùgnø
k
a20 X 2 (an + b2n ). + 2 n=1
^¤,=^ e6=c 6 ^e^e^¥ , e^cc=6e6^ c ¤ dP=6V0
−l
1 l
÷ gg0ùg=jø
∞
2
9dc=cfe6ªC
¬t©
,e¥ þc c¢4^ctc±*¤0f
?&
e6¤,pc? 7 ^ ^cc » ^¤,^=¤ c¤±ª^,d9=ce6± ¨÷ ª0g g0f ùg=« j ø¥ §f (x) p=f¡[e¥ w¥
,eAª¦6cCº
eðd9¦ccC
e60¬ µ f0p¡[
c^c[I¤0
c X Xb ÷ gg0ùZg :ø a [c=C(
=¥ f c¬ ^c« e6§. º÷ = ø[ ¤ 0¦ cC
dëf*cc=7^ cd#ªºª0¤,= ^ ¢ Q=d9f ª ce6²0 cc=7^ c±ô¨(c¤ d#ªQ I¹(=¤ e^^? , −l
∞
∞
2 n
2 n
n=1
n=1
∞
a0 α 0 X 1 + (an αn + bn βn ) = 2 l n=1
Zl
f (x)g(x) dx.
÷ gg0ùg=oø
^e^¬ a b ,αe^c c=β¥4 f0e^cñ6¨(^,¨( c « ^»¤ ^c cd96¤ ^eCf 0¦¯¤0
c4®¯ª0¤ ¬&¨ª0 f0µ « ± Çf
(x) g(x) § ¾ ^ e [
¤ ♦ 6 c±hô ^ 6¥
c^±w TcC
e^ òe6·7^dT[ c^ ¤,¤ =¥
^¥ e6^ ô§c¦he6=,p¢M eŤ cd9e^6 ¡¤,=ª ¥f0
[0, Tl]d9 p=f¡7
, <~ ' ( D D Ex P xIóº cd9c=7¬¢²¤,=^ e67¹(=¤ e^6? ,[
,¨ª0 f « yM ¤ dP6¤,Vq?g0 ,=±~e6ª0d9d#ªô¤0
n
n
n
−l
n
∞ X 1 . 2 n n=1
@9 x ¾t ¤ d9^¤ (q?g0 cp ,ª0 ^ cI¤,p6 cQ¡^ ɨª0 f « x=
X (−1)n+1 2
sin nx,
e^cA =e^ cf c=c¤ cd#ª c[¨(c¤ d#ªQ V÷ gg0ùgnø§,?¦cC
d 0
n
1 π
Zπ
−π
f (x) = x
x ∈ [−π, π],
¤ 0
y®¯ª0¤ ¬ ÷ gg0ùg=ø
∞ X 4 x dx = n2 n=1 2
∞ X π2 1 = , 2 n 6 n=1
÷ gg0 iø
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ce^c,?
=6eV÷ÐqQoúqQø¥ ¨e6ª0= ª♦f 6y« ó§ f±y^p¤ Q¥=6¦7cC ^
,ce6ª~c==^ò ^ ·7cf0^ cd90=cQ
¤ ¡Vc6ª0 §d9ct¦~^¤, =p¤ e^c e6¤ ¾c¤,e6 = ¤, e6=^ ± y fôc¬º±ôñ^¤ ,?d9t ^e¥^ ,cª0¤ ,MT==dT±= f f0cc±yd9 0^ ¤ ^¥f ¦ e^cC
~c=Ce^,cQc= 6§p¦ µ f « v(x) ,pCT ^=e¥6 heÅþe6ª f 7?^
e6¤,=pª^ y c» ^^¤,?^^ ¤ ¤ª¥µ d9c±ô,[mcc=d9¤ 06 C6f f0 e^ [a,c=Cb],Qe( ,p^e^þcd ¨ª0ρ(x) > 0 t ∈ [a, b] MOL 0
1
Zb
÷ gg0 q?ø
ρ(x)|v(x)|2 dx.
cM ^^¤ ¤ª0^ d9~ò ¦I¤ ,c e6[a,c b] f0 cd90 ^f e^ c=C,Q §¦¯¨ª0 f0µ
«~ ±~cQ¡=ªC^
e6^dcÉc=e^c=¥C¦(,?f ,p?
¬ ¤,pL ([a, b], ρ(x), C) L¤ ¥
¥ , ¢Meðþe^f0? , ¤ c¶ ¤ cpµ c
4 ½ , ? c ^
c
¤ c 6 e , ¤ =
6 e L ([a, b], ρ(x), C) C¥
^ I~ c¤ dP ó§f0? , ¤ Tdh ¤ cC¥
^ ^d~
=ª¦f0cd90 ^f e^ c=C,Q ò¦I¨ª0 f « ± v (x) v (x) =ªC
^d,p^Tp¬V e¥ c hv (x)|v (x)i c ¤ ¥
¥ , ^d9cI¤,=^ e6cd a
2
2
2
1
1
A
2
2
hv1 |v2 i = hv1 (x)|v2 (x)iρ =
Zb
÷ gg0 gø
ρ(x)v1∗ (x)v2 (x)dx,
¨ª0 f « f cd90 ^f e^ cVe^c ¤¡[^ ,pô¨ª0 f « v (x) =ªQ
^dþ,pCTp¬ e¥ c v(x) a
c¤ d94 c±ôf cd90 ^f0e^ c=C,Q c±h¨ª0 f «
vn∗ (x)
n
v u b uZ q u kvk = kv(x)k = hv(x)|v(x)iρ = t ρ(x)|v(x)|2 dx > 0.
÷ gg0 ø
a
<~ ' ( D D Ex Q xkPe6== c¬©e^c± e6©e^f0? , ¤ c^c© ¤ cC¥
^ ¼÷ gg0 gø¥P=,? cpµ ^ TIe^c± e6=d÷ gg0ùnø¥ 6·7¬VeC=d9ce6c?¥ ¬ c ó§ e6^dPf cd90 ^f0e^ c=C,Q §¦h¨ª0 f « ± {u (x)} ,pCT=6eÅhc¤c c¤ d90µ ¤ c= c±ô, ^¤ ? a < x < b ,6e¥ n
Zb
∞ k=1
÷ gg0 d3 ø
un (x)u∗m (x)dx = δmn ,
A
u (x) 4 ¨ª0 f « ,f cd90 ^f0e^ ce^c ¤ ¡^ ,pô¨ª0 f « u (x) <~ ' (D D xUT;x(¹cfpQp¬ ce^ e6^dPf0cd90 ^f e^ c=C,Q ò¦h¨ª0 f « ± a
∗ n
n
n 1 o inx √ e , 2π
p , 6eð~c¤c c¤ d9 ¤ c= c±~e(^e^cd
n = −∞, ∞ ρ(x) = 1
,c=¤ 6Cf
[−π, π]
RË}«QÑR«¾VIÍÐÒÒÔ9ÕÙÚpÛf
PÝpÙu¿Í @9 x ¹¤
MOLR7
n 6= m m, n = −∞, ∞
Jnm =
Zπ
−π
½4,? c^ c ¤
n=m
Zπ 1 1 −imx 1 inx √ e √ e dx = ei(n−m)x dx = 2π 2π 2π −π π −i ei(n−m)x = 0. = 2π(n − m) −π Zπ
Jnn =
1 2 √ dx = 1, 2π
cô¤ ^c=? ce^¬ cfpQp¬ °¤c c¤ d9 ¤ c= TPe^ e6^d9»¨ª0 f « ±Icp ?
=¢Mɤ0
cdôQ=d9^,p¥ ¬ §¦Ie^c± e6 f c=c¤,¤ pT6 7cQ=¡ªQ^
, ª ¶*^¨ª0ª0 ^f « c=6
6=÷ gÇg0
úq?^ge^ø¬ ¡[ctI c=cp d9 6c± dTc;¤c c¶cf ¤ cd9ñ^ ¨(¤ ¨(c =« ^c±*Àe^ ®¯e6ª0^¤ d9¬ C f cd90 ^f e^ c=C,Q ò¦h¨ª0 f « f (x)±~T e¥ , ¢MeÅh c¨(c¤ d#ªC −π
n
Cn =
Zb
f (y)u∗m (y)dy,
Zb
f (y)u∗n (y)dy,
÷ gg0 nø
f c=¹c¤,cCp
~e6=e(pc= w^÷0g
g0 cne6ø§¬h¢ ÷ gg0òúq?g^ø¥f=Q6=y y·7÷^gd g0úq?gø&eª0 6cd¼÷ gg0 d3 ø¥ a
f (x) =
∞ X
un (x)
n=0
a < x < b.
÷ gg0 jø
®7 cd ¤ dP? ¬ c¶Cd9^ ô÷ gg0 jøÉ c¤ 0
cf! ^^¤ ¤ c= !e6ª0d9d9 ¤ c= cp ,ªµ f (x) =
Zb
a
f (y)G(x, y)dy,
G(x, y) =
∞ X
un (x)u∗n (y);
÷ gg0 r:ø
n=0
a
a < x < b, a < y < b. çd×@â@Ö ådÙrÏrrØ@ÖVÙuÐbÜÐ9 SRS ê WG? ä¸ÖVÛrâ@Ò×uÒÜuÒÙrØ@Ò]Ó¥×uÒÜ8báRÐ / ådÙrÏrrØrØ Ú àRØZ×@Ø@ÓÚ æráWÖ àÛrâ@ÖRäáRâuÐÙ@äáRàYÒ Ø]3 âuÐVÒá¸â@ÖÜ8 ê
à
âuÐàRÙrØrà ådÙrÏrrØrç ÖRÖVáRG(x, Ù@ÖVéy) ÒÙrØ@Ò
L2
δ(x − y) =
∞ X
n=0
un (x)u∗n (y),
δ(x − y)
a < x < b,
a < y < b,
SRS ê àRI
ä ÛrâuÐàWÒ×rÜ@ØràWÖRÒ¦×rÜ@çÜu0RÖVßyådÙrÏrrØrØ ÚWÙuÐÝ]ÞàYÐVÒáWäçåWäÜuÖVàRØ@Ò]ÓÛ@ÖÜ@Ù@ÖVáRÞ¨×rÜ@ç¸ÖVâráWÖ / Ù@ÖVâ@ÓØrâ@ÖVàYÐÙrÙ@ÖVß äØ@äáWÒÓÞ {u (x)} àfÏZ(x) ÜÐVä]∈ä]Ò LL2 ê 2ÐÏrØ@Ó?Ö 0VâuÐÝOÖRÓÚOÛ@ÖÜ@ÙrÞÒÖVnâráWÖVÙ@ÖVâ@ÓØrâ@ÖRàYÐÙrÙ@2 ÞÒ-äØ@äáWÒ]ÓÞFådÙrÏrrØ@ß Û@ÖVâ@ÖOÑ×Ð u á {un (x)} Ûrâ@Ò×uäáVÐàVÜuÒÙrØuÒ×uÒÜ 8báRÐ / ådÙrÏ rrØrØÍà àRØZ×u Ò SRS ê àRI ê ÖRÖVáRÙ@ÖVéÒÙrØ@I Ò SRS ê àRI éØrâ@ÖVÏZÖÊØ@äÛ@ÖÜ 8RÝ]åYÒáWä
ç ^ à Ø@ÝØ@æ@Ò]äÏZÖVßÜ@ØráWÒâuÐáRådâ@ÒRÚ?3×uÒÒv3OÖÊæuÐVäáWÖ ÛrârØrÙrØ@ÓA Ð u á£ÝbÐÖVÛrâ@Ò×uÒÜuÒÙrØ@ÒÛ@ÖÜ@Ù@ÖVß à£Ûrâ@ÖRäáRâuÐÙ@äáRàWÒ L äØ@äáWÒ]Ó¿ Þ ådÙrÏ rrØrß Ø ÙuÐÝ]ÞàY Ð u á 2 åWäÜuÖVàRØ@Ò]ÓÛ@ÖÜ@Ù@ÖVáRÞ±ê
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í e¥ ,ª0,Ç ==§d9eð6¦cC
dT #d9ce6c!¬¤0¤0
º
º÷÷ ggg0g0 rr::ø ø d9cc? ¡ dP6!=6weðcT4 c cc=d¼7e^^d9 T ce¥d! ôe^d9¤,T=eAe¥¦ cC=
°¬ eÅ ·79^¾Àd!cQñ6
c d ,= cp ^ e^ cp ¬=^ª^d9ò¦ôd96cC
c[e6ª0d9d9 ¤ c= ¶¤,=eA¦ cC
,70¦eÅô¤ 0
c 4 d96cC
½4f c=¥c ,¤ !c±y÷|e^dTe¥ ú ¥
,,ª=6 [¤ d9^ ^¤ ¤ ^ ¤ û T?üýø¥ 0c(e6p¬Ce6=ª0 d9 d94²d9e66^cC
^¯, cceA^¦ccQ
¤ 07
f0#^=ce^¤ e^^d9d9c=¤ ½4 d¥ , e^ ËÅcæ åd9å,c ¥? µ ¥ ¬ T±he6^ ^ c±w¤0
X ÷ gg0 ø u (x)u (y)z , G(x, y, z) = 1
MOLRB
∞
∗ n
n
n
n=0
a < x < b,
a < y < b,
|z| < 1.
e¥ !¤0
÷ gg0 øTeA¦cQ
eðycT cd*e^d9Te¥ =;c¶¤,=^ e6c÷ gg0 oøT c dP=6eÅ
e¥/ ¥
,ª0¢M7 d*c¤,?CcdTl( ,y ¢4c±~¨ª0 f « lim
z→1+0
Zb
ϕ(x) ∈ L2
ϕ(y)G(x, y, z)dy = ϕ(x).
÷ gg0 3iø
e¥ !ye^cc= c=·7^ þ÷ gg0 ø f cd90 ^f e^ c e¥ c c¶ ¤ ¥
¥ © ¤ ÷ gg0♦3i/ øT e¥ , 6eðy c ¢4c±f ¤ zc±If cd90 ^f0e^ c±V0 ce^f ce6 0 Éf=eCpz¥→ ¬ 1c± fô¥
¾² ¤ 66 ªQ c¬Q±h=pcf ¯¤ª d9¡VcQ¡V c e6c7
p¬67IcC
ª¶¨(c¤ d#ªQ ¤ cfªy^c¤ ^d9 l¯ ¤ 0¦ [qW:0ùg a
t·?x((òªd9eÅ ccQ¡[^ e6d l¯ 2π¤ µ¸0 ¦ ^ ¤ » cC÷ý?
= P^fe^ªf e^0c¦I ¨c?ª0µ¸ dPf c=« c=± cC ªQ
c§p¦þ ¥ µ %& c¤ ¢M'70¦þD D ,x>=!c=¶ U¤ 6 Cf0 - +[−π, d9 ^¢M70¦¶f c ^ c e¥ c7c π]^f¶¤,pC¤ T7 ^¤ c^c¤ cC
ø¥e^ ¤,=¥
ce^cc= c=·7¥µ Zπ
1 ϕ(y)G(x, y)dy = [ϕ(x + 0) + ϕ(x − 0)] 2
−π
∞ 1 1X G(x, y) = + (cos nx cos ny + sin nx sin ny). 2π π n=1
Ue^d9T e¥ = P = =+-ff0 =f
x ,#þ0
^^e6c»c? h7 T±ô 7cp ¤,= 6c±¶eð*,= e¥^c©C¦÷ cCg
g0 3 d9q?Tø¥0±*¤, =eA¤ ¦cQC
,=feð~eA¦ 7cC
c Td9c e6 c d ¹¤ ce6ª0d9d9 ¤ª^d^^cd96cC
cd½4¥ , =e^e^d9c=¤ de^ cd9c¥p¥ ¬ª0¢ ¨ª0 f « ¢ G(x, y, z) =
c[ ¤
∞ 1 1X n z (cos nx cos ny + sin nx sin ny) = + 2π π n=1 ∞ 1 1X n = + z cos n(x − y). 2π π n=1
|z| < 1
~ ¤ cCcp ¬ cd α e^ ¤,=¥
cVe^cc= c=·76 ,
∞ ∞ i X 1h 1 X n z n (eiαn + e−iαn ) = z cos nα = 1 + + 2 n=1 2 n=1 iα 1 1 − z2 ze ze−iα i = = 1+ + , 2 1 − zeiα 1 − ze−iα 2(1 − 2z cos α + z 2 )
RË}«QÑR«¾VIÍÐÒÒÔ9ÕÙÚpÛf
PÝpÙu¿Í
? =
MOLWG
÷ gg0 3 q?ø
∞
1 X n 1 − z2 + . z cos nα = 2 n=1 2(1 − 2z cos α + z 2 )
óP ¥
cp¥ ¬= c
÷ gg0 3 gø =e^e^d9c=¤ d ¤ cCcp ¬ª0¢ µ¸ ^¤ cC
^e^fª0¢ö¨ª0 f « ¢ ªQ
cp 6c¤0µ ¢M¯ª0¢ ,c=¤ 6Cf [−π, π] ªe¥ c2π dl¯ ¤ 0¦ =;°c=C,Q dþ ^¤ ϕ(x) 6 1 − z2 . 2π(1 − 2z cos(x − y) + z 2 )
G(x, y, z) =
ϕ(x, z) =
Zπ
G(x, y, z)ϕ(y)dy =
−π π−x Z
=
G(x, t + x, z)ϕ(t + x)dt =
Zπ
G(x, t + x, z)ϕ(t + x)dt.
Çc
e¥^ e^¥¬
d9 ±hc e^ cp^ ^¬=¤,C?c !?, 7
e^¬ ^¤ cQ
ce6¬¢²¨ª0 f « ± −π
−π−x
ϕ(x, z) =
Z0
ϕ(x)
G(x, y, z)
pCc¬^d
G(x, x + t, z)ϕ(x + t)dt +
−π
+
Zπ
G(x, x + t, z)ϕ(x + t)dt =
0
=
Zπ
G(x, x + t, z)ϕ(x + t) + G(x, x − t, z)ϕ(x − t) dt.
ó*ª0 6cdþ c^c[0
¨ª0 f « 0
G(x, y, z)
1 ϕ(x, z) = π
A
(cc=C,Q ^ c
Zπ 0
÷ g=g0 3 gø§ cp ,ª0 d
g(x + t)(1 − z 2 ) dt, 1 − 2z cos t + z 2
÷ gg0 3ø
1 g(x + t) = [ϕ(x + t) + ϕ(x − t)]. 2 z
Ç =d96 dT,c
,~e^¥¦ T cp 6eÅ y
,¶ ¢4c^c
1 π
Zπ
÷ gg0 3Z3 ø
e^ ¤,=¥
c 0
ε ∈]0, π[
1 − z2 dt = 1 1 − 2z cos t + z 2
1 lim z→1−0 π
Zπ ε
g(x + t)(1 − z 2 ) dt = 0, 1 − 2z cos t + z 2
÷ gg0 3 nø
× ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í pfô=fôªQf0 =¢¯fô ñ6cdºe¥ ,ª0,=C,=d9^,p¥ ¬
¤ cô cp cQ¡V¥ ^ ,7 e¥ 6 ¬e6¤ ^d9eð l( , ¢4c6cw e¥ T^¤ ^dp=f c ε ∈]0, π[ cÀ
,»e^¥¦ t ∈ [0, ε[ T cp 0 ce^¬ ^¤,p^ e6ρc > 0 ÷ gg0 3jø |g(t) − g(+0)| < ρ. bcA
[e^0 ,ª©÷ gg0 3Z3 ø&»÷ gg0 3jø&e^ ¤,=¥
e¥ ¥
,ª0¢Mphc« ^ f0 1
MOLRI
1 Zε g(t)(1 − z 2 ) dt − g(+0) < 2 π 1 − 2z cos t + z 0
1 Zε [g(t) − g(+0)](1 − z 2 ) dt = < π 1 − 2z cos t + z 2 0
=
1 π
Zε 0
? =,e^ ¤,=¥
c=« ^ f
1 ≤ρ π
|g(t) − g(+0)|(1 − z 2 ) dt ≤ 1 − 2z cos t + z 2
Zε 0
1 − z2 dt ≤ ρ, 1 − 2z cos t + z 2
1 Zε g(t)(1 − z 2 ) dt − g(+0) < ρ, 2 π 1 − 2z cos α + z
7Q= eðpc= z °4e^¢§
ôe7ª0 6cdK÷ gg0 3 nøÉc ¤ ¥
¥ ^ ©¨ª0 f « cp ,ª0 d 0
lim
Zπ
z→1−0 −π
1 G(x, y, z)ϕ(y)dy = [ϕ(x + 0) + ϕ(x − 0)], 2
G(x, y, z)
÷ gg0 3@:ø
cô¾º¤ ¤,^?6c=
¥? ce^¬ó§ ^c« f,p?Q p¬¬ T¨ª0 f « ¢7d9( cd»=0
4 cf0p¡^d©e^ ¤,=¥
0µ d9ce6p♦ ¬hªe¥ c~»¨ ª0cp f « c= ±ô£á¯÷ =g^g0^ ¤ o¤,ø4
,» cp cd9c~áI6¡=0
¤, P (x) ¨ª0 f « ± C ¤0µ Un (x)
w°±
n
Iν+n,n (x)
çy+- !y+ %&+ Ò# $Ê,
*
%&D> N, D, Z['Y @D N=@H >?x>=,Rpx¥:=Zè=,GPQ\0D>R¸DY6`=@Y¸@EPRpD,E[OD,_N,N,F Zx
f (x) dx =
Zx
x
x
f (x)
(>CB
Z ∞ Z X a0 πnx πnx dx + dx + bn sin dx . an cos 2 l l n=1
D>É`>Q\dY6D ÷ g=úq?ø
U cp cQ¡V [ [+- ^d
xp¾§ce^ cp ¬=^ª^d9eðIcc=7^ Td~ª0¤,= ^ ^d~Q=d9f ª ce6¶÷ gg0ùg=oø¥ 1, [x , x]; ÷ g=ùgø g(x) = x0
x0
x0
x0
0
0, [−l, l] \ [x0 , x].
Y©=Ë+éòÒtQÍÐÜAÙ0Ö=ÙÑ¥Ø=ß^ÒÖ Í ðÙ0ÖÜCÑ¥Ò Ñ6ÞÍ]ðÙÖÓ=ÍÅÎÐÏ ÖZ7ÙÚpÛÑ¥Ø&
PÝpÙu¿Í
MOLRL
bcA
1 l
Zl
1 f (x)g(x) dx = l
Zx
∞
a0 α 0 X f (x) dx = (an αn + bn βn ). + 2 n=1
°4e^¢§
eª0 6cd¨(c¤ d#ªQ ~
,~f cñ^¨(¨( « ^c¶®¯ª0¤ ¬(¨ª0 f « t÷ g=ùgø x0
−l
1 α0 = l
Zl
1 g(x) dx = l
Zl
πnx 1 g(x) cos dx = l l
cos
−l
Zx
x0
Zl
1 πnx dx = l l
Zx
βn =
¤ 0¦cC
df~¤,=6 e6=ª Zx
f (x) dx =
1 l
sin
dx,
x0
−l
1 αn = l
Zx
g(x) sin
πnx dx l
x0
−l
Zx
πnx dx, l
Zx ∞ Zx X πnx a0 πnx dx + dx + bn sin dx , an cos 2 l l n=1
e^c,?
=¢M76d9ª(e÷ g=úq?ø¥6l¯cf0pQp¥ ¬e6ce^cc= c=·7^ ô÷ g=úq?ø; ¤ c¥
^ cMcp ¬f c4, ce^ cª0¤,= ^ ôQ=d9f ª ce6 6¯ e^ c? ¬=Cc= h ¤ ¥
cp cQ¡^ ±hc¦ =¤,=f^¤ eA¦cQ
d9ce¥©¤ 0
÷ gg0ùg=ø¥ ¹cñ6cd#ª!d9c?¡V cVª ^¤ ¡[
p¬ .c~¤0
*÷ g=úq?øMeA¦ cC
eð 6Q=¹ e^¤ cd9d9c6¡c=ª eAc¦ fcC
d9ce6Twª e¥ eA¦ccC
(c^^cVc¤ ¤0^d9
h©÷d9gc?g0¡Vùg= c§ø¥Q=d9c^ ô¤ ¬M^ ¤ ccd9? 6¡ce^ª ¬
f ccf0d pQp¬ m¤ ¬¤ =c♦,d9¤,¯=e6e^cdP^pc ;¤ ^c=¤ ^^d9dP[−l, §ô¦hcl]e6,p[=6 ¤ eÅcd9e^6 ¡¤,ª =f ¥
c±w
,!e^ ªe¥µMf ce^ ªe¥µ¸¤ 0
ch[0,®¯2l]ªµ [0, l] <~ ' ( D H >x =,x(¹cfpQp¬ cy
,w ¢4c±=e^cp ¢M cô ^^¤ ¤ ªCd9c±,¶ ¤ cd9¥µ ¡ª f [−l, l] ¨ª0 f « f (x) ¤0
Xb ÷ g= ø , n A
b 4 f cñ^¨(¨( « ^ ®¯ª0¤ ¬(ñ6c±h¨ª0 f « eA¦cC
eÅ cC@
h9C ,=f c dþxT¾´e6ª0d9¨(d9c¤ d#,ªC cp» c?÷ ¡Vg= úq?ø¯T cp dÀ ^^¤ ¤ c= hT¤,p¡^ ± Pe6cQ70¦ x =0 Zh i ÷ g= 3 ø l Xh a πnx b πnx a i dx = cos sin − −1 . f (x) − x0
x0
x0
x0
∞
n
n=1
n
0
x
∞
2
π
e¥ ~^e6hcc=C,Q ^ 0
/
2l π
∞ X n=1
n
n=1
F (x) = A0 =
n
n
0
Zx h 0
bn , n
l
n
a0 i f (x) − dx, 2
An = −
lbn , πn
Bn = −
l
÷ g=ùnø lan , πn
1
SRNRN
cw÷ g= 3 ø&d9c?¡V c7Q= eCp¬
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í
∞ A0 X πnx πnx F (x) = . An cos + + Bn sin 2 l l n=1
ó§cA =e^ c^c¤ ^d9g=úq ¤ 0
÷ g= jø§eA¦ cC
eðwf~¨ª0 f « ¨( « ^ A f c ^ ^h ,e¥ ¥
cp¥ ¬ c
F (x)
÷ g= jø c[cA
f cñ^¨µ
0
÷ g=0;:ø c=ñ6cf^ªCc~
!¤0t
òl(^ ,f0=6ñ6ceð^¦c~cC
¤ 0d9
tc®¯e6ª0¬¤ ¬¤h0
÷ g=÷ jg=øM ø¥¤ #c Ç =d9^6^¤ ,dT¤ª^d c,d9¶cQ ¡V cw^¤ ,=?± e6ª0d9d#ª [−l, l] cp ,ª0 d ∞
2l X bn = A0 < ∞, π n=1 n
Zl
F (x)dx =
Zl
b#=f dºc¤,pCcdT
−l
−l
Zl Zl ∞ X A0 πnx πnx dx + dx + Bn sin dx = A0 l. An cos 2 l l n=1 −l
−l
1 A0 = l
e^cc=6e6^ ,c y÷ g=;:ø§e¥ ¥
,ª6
÷ g= oø
F (x)dx
−l
∞ X bn n=1
Zl
π = 2 n 2l
Zl
÷ g= ø
F (x) dx.
®, (c¤ d#¤ ªQ ô^¥÷ pg=~ f~oøPe6 ª0cpd9Cd9 cp ,¤ c 6[= ¤ ¢ö¶÷ g=;:^^ø¥¤ ¤,c== ~¤0
7,?¦cC
¬[f cñ^¨(¨( « ^ 7 A <~ ' ( D H Ex D x((=±I¤,p6 cQ¡[^ 9¨ª0 f « f (x) = x ^Q?
= c±(,T ¤ cd96¡ª f , [¤ 0
!®¯ª0¤ ¬= [−π, π] ^@ w9 e^^ cdKc=f c6xñ^¨(b#e6=¨(f = cª« T¢M ¤,7^p6¦w cc? ¡a^ ^ ^¤ÉbQ, P=cb#±0¶
=^ f ce^cI ±þcp( e^ ¬=c^¤ e^ª c6d9©^eŤ h I6
ªQqQ=
ª0c f ¤,^ þp^ c^ce6dT7¤ &¥
e^c»^^^ ¤ ,¤ ¤,TcdtTp= T e¥ e¥¥c µµ , ¤ =e6 Qd9 ^dT¤ 4/ q?e¥g0 tCc4ce^ e^cpf0 c¬=d9Ccc&¤,pp6¬ eðtcQ¡[^C = ^e6&d9,cQT¡[d ¥4¤,pò6 cQ¬¡,^= ±0
^^dö c4÷ g^g0^c4ùg= ø¥^ ^cp¤, , ª0¤ c^ = T d ^dT l¯^± e6¥ ¬ c c^c¤ ^d97g=úq( d9^^d −l
0
2
n
0
Zx 0
n
x
x dx =
X (−1)n+1 2 Z n
sin nx dx
0
∞
A0 X x2 = + An cos nx + Bn sin nx . F (x) = 2 2 n=1
÷ g=úqQiø
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PÝpÙu¿Í
(¨/ e¥c ¤ ºd#ªCª0 =^de6÷¬ g=&ùnø§c»,=±0¤,
p^6d c?¡^ ÷ gg0ùg=ø mcñ^¨(¨( « ^
An = −
A0
SRNZM
a0 = an = 0 bn = (−1)n+1 2/n
(−1)n 2 (−1)n+1 2 = , n2 n2
9c© c
Bn = 0.
d9cQ¡V c[,=±h0 ~ c[¨(c¤ d#ªQ V÷ g=;:ø A0 = 2
∞ X (−1)n+1 2
n2
=4
If c=c¤ c±~eª0 6cdC^e6 c±he6ª0d9d9 n=1
∞ X (−1)n+1
÷|e^dT, ¤ d9^¤qQ ø§e¥ ¥
,ª6
n=1
A0 = 4
0 ~ c[¨(c¤ d#ªQ V÷ g= oø 1 A0 = π
n2
Zπ
∞ X (−1)n+1 n=1
=
n2
,
π2 12
π2 π2 = , 12 3
1 F (x) dx = π
Zπ
x2 x3 π π2 dx = = . 2 6π −π 3
¾ f0c ^ cdwc^& cQ
e¥== cf0Me^¥¦I,=±0
^ §¦7f cñ^¨(¨( « ^c7÷ g=úqQiø
=6 ¤,p6 cQ¡^ −π
−π
∞
π 2 X (−1)n 2 x2 = + cos nx 2 6 n2 n=1
0
x2 =
∞ X π2 (−1)n cos nx, +4 2 3 n n=1
e^c,?
=¢M76Ie( cp ,ª0 ^ Td*[ ¤ d9^¤ qQ ¤,p6 cQ¡^ ^dT <~ ' ( D H Ex H x ?eð¦cC
,~IC^e6 c^cy, cp ,ª0 ^¤ cC
[0, π] ¤,p6 c?¡^ ∞ X nπ 1 −x/2, 0 ≤ x < π/2; cos sin nx = (π − x)/2, π/2 < x ≤ π, n 2 n=1
,=±~e6ª0d9d#ªô e¥ cc^c¤0
ôe6ª0d9d#ªô¤0
V®¯ª0¤ ¬
∞ X 1 nπ cos 2 n 2 n=1 ∞ X 1 nπ cos sin nx. 3 n 2 n=1
÷ g=úqq?ø ÷ g=úq?gø ÷ g=úqQø
1 ×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í SRNRS @9 x0#0
V®¯ª0¤ ¬(÷ g=úq=q?ø. ¤ c ^^¤ ,¤ª^d4 ¤ ¥
¥ ?¦c=ªQ ,I
c ^f c=c¤ c^c x < π/2
~ cp ,ª0 d Zx 0
=−
0
Zx ∞ ∞ X X 1 1 nπ nπ cos sin nx = cos sin nx dx = dx n 2 n 2 n=1 n=1 0
∞ X n=1
∞ x X 1 nπ nπ 1 cos cos nx = − cos (cos nx − 1) = 2 2 n 2 n 2 0 n=1 Zx x2 x2 x x dx = − = − − = 2 4 0 4 0
÷ g=úqV3 ø e¥ cC c^cw¤0
»÷ g=úq?gø¥.¤,=^ e6c÷ g=úqV3 ø ¤ c ^^¤ 0µ ¤ª^dþi,c[K ¤ ,c=d9±6¡»ª e6ª0f d9 d#[0,ªπ/2] c
=6 0
∞ ∞ X x2 X 1 1 nπ nπ = cos cos nx − cos . 2 2 4 n 2 n 2 n=1 n=1
Zπ/2 0
∞
X 1 nπ x2 dx = cos 2 4 n 2 n=1
Zπ/2 Zπ/2 ∞ X 1 nπ cos nx dx − dx cos 2 n 2 n=1 0
0
∞ ∞ nπ nπ π3 X 1 nπ π X 1 cos cos = sin − , 3 2 96 n=1 n 2 2 2 n=1 n 2
c=fªC
eª0 6cde^cc= c=·7^
cos
d9^^d
nπ 1 nπ sin = sin nπ = 0 2 2 2
÷ g=úq?nø ic ,=±e6ª0d9d#ª¤0
h®¯ª0¤ ¬h÷ g=úqQø¥ ¤,=^ e6c»÷ g=úqV3 øM ¤ c ^^¤, ¤ª^d 67(¤,pI,[ ¤ cd96¡ª f [0, x < π/2] ,bcA
0
∞ X 1 nπ π2 cos = − . n2 2 48 n=1
Zx 0
∞
X 1 nπ x2 cos dx = 4 n2 2 n=1
Zx 0
Zx ∞ X nπ 1 cos nx dx − cos dx 2 n 2 n=1 0
∞ ∞ X nπ nπ x3 X 1 1 cos cos = sin nx − x . 3 2 12 n=1 n 2 n 2 n=1
¹cC
e6p= ¨(c¤ d#ªQ ,ª÷ g=úq?nø§[ cp ,ª0 ^ cI¤,=^ e6c ,p±0
^d ∞ X nπ x3 xπ 2 1 cos sin nx = − , 3 n 2 12 48 n=1
0<x<
π . 2
÷ g=úqQjø
Y©=Ë+éòÒtQÍÐÜAÙ0Ö=ÙÑ¥Ø=ß^ÒÖ Í ðÙ0ÖÜCÑ¥Ò Ñ6ÞÍ]ðÙÖÓ=ÍÅÎÐÏ ÖZ7ÙÚpÛÑ¥Ø&
PÝpÙu¿Í icõ,p±*e6ª0d9d#ª ÷ g=úqQø7© ^¤ ?
¤ c ^^¤ ¤ª^d*[ ¤ cd96¡ª f Ic=
c x > π/2
SRNQà
π/2 < x ≤ π x=π
P¤0
®¯ª0¤ ¬t÷ g=úqq?ø
Zπ ∞ ∞ π X X 1 1 nπ nπ cos cos cos nx = sin nx dx = − 2 n 2 n 2 x n=1 n=1 x
0
∞ X 1 nπ =− cos (cos nx − cos nx) = 2 n 2 n=1 Zπ (π − x)2 (π − x)2 π π−x dx = − = =− 2 4 4 x x
÷ g=úqW:ø
∞ ∞ X nπ nπ (π − x)2 X 1 1 cos cos = cos nx − cos nπ. 2 2 4 n 2 n 2 n=1 n=1
¹cp cQ¡V h÷ gpúqW:ø
x = π/2
cp ,ª0 de^cc= c=·7^
∞
÷ g=úqQoø
∞
X 1 nπ π2 X 1 2 nπ cos cos = − cos nπ. 2 2 16 n=1 n 2 n 2 n=1
¹¤ ^c¤,p^ª^dþ ^¤ =ª0¢ e6ª0d9d#ªô[ ¤,=c±ô,=e6t÷ g=úqQoø§e¥ ¥
,ª0¢M7 d*c¤,pCcdT ∞ ∞ ∞ ∞ X X 1 1 1 + cos nπ X 1 1 + (−1)n X 1 2 nπ cos . = = = n2 2 n2 2 n2 2 (2k)2 n=1 n=1 n=1 k=1
°4e^¢§
eª0 6cd¼÷ÐqQoúq?gø§,=±0
^d
÷ g=úqQø
∞ ∞ X X 1 π2 1 2 nπ cos = = . 2 2 n 2 2k 24 n=1 k=1
¹cC
e6p= w÷ g=úqQø§h÷ g=úqQoø¥ cp ,ª0 d
÷ g=ùg=iø
∞ X nπ π2 π2 π2 1 cos cos nπ = − = − . n2 2 24 16 48 n=1
¾*e^c¢¼c ^¤ ¥
¬ , cC
e6== cf0h÷ g=ùg=iø§h÷ g=úqW:ø& ¤ cQ
^^cf~0
,ª ∞
nπ (π − x)2 X 1 π2 cos = cos nx + , 4 n2 2 48 n=1
π < x ≤ π. 2
¹¤ c ^^¤ ¤ c=¶ cp ,ª0 ^ cT¤,p¡^ 7,V ¤ cd96¡ª f c= cp ,ª0 d 0
Zπ x
∞
X 1 (π − x)2 nπ cos dx = 4 n2 2 n=1
Zπ
π2 cos nx dx + 48
x
∞ X nπ (x − π)3 π 2 1 cos sin nx = − (x − π), 3 n 2 12 48 n=1
x > π/2 Zπ
÷ g=ùg0q?ø
c
x=π
dx
x
π < x ≤ π. 2
÷ g=ùggø
SRN 0
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í 1
°=³¥
h÷ g=úqQjø§©÷ g=ùggø¥ d9cQ¡V cQ= eCp¬
3 xπ 2 π x ∞ − , 0≤x< ; X 1 nπ 12 48 2 cos sin nx = 2 3 3 n 2 π (x − π) π (x − π) n=1 − , < x ≤ π. 12 48 2
÷ g=ùg=ø
2 π2 π x ∞ − , 0≤x< ; X nπ 1 4 48 2 cos cos nx = 2 2 2 n 2 (x − π) π π n=1 − , < x ≤ π, 4 48 2
÷ g=ùgY3 ø
Ç =d96 dT,cd9 cª cªe6=p c0 w¤,=^ e6c
f c=c¤ c( cp ,ª0,=6eÅhc=³¥
^ ^d÷ g=úqV3 ø¥÷ g=úq?nø§©÷ g=ùg0q?ø¥ w À
ß#>ºyed !y+ %&+ Ò# $Ê,
*
&% 'DtJx>=,x¥è,Q\0D>R
:=(Y¥B³Z_ Y [−l, l] OD,_N,N,F f (x)GPZ[:=_Nu0G&\0Z[> f (−l) = f (l)G-(>CB D>[`>Q\dY6D,D>?Rp:=Z03 U¤,p6 cQ¡ ^ +-
x / e¥ [ ¤ ¥
cp cQ¡^ ò^c¤ ^d9ºe^ ¤,=¥
c pcd9 d9^^dw
∞ πnx a0 X πnx an cos f (x) = , + + bn sin 2 l l n=1 ∞ X πn πnx πn πnx 0 f (x) = . bn cos − an sin l l l l n=1
÷ gY3 úq?ø
÷ gY3 ùgø eóf0
c¤ñ^ª0¨(^¨(c±¶ « e6 c^¤ pc= d9
a ,y b¨ª0 f « f (x) d9cQ¡V cIQ= e^p¬e^ce6^ c¤,p6 cQ¡^ X ÷ Yg 3 ø a πnx πnx f (x) = , + + b sin a cos 2 l l f c=c¤ TIc ¤ ¥
¥ , ¢Meðwe6p=0
=¤ Édc¤,pCcdº c¨ª0 f « f (x) 0
0 n
0 n
∞
0 0
0
0 n
0 n
n=1
0
1 a00 = l
Zl
f 0 (x) dx,
−l
1 a0n = l
Zl
f 0 (x) cos
πnx dx, l
−l
b0n =
1 l
Zl
f 0 (x) sin
πnx dx. l
¾§T e¥ ^ ¯ ^¤ c^c ^^¤,? eª0 6cdþªe¥ c ±~^c¤ 6dP
=6 −l
1 a00 = l
Zl
−l
l 1 1 f 0 (x) dx = f (x) = [f (l) − f (−l)] = 0. l l −l
÷ gY3 3 ø
4Ë ÞÖD££ÉÍ ÙÍÐÒ[Ö=ÙÑ¥Ø=ß6ÒÖ Í ðÙ0ÖÜCÑ¥Ò Ñ6ÞÍ]ðÙ0ÖÓ=ÍÅÎÐÏÖr¯ÙÚpÛÑ¥Ø&
PÝpÙu¿Í l¯ ceÅ ¥
0¦h ^^¤,? T e¥ d c[,=e6? dT
¼
a0n
SRNR7
Zl πn 1 πnx l πnx = f (x) cos f (x) sin dx = + l l −l l l −l
πn 1 = [f (l) − f (−l)] cos πn + 2 l l
Zl
f (x) sin
πnx dx = l
−l
Zl πn 1 πnx πn = f (x) sin dx = bn , l l l l
÷ gY3 ùnø
−l
Zl l 1 πnx πn πnx b0n = f (x) sin dx = f (x) cos − l l −l l l
−l
Zl πnx πn πn 1 f (x) cos dx = − an . =− l l l l
÷ gY3 jø
=6V, ¯^ e^¤ ¢ ^¤ T e¥ ccI=ª0 ¢ ^¤ c e^cC¬
,¥ µ ^e f c4¾* e^¤ 0cQ ,
ªycp ,ª¡e¥ ^c ¨fª0(−l) f « = ff(l)(x) f0ec= ¤ c¤ cd9c6¡cª ^e^f0 ^ [−l, f cñ^¨(¨( « ^c a a b y÷ gY3 3 ø 4 ÷ gY3 jø& cp ,ª0 d l] ÷ Yg 3 ; :ø πn πn a = 0, a = b , b =− a . l l ¾*¤ 6^ªQ ¬Qpp¯¤0
K÷ Yg 3 ùgø§»÷ Yg 3 ø&e^c,?
=¢M? b#=f dºc¤,pCcdT d9 cfpQ? ,c
,~c ¤ ¥
¥ ^ §¦hf0 =e^e^c¨ª0 f « ± 0e6¦¼=ª0 ¢M¤ 7c0C¦hcC
dþ T¤0º
d9cVcQ¡V®¯ ª0c*¤ ¬ =cp ,ª0 ¬ c0 ^ Td
¨(¨(^¤ ^ « ¤ c= ^d e^cc=f(x) 6pµ c0♦ ^¹ cCc
^ cþ^¤
f ¨(^dT¨(,^¤ ^c[ « ^¤ cc¤ ^=dP yYg3 ú¤ q(0¤,
=*e^f ®¯¤ ª0ò¤ ¬==64y c¤ »« 6= ¤,,=? ¬ ¤ª0ª¢ 6c=eAC¦ d9cCcQ
¡V d9 cce6e6 ¬ ¤ cC¥
= ¤,¨(=¨(^ ¤ ¤^ª 6« ~ C¤ ,c=f=y ¤,c=^c^¤ 0e6
4ôf7¶ ¤ Tc¤,Cp¡cC
^ c±þf÷ gY(x) Ql¯¤ª0^ d9e¥ c=d9 ?^c¤ 6d& 3 ùgø¥l( ,wªe6p= cp ^ ©eA¦cQ
0µ d9ce6»¤0
»÷ Yg 3 ùgøf f (x) e¥ Å
,ª6!ce^ cp ¬=Ccp¬eð»e^¨(c¤ d#ªC ¤ c= Td9t¤,= ^
¤ ¨(C¨(,=^¤ f0^= d9« 7 e𤠦ccC
p d9 ce6¤,7=l¯ c d97^¤ ( l¯cþ eA¤ ¦0cQ¦
, T7=0,¦?eð c^¤ 0
cc4 Mcd#=ª= f0c=d9f7^ñ¥¤ c4cþ
eA¥¦ cC=
,67eð ± ¤ eð ¤0
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2k − 1
.
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n X sin(2k − 1)x
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k=1
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
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h sin 3x sin(2n − 1)x i SR7 . = 2 sin x + + ··· + 3 2n − 1
ê MON
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S2n−1 (x) = 2
Zx 0
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sin 2nt dt. sin t
0
hZx sin 2nt i0 sin 2nx dt = =0 [S2n−1 (x)] = sin t sin x 0
0
@Ù ÒÛ@ÖRäâ@Ò×uäáRàWÒÙrÙ@ÖäÜuÒ×@åYÒáÚræráWÖ¸äåZÓÓAÐ S (x) Ú@Ö?0VâuÐÕÐçräv8 à ÙråbÜ8 à áWÖVærÏdÒ x = 0 ÚrØ@ÓÒ]Òá 2n−1 G]ÏdäáRâ@Ò]ÓåZÓÞ³àÍáWÖVærÏZÐbã (n) Ú ê p±Òv3]ÏZÖÍåYäáVÐÙ@ÖVàRØrá?8@Ú æráWÖáWÖVærÏrØ ÛrârØ Ø ÓAÐÏZäØ@m ÓåZ= ÓAÐZ1,ÚtÐnÛrârØæ@ÒáRÙrÞã m Å áWÖVærÏrÐVÓØ ÓØrÙrØ@ÓxåZmÓAÐZê Ù@Òæ@ÒáRÙrÞã m çZàVÜ@+ç u áWäçÍxáWmÖVærÏZ=ÐVÓmπ/2n ×@âr!å 3OÖVßäáWÖVâ@ÖVÙrÞ±ÚrÝbÐVÓÒÙ@ÖVß 2nt = τ æuÐVäáRØrærÙrå!u äåZÓÓåq SR7 ê MRM ÓÖbÑÙ@ÖÛrâ@Ò×uäáRÐàRØrá?8¸ÏrÐÏ S2n−1 (x) =
Zx
1 sin 2nt dt = sin t 2n
0
ØZÜ@Ø
Z2nx
SR7 ê MOS
SR7 ê M¥à
SR7 ê M 0
sin τ dτ sin τ /(2n)
0
1 S2n−1 (x) = 2n
Z2nx
sin τ dτ = sin τ /(2n)
0
1 = 2n ··· +
3
×uÒ
Zπ
sin τ dτ + sin τ /(2n)
0 Zkπ
r ÒÜtÐç£æuÐVäá? 8 ×@â@Ö?V0 Ø ê Å u . k Ü@çÛrâ@ÖVØrÝ]àWÖÜ8RÙ@Ö43OÖ ØrÙráW2nx/π Òv3]âuÐbÜÐØrÝ¥ SR7 (l+1)π Z
π
sin τ dτ + sin τ /(2n)
sin τ (−1)l dτ = sin τ /(2n) 2n
sin τ dτ + . . . sin τ /(2n)
Z2nx
kπ
(k−1)π
1 2n
Z2π
sin τ dτ , sin τ /(2n)
ê M¥à Ø@ÓÒ]Ò]Ó
Zπ
sin u du = (−1)l Sl , sin(u + lπ)/(2n)
0
lπ
ÖVáRÏråb×ÐäÜÒ×uåWÒáVÚræráWÖ Sl > 0,
.Ü@çÛ@ÖRäÜuÒ×@Ù@Òv3OÖ ØrÙráWÒv3]âuÐbÜÐØrÝ SR7 1 2n
Z2nx
kπ
ê M¥à
r× Ü@ç l ≤ k − 1. ÓÖbÑÒ]ÓÝbÐÛrØ@äOÐá?8
Sl > Sl+1
sin τ dτ = (−1)k S¯k , sin τ /(2n)
SR7 ê MO7
SR7 ê MOB
$ T=Ë+¢ß6ØQÒ Ñ6ÞÍ Ù0Òß6Ú¶Î¯Ñ Û0Ö ÞÑQÎ]O¿ðÙ0ÖÜCÑ¥Ò Ñ6ÞÍ]ðÙÖÓ=ÍÅÎÐÏ ÖZ7ÙÚpÛÑ¥Ø&
PÝpÙu¿Í SZMRM ÛrârØræ@Ò]Ó S¯ < S ê 7"â@ÒÝåbk Ü 8áRÐáWkÒ æÐVäáRØ@ærÙr!å uyäåZÓÓå S (x) ÓÖOÑÙ@ÖÛrâ@Ò×uäáRÐàRØrá?8£ààRØZ×uÒ Ý]ÙuÐÏZÖVæ@Òâ@Ò×@å / 2n−1 u ÕvÒ 3OÖRä
çâYçd×Ð SR7 ê MbG? S2n−1 (x) = S0 − S1 + · · · + (−1)k−1 Sk−1 + (−1)k S¯k å+0VÞàY Ð u ÕØZãÚä]4Ö 3
ÜÐVäÙ@ Ö SR7 ê MO7 ÚAäÜ Ð 3ÐVÒ]ÓÞãê 7³áWÖVærÏZÐbã x = x(n) = mπ/2n r@ÒÜÐçæuÐVä?á 8 ×@â@?Ö 0VØ 2nx/π ä]ÖVàRÛuÐb×tÐVÒáä m ÚZÛ@ÖRäÏdÖÜ 8RÏrå 2nx(n) /π = m Ø ÚräÜuÒ×umÖVàYÐáWÒÜ 8RÙ@ÖrÚuäåZÓÓAx Ð SR7 ê Mb?G m à G]áRØZã£áWÖVærÏrÐbãÛrârØrÙrØ@ÓAÐVÒá¸Ý]ÙuÐ]æ@ÒÙrØrç * SR7 ê MOI (n) S2n−1 (xm ) = S0 − S1 + · · · + (−1)m Sm . .Ü@ç£Ùu Ð 3Ü@çd×@Ù@ÖRäáRØàRÞÛrØréÒ]ÓÙ@Ò]äÏZÖ Ü 8RÏdÖ Û@ÒâràRÞx ã G]ÏZäáRâuÒÓAÐbÜ 8RÙrÞãÍÝ]ÙuÐ]æ@ÒÙrØrß S (x(n) ) w m
2n−1
(n)
S2n−1 (x1 ) = S0 , (n)
(n)
S2n−1 (x2 ) = S0 − S1 ,
S2n−1 (x3 ) = S0 − S1 + S2 ,
...
5ÊáVÐÏ ÚWÙ@Ò(0RÖÜ8RéÖRÒ¦Ø@ä]äÜÒ×ÖVàYÐÙrØ@Ò¦Û@ÖVÏrÐÝ]ÞàYÐVÒáÚdæráWÖ±ÛrârØ¥ØrÏ
ä Ørâ@ÖVàYÐÙrÙ@ÖRÓ n ØØrÝOÓÒÙ@ÒÙrØrØ x ÖVáÙråbÜ@ç£×uÖ π/2 äåZÓÓAÐ S (x)/ àWÒ×uÒáä]Ò(0bçäÜuÒ×@å!u ÕØ@Ó Ö?0VâuÐÝOÖRÓ´wuâuÐàRÙuÐçÍÙråbÜu à áWÖVærÏZÒ 2n−1 Ú ÖVÙuУâ@ÒÝ]ÏZÖàWÖVÝ]âuÐVäáRÐVÒá×uÖÍäàWÖRÒv3OÖÓAÐÏdäØ@ÓAÐbÜ8RÙ@Ö?3OÖÝ]ÙuÐ]æ@ÒÙrxØrç =S0 à£áWÖVærÏZÒ x(n) = π/2n Ø ÝbÐáWÒ]ÓÙÐ]ærØrÙuÐVÒáÍÏZÖÜu(Ò 0WÐ?á 8Wä
çÚ ÛrârØ@?Ö 0Vâ@Ò 0/ áVÐç à áWÖVærÏr1Ðbã x(n) = mπ/2n G]ÏZäáRâ@Ò]ÓAÐbÜ 8RÙrÞÒ Ý]ÙuÐOæ@ÒÙrØrçÚYÛrârØræ@Ò]Ó ÓAÐÏdäØ@ÓAÐbÜ 8RÙrÞ¦Ò±mÝ]ÙuÐ]æ@ÒÙrØrçÍ+å 0VÞàY Ð u áÚuÐÓØrÙrØ@ÓAÐbÜ 8RÙrÞÒ±àWÖVÝ]âuÐVäáRÐ / 9 e=gY3 u áÛ@Ö ÓÒâ@Ò¸Ûrâr Ø 0Ü@ØrÑÒÙrØrç x Ï π/2 ê ƱРÖVáRâ@ÒÝÏZÒ [π/2, π] ÏZÐâ / áRØrÙuÐ àR´Þ 3
Ü@çd×@Ørá£ÏZÐÏ ÝOÒârÏrÐbÜ 8RÙ@ÖRÒÖVáW?Ö 0VâuÐÑÒÙrØ@ÒWDê ƱРÖVáRâ@ÒÝ]ÏZÒ ÑÒ ÏZÐâráRØrÙuÐÙ@Òæ@ÒáRÙrÞÓÔ?Ö 0VâuÐÝOÖRÓgÛ@ÖVàRáWÖVâdçrÒáWä
çÍäÊÖVáRâ@ÒÝ]ÏrÐ [0, π] ê+½Ø@äRê S 0ØZÜrÜuÊäáRârØrâråWÒá [−π, äÏZÐÝbÐ0]ÙrÙ@ÖRÒ±àRÞéÒÊ×rÜ@ç n = 6 ê ½ÐVä]ä]ÓÖVáRârØuÓyáWÒÛ@Òâ 8òÛ@ÖVàWÒ×uÒÙrØuÒÍæuÐVäáRØrærÙ@ÖVßäåZÓÓÞ Ø Ò]x Ò G]ÏZäáRâ@Ò]ÓåZÓÖVàÛrârØ n → ∞ ê ƱÐ]ærÙ@Ò]Ó³äÛ@ÒâràW4Ö 3OÖ Å ÙuÐ Ø 0RÖÜ 8RévÒ 3OÖ ÓAÐÏdäØ@ÓåZÓAÐ µ(n) àáWÖVærÏZÒ x(n) = π/2n ÚÝ]ÙuÐ]æ@ÒÙrØ@Ò 1 1 ÏdÖVáWÖVâ@4Ö 3OÖrÚuä]4Ö 3
ÜÐVäÙ@ Ö SR7 ê MOS Ú@âuÐàRÙ@Ö Å (n) µ1
=
(n) S2n−1 (x1 )
.Ü@ç£àRÞærØ@äÜuÒÙrØrçÍÛrâ@Ò×uÒÜÐ
(n)
µ1
ÛrârØ
(n) µ1
ØáWÖ43
×tÐ
Zπ
SR7 ê MOL
SR7 ê SRN
SR7 ê SZM
SR7 ê SRS
sin τ dτ. sin τ /(2n)
0
n→∞ =
Zπ
ØrÙráWÒv3]âuÐbÜf SR7 ê MOL Ûrâ@Ò×uäáRÐàRØ@ÓÔààRØZ×uÒ
sin τ τ /(2n) dτ τ sin τ /(2n)
0
(n) lim µ n→∞ 1
È
1 = S2n−1 (π/2n) = 2n
×uÒ]äv8¸äØ@ÓàWÖÜuÖRÓ
ï$ð (x)
=
Zπ
sin τ τ /(2n) dτ = τ sin τ /(2n)
Zπ
sin τ dτ = ï$ð (π) = µ1 . τ
Ö?0RÖVÝ]0ÙuÐ]æ@ÒÙuÐ äÛ@ÒvrrØuÐbÜ8RÙuÐç@0ådÙrÏrrØrç
ï$ð (x) =
Zx
sin τ dτ, τ
ï$ð (0) = 0,
0
ï$ð (∞) =
π , 2
ÙuÐÝ]ÞàYÐVÒ]ÓAÐç ØrÙ@áWÒ3]âuÐbÜ8RÙrÞÓ¥äØrÙråWä]ÖRÓòñ Mvó ê 7 â@ÒÝ]åbÜ8báRÐáWÒ ÓÖbÑÒ]Ó"äÏrÐÝbÐá?8@ÚæráWÖÝ]ÙuÐ]æ@ÒÙrØ@Ò à áWÖVærÏdÒ Û@ÒâràWÖ43OÖÓAÐÏZäØ@ÓåZÓAÐ ÛrârØ n → ∞ ÖVáRÙu×8Ù@Ò äáRâ@Ò]ÓØráWä
ç£ÏÙråbÜuÚZбâuÐàRÙ@ÖÝ]ÙuÐOæ@ÒÙrØu ØrÙráWvÒ 3]âuÐbÜ 8RÙ@Ö43OÖ¸äØrÙråYäOÐàáWÖVærÏZÒ x = π ÚZáVê ÒRêrÛrârØ0Ü@ØrÑÒÙrÙ@Ö SR7 ê QS à π µ = ï$ð (π) ≈ + 0,281, 1
2
SZMOS Uä]Óê ÚrÙuÐÛrârØ@ÓÒâ ÚÛñ Mó¼ ê sÖRäÜuÒ×@!å u ÕØ@ÒÝ]ÙuÐOæ@ÒÙrØrç
1 (n)
µm
ÛrârØ
n→∞
à áWÖVærÏZÐbãxG]ÏZäáRâ@Ò]ÓåZÓAÐØ@ÓÒvu á Ûrâ@Ò×uÒÜ@Þ
(n) = ï$ð (mπ), µm = lim µm n→∞
sÖRäÏdÖÜ8RÏråÝ]ÙuÐOæ@ÒÙrØrç
(n)
äáRØ
µm
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í
àáWÖVærÏrÐbãàWä]Òã¥G]ÏdäáRâ@Ò]ÓåZÓÖVà ÏdÖÜuÒ(0Üu áWäçÖVÏdÖÜuÖ νm = µ m −
π , 2
π/2
×rÜ@ç£ÏZÖVáWÖVâ@ÖVß
ν2 ≈ −0,153,
ν3 ≈ 0,104,
ν4 ≈ −0,073,
...,
|νm | > |νm+1 |, π π lim |νm | = lim ï$ð (mπ) − = S(∞) − = 0. m→∞ m→∞ 2 2
ÚdáWÖâuÐÝ]Ù@Ö
m = 1, ∞,
Ö?0VâuÐÝ]å!u áÝ]ÙuÐÏZÖVæ@Òâ@Ò×@å!u Õå!uÊäç£ÛuÖRäÜÒ×ÖVàYÐáWÒÜ8RÙ@ÖWäá?8&Uä]ÓêráRÐ40Ü@Ørrå£Ý]ÙuÐ]æ@ÒÙrØrß ν1 ≈ 0,251,
SR7 ê S 0
m = 2, ∞.
ï$ð (x)
àhñ Mvóô
SR7 ê SR7
/
SR7 ê SRB
2 ÐÏrØ@Ó?Ö 0VâuÐÝOÖRÓÚràWä]ÒÊäÏrÐÝbÐÙrÙ@ÖRÒàRÞéÒ Û@ÖVÝ]àWÖÜ@çrÒá¸Ö]ãZÐâuÐÏráWÒ râ ØrÝOÖVàYÐá?8gäãdÖO×@Ø@ÓÖRäá?8æuÐVäáRØrærÙrÞã¨äåZÓÓ S (x) Ï ådÙrÏrrØrØ / ÙuÐ ÖVáRâ@ÒÝ]ÏdÒ [0, π/2] äÜuÒ×@å!u ÕØ@Ó Ö?0VâuÐ2n−1 ÝOÖRÓê2 ÖVærÏråòâuÐÝ]ârÞ / fàYÐ (x) ådÙrÏrrØrØ R à Þ u × Ò @ Ü @ Ø Ó A Ó b Ð u Ü V Ö ß ÖVÏrâ@Ò]äáRÙ@ÖRäá?8?u ê7 f (x) â@ÒÝ]xåbÜ = 8áVÐ0áWÒ Û@ÖÜ@ådærØ@Óy ×@àYÐÛ@ÖÜ@ådØrÙráWÒâràYÐbÜÐ [0, δ[ Ø ]δ, π/2] lê δ ƱРàRáWÖVâ@ÖRÓ¨Û@ÖÜ@ådØrÙráWÒâràYÐbÜuÒâdçd× äãdÖ]×@ØráWäçâuÐàRÙ@ÖRÓÒârÙ@ÖrÚtÛ@ÖRäÏZÖÜ 8RÏrå G]ÏZäáRâ@Ò]ÓåZÓÞ±ÚYÛ@ÖVÛuÐb× Ð u ÕØ@Ò¦vä u×ÐZ$Ú 0Ü@ØrÝ]ÏrØ¸Ï àäØZÜ@åä]ÖVØrÝOÓÒ / π/2 r â @ Ø Ó R Ö ä R á Ø R 0 Ö Ü R 8 é Z Ø ã ] Ý u Ù O Ð @ æ Ò r Ù r Ø ß Ø ê 4 Í u â Ð 4 r Ø r Ï Ø u æ V Ð äáRØrærÙrÞãäåZÓÓ 9 e=;gn Ûrâr Ø 0RÖÜ 8RéØZã n äÏZÖÜ 8ò!å 3OÖ]×@Ù@mS Ö 0Ü@nØrÝ]ÏdÖ ÛrârØ@ÓÞÏr Ð u áòÏÛrâdçrÓÖVß EârØ@äRê SR7 ê}ƱУÛ@ÒâràWÖRÓÑÒÛ@ÖÜ@ådØrÙráWÒâràYÐbÜuÒ Ú 3
×uH
Ò ådÙrÏ rrØrç f (x) äÏrÐ]ærÏZÖRÓ fÛ@Ò(x) â@ÒãY=ÖO×@π/2 ØráÊÖVá¦ÙråbÜ@ç±Ï π/2 ÚOâuÐàRÙ@ÖRÓÒârÙuÐVçä
ãYÖO×@Ø@ÓÖRä?á 8ÊäåZÓ[0, Ón Þ δ[ SR7 ê MRM I Ï G]áWÖV> ß ådÙrÏ rrØrØ ÚbÒ]äáWÒ äáRàWÒÙrÙ@ÖrÚtÙuÐârådéÐVÒáWä
ç ê ÆÖ£äåZÓÓAÐZÚtàÖVáOÜ@ØrærØ@ÒÖVá f (x) ÚtÓÒÙrçrÒáWä
ç Ù@ÒÛrâ@ÒârÞàRÙrÞÓ"?Ö 0VâuÐÝOÖRÓ/ ê ½ÐàRÙuÐçÛrârØ@ÓÒârÙ@Ö à 0Ü@ØrÝ]ØáWÖVærÏrØ ÚVÖVÙuÐÊä-àWä]ÉÒ 0RÖÜ 8RéØ@Ó âuÐÝOÓAÐbãdÖRÓ ÙuÐOærØrÙuÐVÒáÏZÖÜuÒ / π/2 0WÐá?8Wä
ç£ÖVÏZÖÜuÖeG]áW4Ö 3OÖ ÝÙÐ]æ@ÒÙrØrçÛ@ÖÓÒâ@Òx Ûr=ârØ δ 0Ü@ØrÑÒÙrØrç x ÖVá x ≈ δ ×uÖ x = 0 !ê 7ä]W Ò G]ÏdäáRâ@Ò]Óå / ÓÞ"äÓAÐbÜ@ÞÓØ m Úrä
×@àRØ 3Ðçrvä 8¸àVÜuÒàWÖrÚ 3]ârådÛrÛrØrâr!å u áWä
çÍÖVÏdÖÜuÖáWÖVærÏrØ x = π/(2n)| Ú ä]ÖRÖVáRàWÒáWäáRàR!å u ÕÒßÛ@ÒâràWÖRÓË å m = 1 ÙuÐØ 0RÖÜ 8RéÒ]Óå ÓAÐÏZäØ@ÓåZÓåWÉê 1sÊâ@Ò×uÒÜ 8RÙ@ÖRn→∞ Ò Ý]ÙuÐ]=æ@Ò+0 ÙrØ@Ò æuÐVäáRØrærÙ@ÖVß äåZÓÓÞH à G]áWÖVßÍáWÖVærÏdÒ±ÓÖbÑÙ@ÖÛrâ@Ò×uäáRÐàRØr?á 8¸ààRØZ×uÒ S∞ (+0) =
π + ν1 − ν2 + ν3 − . . . 2
AÄ äÜ@ØäåZÓÓå Ý]ÙuÐÏdÖVæ@Òâ@Ò×@å!u ÕÒß@äçòÛ@ÖRäÜuÒ×uÖVàYÐáWÒÜ8RÙ@ÖRäáWØ ν SR7 ê SR7 4Ö r@ÒÙrØ@á?8ÍÛ@ÒâràRÞÓ¥äÜÐ / 3ÐVÒ]ÓÞÓÚ@áWÖÛrârØ0Ü@ØrÑÒÙrÙ@ÖRÒ±Ý]ÙuÐOæ@ÒÙrØ@Ò ÓÖOÑÙ@Ö ÝbÐÛrØ@i äOÐá?8 S∞ (+0)
S∞ (+0) =
π π + ν1 = + 0,281. 2 2
7â@ÒÝåb Ü 8áRÐáWÒÙuÐÛ@ÖÜ@ådØrÙráWÒâràYÐbÜuÒ δ[ Ûrâ@Ò×uÒÜ 8RÙ@ÖRÒ¦Ý]ÙuÐOæ@ÒÙrØ@Ò¦æuÐVäáRØrærÙ@ÖVߣäåZÓÓÞ à áWÖVæ / dÏ ÒYõ N ÛrârØ@ÓÒârÙ@ÖÙuÐ N Ú SRIZM 0RÖÜ8RéÒRÚVæ@Ò][0, Ó ÚVØáWÖÜ8RÏZÖÊÝbÐáWÒ]Ó Ù@ÒÛrâ@ÒârÞàRÙrÞÓÖ?0VâuÐÝOÖRÓ â@ÒÝ]ÏZÖ åZÓÒÙ8RéÐVÒáWä
ç×uÖÙråbÜ@çà áWÖVærÏdÒ x = 0 Eârπ/2 ØuäVê SR7 êÆÖ*G]áWÖØÖVÝ]ÙuÐOæuÐVÒáÚ@æ@áWÖ*3OÒ]ÖRÓÒáRârØræ@Ò]äÏrØuÓ ØrÝOÖ?0VâuÐÑÒÙrØ@ÒOÓÛrâ@Ò×uÒÜÐæuÐVäáRØrærÙrÞãäåZÓÓgâYçd×Ðx SR7 ê L çZàVÜ@çrÒáWä
çÜuÖRÓAÐÙuÐçÚZØrÝOÖ?0VâuÐÑÒÙrÙuÐç ÙuÐÍârØ@äRê QS à ìÚ íVÚ àWÒâráRØ@ÏZÐbÜ 8RÙrÞÒ£Ü@ØrÙrØrØgÏdÖVáWÖVâ@ÖVßåb×rÜ@ØrÙ@ÒÙrÞ}Û@Ö äâuÐàRÙ@ÒÙrØu.ä π/2 ≈ 1,57 ÙuÐ àWÒÜ@ØrærØrÙrå Ú ä]ÖRäáRÐàVÜ@+ç u Õ!å u#ÛrârØ@ÓÒârÙ@Ö MOIö êƱÐÍÖVáRâ@ÒÝ]ÏZÒ ÏZÐâráRØrÙÐÍÝOÒâ π]ÏZÐâráRØrÙuÐ Ù@Òæ@Òá / ÏZÐbÜ 8RÙ@ÖÖVáW?Ö ν0V1âuÐ≈Ñ0,28 ÐVÒáWäçäÖVáRâ@ÒÝ]ÏZÐ [0, π/2] 4ê 7&äàW4Ö u¨ÖVæ@Òâ@Ò× 8@ÚRäÖVáRâ@ÒÝ]ÏrÐ [π/2, ÙrÞÓg?Ö 0VâuÐÝOÖRÓÖVáW?Ö 0VâuÐÑÐVÒáYä
çÙuÐÖVáRâ@ÒÝOÖVÏ [−π, 0] !ê 7ä]Ò àWÓÒ]äáW Ò G]áWÖä]ÖRÖVáR[0,àWÒπ] áWäáRàRåYÒ* á 3]âu4Ð ØrÏråY/ Ú ØrÝO?Ö 0VâuÐÑÒÙrÙ@ÖRÓåÙuÐârØ@äRê QS à ìÚ íVÚræráWÖØÍáRâ@(Ò 0RÖVàYÐbÜuÖRvä 8 ×uÖVÏrÐÝbÐ?á 8@ê
+{?Ë}|Ï0ÑðÙÑQÎ]O¿[Î> ÑÐÛ0Ö ÞÑQÎ]¥ÖòðÙ0ÖÜCÑ¥Ò Ñ6ÞÍ]ðÙ0ÖÓ=Í¥ÎÐÏ0Ñ¥ÜCÑÉÙÚpÛ,ßx
PÝpÙu¿Í w ~± ) +-,§ * %& + ´ + %&+ Ò ! #$Ê,
SZM¥à
è4Ö G¾Ø rrØ@ÒÙráRÞ âYçd×e Ð Ã ådâ 8WÒRÚWÏZÐ Ï G]áWÖ±äÜuÒ×@åYÒá ØrÝáWÒ]ÖVâ@Ò]ÓÞ ½Ø@ÓAÐÙÐ M 0@ê M ÚYäáRâ@Ò]ÓçZáWä
ç¸Ï rÙ åbÜu ÛrârØ n → ∞ ê áWÖVærÏrØ Ý]â@ÒÙrØrçàRÞærØ@äÜuÒÙrØr߸àRÞ´3OÖ]×@Ù@ÖrÚWæráWÖ?0VÞ ÖVÙrØ äáRâ@Ò]ÓØZÜ@Ø@äv8 ÏÙråbÜu ÏZÐÏÓÖbÑÙ@Ö0VÞäáRâ@Ò]ÒVÚVáWÖ43
×ЦâYçd×*0Våb×uÒá0VÞäáRâ@Ò]ÒäãdÖ]×uØrá?8Wä
çê47 ÛrârØ@ÓÒâ@Ò MOI ê M ÓÞ&åYäáVÐÙ@ÖVàRØZÜuØ ×rÜ@ç ÏdÖVÙrÏrâ@ÒáRÙ@ÖVß9ådÙrÏrrØrØ ÚuæráWÖ£äÜÐ 3]ÐVÒ]ÓÞÒÛ@ÖVâdçd×@ÏZÐ 1/n ÛrârØ n → ∞ àÏZÖ4G¾ØrrØ@ÒÙráVÐbã à ådâ 8WÒRÚV×ÒÜt Ð u ÕØ@Òâdçd×ÓÒ×rÜuÒÙrÙuÖ ä
ãYÖO×@çZÕØ@Óä
çÚWàWÖVÝ]ÙrØrÏr Ð u áÚdÒ]äÜ@ظØ@äãdÖ]×@ÙuÐçådÙrÏrrØrçØ@ÓÒ Òá¸âuÐÝ]ârÞàRÞ±ê 7-ÞçräÙrØ@ÓÚÏZÐÏ àVÜ@Ør+ç u á¸âuÐÝ]ârÞàR¿ Þ ådÙrÏ rrØrØÍØ Ò]Ò±Ûrâ@ÖVØrÝ]àWÖ]×@ÙrÞã ÙuÐãZÐâuÐÏráWÒâ / Û@ÖVàWÒ×uÒÙrØrçÏZ4Ö G¾Ø rrØ@ÒÙráWÖVà à ?Ö 0VÕÒ]ÓÔäÜ@ådæuÐVÒRê ëìYíuî ìdï F ð ;÷¼ó Cu* ó ø yù)ú" û+) (ü #¾ ù)ú" ' 2l ýþÿ ü ! f (x) ÿ #I% )yû! % @ù?" 4% #¾ )vù?")ú " % #>%(') *) (e % !'^ xù?" 4% Û'^ x ù "+ ü # o# ÿü ( û ÿ )vù?") " % # l yü ÿ "$) (k − 1) ü ) k @í4) ( ü )ÿû ) f (x) ) ÿ % ÿ *) ÿ ÿ '^þ #¾ )I ÿ ù "+ ü # ý}ý *)úÿ %4þ ) û! an bn ýþù?ÿ " k n→∞ ÿ ÿ ÿ 1/n lim
n→∞
ù?" û+)ú'¶% )´"+
SRB ê M
an bn = lim an nk = lim = lim bn nk = 0, n→∞ n→∞ 1/nk n→∞ 1/nk ∞ X
n=1
(
nα (|an | + |bn |),
, í ðZÿYð-ZìRúûuö-+uí ó}sÊåYäá?8HådÙrÏrrØ@ç
f (x)
ÙuÐ
[−l, l]
SRB ê S
α = 0, k − 1,
Ø@ÓÒ]Òá âuÐÝÜuÖbÑÒÙrØ@Ò SRB ê à
∞
f (x) =
a0 X πnx πnx . + + bn sin an cos 2 l l
Wá Ò]ÖVâ@Ò]ÓÒ£ÖâuÐàRÙ@ÖRÓÒârÙ@ÖVßÔä
ãYÖO×@Ø@ÓÖRäáRØ ÚâYçd×» SRB ê à ä
ãYÖO×@ØráWäçâuÐàRÙ@ÖRÓÒârÙ@Ö Ø×uÖ ÛråWäÏrÐVÒáÛ@ÖVæZÜuÒÙrÙ@ÖRÒ×@رÒâ@ÒÙrrØrâ@ÖVàYÐÙrØ@ÒRÚÛuÖRäÏZÖÜ8RÏråâdçd× ×rÜ@ç f 0 (x) ä ÏdÖ4G¾ØrrØ@ÒÙráRÐVÓØ / Ø w a0n b0n ∞ X SRB ê 0 πnx πnx 0 f (x) = + b0 sin a0 cos n=1
4Ö 3
ÜÐVäÙ@Ö
n
n
l
l
áVÐÏrÑÒ±ä
ãYÖO×@ØráWä
çâuÐàRÙ@ÖRÓÒârÙ@Ö¸à äØZÜ@å£áWÖVßÍÑÒáWÒ]ÖVâ@Ò]ÓÞ±êuèÖ4G¾ØrrØ@ÒÙráRÞ Ïd4Ö G¾Ø rrØ@ÒÙráRÐVÓØ a0 Ú b0 ä]ÖRÖVáRÙ@ÖVéÒÙrØrçrÓ6 Ø S 0@ê ?G w n=1
n
Ú
a n bn
äàRçZÝbÐÙrÞ ä
n
an = −
l 0 b , πn n
bn =
SRB ê 7
l 0 a . πn n
sÖRäÏdÖ Ü 8RÏrå¥ådÙrÏrrØrç Ö?0ÜÐb×ÐVÒáÙ@ÒÛrâ@ÒârÞàRÙrÞÓØÛrâ@ÖVØrÝ]àWÖO×@ÙrÞÓØ£×uÖ 3OÖ Û@Ö âYçd×@ÏrÐàRÏZÜu ærØráWÒÜ8RÙ@ÖrÚ@áWÖfÖV(x) Û@ÒâuÐ rrØuÉ×@رÒâ@ÒÙrrØrâ@ÖVàYÐÙrØrçÍÓÖbÑÙ@ÖÛ@ÖVàRáWÖVârØrá?(k8 − 1)/ 00
f (x) =
∞ X
a00n cos
n=1
ÛrârØræ@Ò]Ó
/
SRB ê B
πnx πnx , + b00n sin l l
RS B ê G? Ø£âdçd×9 SRB ê B ä
ãYÖO×@ØráWä
ç£âuÐàRÙ@ÖRÓÒârÙuÖrê Ö43
ÜÐVäÙ@ÖåYäÜuÖVàRØrçrÓgáWÒ]ÖVâ@Ò]ÓÞ±Ú@ÖVÛ@ÒâuÐ rrØu¥×@رÒâ@ÒÙ / rrØrâ@ÖVàYÐÙrØrçÔÓÖbÑÙ@Ö Û@ÖVàRáWÖVârØrá?8â@ÖVàRÙ@Ö âuÐÝWÚÛ@ÖVÏrÐ ÓÞ#ÙuÒ£Û@ÖÜ@ådærØ@Ó âdçd×nà ådâ8WÒ£×rÜ@ç k/ ß k Ûrâ@ÖVØrÝ]àWÖ]×@Ù@ÖVß ∞ X SRB ê I πnx πnx , + b(k) sin f (k) (x) = a(k) cos an = −
l 2 a00n , πn
n
n=1
bn = −
l
l 2 b00n , πn
n
l
1 ×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í ZS M 0 çZàVÜ@ç+u ÕÒß@ä
ç"ÏråWä]ÖVærÙ@Ö / Ù@ÒÛrâ@ÒârÞàRÙ@ÖVß¿ådÙrÏrrØ@Òß ê15ÊÝ9 ÖVâ@Óåbܶ SRB ê 7 ØÀ SRB ê G? Ù@Ò]äÜuÖOÑÙ@Ö åWäáRÐÙ@ÖVàRØrá?8¸äàRçZÝ(8¸ÓÒÑ×@å£ÏdÖ4G¾Ør@Ø@ÒÙráVÐVÓØ a Ú b Ø a(k) Ú b(k) w n n n n ( @ æ Ò R á @ Ù R Ö Ò; (k) (−1)k/2 an /nk , k an = Ù@Òæ@ÒáRÙ@ÖRÒ ; (k) (−1)(k+1)/2 bn /nk , k ( æ@ÒáRÙ@ÖRÒ ; (k) (−1)k/2 bn /nk , k bn = Ù@Òæ@ÒáRÙ@ÖRÒ ; (k) (−1)(k−1)/2 bn /nk , k ØZÜ@Ø ( æ@ÒáRÙ@ÖRÒ ; (k) SRB ê L (−1)k/2 an , k k an n = @ Ù Ò @ æ Ò R á @ Ù R Ö Ò (k) (k+1)/2 (−1) bn , k ; ( @ æ Ò R á @ Ù R Ö Ò (k) SRB ê MON (−1)k/2 bn , k ; bn nk = @ Ù Ò @ æ Ò R á @ Ù R Ö Ò (k) (k−1)/2 ; (−1) bn , k 2ÐÏÏZÐÏÏZ4Ö G¾Ø rrØ@ÒÙráRË Þ Ã ådâ 8WÒ a(k) Ú b(k) Úrä]4Ö 3
ÜÐVäÙ@ÖáWÒ]ÖVâ@Ò]Ó Ò ½Ø@ÓAÐÙuÐ M 0@ê M ÚräáRâ@Ò]ÓçZáWäç n n ÏÙråbÜ u ÛrârØ n → ∞ ÚZáê ÒRê SRB ê MRM = 0, lim a(k) = lim b(k) n→∞ n n→∞ n áWÖ±Ûrâ@Ò×uÒÜ 8RÙrÞߣÛ@Òâ@ÒãdÖ]S × SRB ê L & Ø SRB ê MON ÛrârØ äådæ@ÒáWÖRn Ó SRB ê MRM ÛrârØràWÖO×@Øráq Ï SRB ê M ê 2 ÐÏrØ@Óg?Ö 0VâuÐÝOÖRÓÚZÛ@ÒâràWÖRÒ ådáRàWÒâdÑ×uÒÙrØ@ÒÊáWÒ]ÖVâ@Ò]ÓnÞ¨→ ×uÖV∞ ÏrÐÝbÐÙ@Örê ãYÖO×@Ø@ÓÖRä?á 8âdçd×uÖVh à SRB ê S ?Ö 0Vå / äÜuÖVàVÜuÒÙuÐâuÐàRÙ@ÖRÓÒârÙ@ÖVßÉä
ãYÖO×@Ø@ÓÖRä?á 8?u~âYçd×uÖVj à Ã ådâ 8WÒRÚÛrâ@Ò×uäáRÐàVÜ@+ç u ÕØZÌ ã ådÙrÏ rrØrØ f (x) Ú Ø V á ê
× ê Z Ú u × Ö R à Z Ï Ü u r æ r Ø W á Ò Ü R 8 @ Ù r Ö @ Ú r æ W á Ö Ø R á @ â Ò ( R 0 V Ö Y à b Ð u Ü R Ö ä v ¸ 8 u × V Ö Z Ï Ð b Ý Ð á ? @ 8 ê f 0 (x) f (k−1) (x) t áWÓÒáRØ@ÓÚZæráWÖ àäÜ@ådæuÐVÒ¦Ù@ÒÛrâ@ÒârÞàRÙ@ÖVh ß ådÙrÏ rrØrØÛ@ÖVâdçd×uÖVÏ£ÓAÐbÜuÖRäáRØÏd4Ö G¾Ø rrØ@ÒÙráWÖVà à ådâ ♦8WÒ Ú V Ö R á @ Ù R Ö ä r Ø W á Ò Ü 8RÙ@Ö r Û r â Ø ÚÏrÐ Ï G]áWÖòäÜuÒ×@åYÒáØr6 Ý SRB ê M ÚAÛrâ@ÒàRÞéÐVÒá 1/n n ÖVÝ]ÙuÐOæuÐVÒáÚ@æráWÖà Ò×@ØrÙrØ rråWaê nBáWbÖ Ïd4Ö G¾Ø rrnØ@Ò→ÙráVÐb∞ã a Ú b äÜ Ð 3]ÐVÒ]ÓÞÒRÚ@Ø@ÓvÒ u ÕØ@Ò4Ö r@ÒÙrÏrå 1/n Ú ÖVáWäådáWäáRàR!å u á?ê ƱÐâYçd×@ålä GáRØuÓ Û@ÖO×@æ@ÒârÏrÙ@Ò]ÓÚRæráWÖ áWÒ]nÖVâ@Ò]nÓAÐÊäÛrâuÐàWÒ×rÜ@ØràYÐ ×rÜ@çÛ@ÒârØ@ÖO×@Øræ@Ò]äÏrØZã ådÙrÏ rrØrß ê ÄAäÜ@ØáWÒ]ÖVâ@Ò]ÓAÐ ÛrârØ@ÓÒÙrçrÒáWä
çÏ&Û@ÒârØuÖ]×@Øræ@Ò]äÏrØÔÛrâ@Ö]×ÖÜ@ÑÒÙrÙrÞÓ(äÛrâ@ÖRÓÒÑådáRÏZÐ ådÙrÏ rrØrçrÓ Ú@áWÖ¸ÖVá¸ÙrØZãÍØØZãÍÖO×@Ù@ÖRäáWÖVâ@ÖVÙrÙrØZã Ûrâ@ÖVØrÝ]àWÖ]×@ÙrÞã äÜuÒ×@åWÒááRâ@(Ò 0RÖVàYÐ?á 8 [−l, àRÞÛ@Öl]Ü@Ù@ÒÙrØrçÍåWäÜuÖVàRfØr(x) ß SRB ê MOS f (−l) = f (l), f 0 (−l) = f 0 (l), . . . , f (k−1) (−l) = f (k−1) (l). sÒâ@(Ò ±ÖVâ@ÓåbÜ@ØrâråYÒ]F Ó G]áRå áWÒ]ÖVâ@Ò]ÓåWÚZä×ÒÜtÐàÒ]W Ò 0RÖÜuÒ]Ò åb×u?Ö 0VÙ@ÖVßàÛrâuÐÏ@áRØræ@Ò]äÏrØZãÛrârØZÜuÖbÑÒ / ÙrØrçdãê ëìYíuî ìdï F ð ;÷<ó ;t ó þ ýþÿ ü ! f (x) þ %()v *% "+)v þ % !!'n *) ")' SRB ê M * û ü
Û%(! ´ & û ü #¾'^ "!# " % # % ù?" 4% (k) (*(ü #4û ü ' (k) x i = 1, p û ü )´"$# " % # "! # ü# ) k $ "$) ÿ f (x)ü ù?"δ (xi ) ü i# j n→∞ ) í4) (ü )vû '^#¾ ) þÿû! f (x) ýÛù ý "+ ü # ÿ þ *)úan bn%4)ý ÿ ÿ ÿ ÿ ÿ 1/n (k + 1)
, í ðZÿYð-ZìRúûuö-+uí óè±ÐÏØÛrârØ£×uÖVÏZÐÝbÐáWÒÜ8Wä]áRàWÒáWÒ]ÖVâ@Ò]ÓÞ SRB ê M Ú0Våb×uÒ]ÓgØ@ä
ãYÖO×@Ørá?8ØrÝâuÐÝÜuÖ / ÑÒÙrØrçf SRB ê êD s ÖVæZÜuÒÙrÙrÞÓ¨×@ر Òâ@ÒÙr r Ørâ@ÖVàYÐÙrØ@Ò]Ó¨âdçd×uÖVà@à ådâ8WÒ>ådÙrÏrrØrØ Ø Ò]Ò k/ ß f (x) Ûrâ@ÖVØrÝ]àWÖ]×@Ù@ÖVß V0 ÞÜuÖåWäáRÐÙ@ÖVàVÜuÒÙ@ÖrÚræráWÖÏdÖ4G¾ Ør r Ø@ÒÙráRÞ Ú Ø (k) Ú (k) äàRçZÝbÐÙrÞ¥ä]ÖRÖVáRÙ@Ö éÒÙrØrçrÓØ9 SRB ê L Ú} SRB ê MON ê-B áVÐÖVÛ@ÒâuÐ rr ØrçxV0 ÞÜÐàWÖVÝOÓÖbÑÙ@aÖVnßÍbàn äØZÜ@aån âuÐbàRnÙ@ÖRÓÒârÙ@ÖVßä
ãYÖO×@Ø@ÓÖ // äáRØådÛ@ÖRÓçZÙrådáRÞãâdçY×ÖVàx à ådâW8 ÒRê s ÖRäÏZÖÜR8 Ïrå@ ådÙrÏr r Ørç (k) Ø@ÓÒ]ÒáâuÐÝ]ârÞàRÞ±ÚáWÖ¸Ò]Ò âYçd× Ã ådâW8 Ò±Ù@Ò±çZàVÜ@çrÒáWäçâuÐàRÙ@ÖRÓÒârÙ@Öä
ãdÖ]×@çZÕØ@ÓäçÚ@Ø ÚuäÜuÒ×uÖVfàYÐáW(x) ÒÜW8 Ù@ÖrÚtäàRçZÝ( 8 ÓÒÑ×@åÍÏZÖ4G(I Ø / rr Ø@ÒÙráVÐVÓØ a Ú b Ø a(k+1) Ú b(k+1) Ù@ÒàWÖVÝOÓÖbÑÙ@Ö ÙuÐßráRØäÊÛ@ÖRÓÖVÕe8?u Û@ÖVæZÜuÒÙrÙ@Ö4O3 Ö×@ر Òâ@ÒÙ n Ù@ÒÓÒÙ@Ò]ÒRÚtØZã ÓÖOÑÙ@Ö£àRÞærØ@äÜ@Ørá?@ rr Ørâ@ÖVàYÐÙrØrç nâdçd×nÐ& SRB nê I ê
2 Ò]Ó 8 ÚÛ@ÖÜR8 Ý]åWçrävÍ 8 ÖVÛrâ@Ò×uÒÜuÒÙrØ@Ò]Ó / MOS ê SRB w
à
(k+1) an
(k+1)
bn
1 = l =
1 l
Zl
−l Zl
−l
f (k+1) (x) cos
πnx dx, l
f (k+1) (x) sin
πnx dx. l
SRB ê M¥à
+ {?Ë}|Ï0ÑðÙÑQÎ]O¿[Î> ÑÐÛ0Ö ÞÑQÎ]¥ÖòðÙ0ÖÜCÑ¥Ò Ñ6ÞÍ]ðÙ0ÖÓ=Í¥ÎÐÏ0Ñ¥ÜCÑÉÙÚpÛ,ßx
PÝpÙu¿Í SZMO7 sÊâ@Ò×@Û@ÖÜuÖOÑØrà¸áWÒÛ@Òâ8@ÚrÙuÐÛrârØ@ÓÒâ Ú æ@ÒáRÙrÞÓÚrÝbÐÛrØréÒ]ÓgÛ@ÒâràWÖRÒâuÐàWÒÙ@äáRàWÖà@ SRB ê L ààRØZ×uÒ k k
an n =
(−1)k/2 a(k) n
= (−1)
k/2 1
l
Zl
πnx dx. l
f (k) (x) cos
−l
.Ü@çàRÞærØ@äÜuÒÙrØrçHG]áWÖ43]ÖØrÙráWÒv3]âuÐbÜÐÛrâ@Ò×@Û@ÖÜuÖbÑØ@ÓäÙuÐ]æuÐbÜÐZÚRæ@áWÖÊàÛrâ@ÖRÓÒÑådáRÏdÒ Ø@ÓÒ / [−l, l] ÒáWä
çÍÖ]×@ÙuÐáWÖVærÏrÐâuÐÝ]ârÞàYÐ ÚZáWÖ43 ×Ð x1
1h an nk = (−1)k/2 l
sÊâ@ÖVØrÙráWÒv3]ârØrâ@ÖVàYÐVà¸Û@ÖæuÐVäáçrÓ´w
Û@ÖÜ@ådærØ@Ó
U = f (k) (x),
Zx1
πnx dx + f (k) (x) cos l
−l
dV = cos
πnx , l
Zl
f (k) (x) cos
x1
dU = f (k+1) (x) dx,
V =
πnx i dx . l
l πnx sin , πn l
Zx1 l k/2 h l l (−1) πnx πnx (k) a n nk = f (x) sin dx + f (k+1) (x) sin − l πn l −l πn l −l
+
l (k) πnx l l f sin − πn l x1 πn
Zl
x1
f (k+1) (x) sin
πnx i dx = l
i πnx1 1 n l h (k) f (x1 + 0) − f (k) (x1 − 0) sin − = (−1)k/2 l πn l Zl πnx o l dx . f (k+1) (x) sin − πn l −l
ÄAäÜ@ØÖ?0RÖVÝ]ÙuÐ]ærØrá?8
Øådæ@Ò]äá?8 SRB ê M¥à ÚráWÖ ØZÜ@Ø
SRB ê M 0
δ (k) (x1 ) = f (k) (x1 + 0) − f (k) (x1 − 0)
h 1 πnx1 l (k+1) i an nk = (−1)k/2 − δ (k) (x1 ) sin − a πn l πn n h δ (k) (x ) sin(πnx /l) i SRB ê MO7 l 1 1 (k+1) + a . n πnk+1 πnk+1 çZàVÜ@çrÒáWä
ç60RÒ]äÏZÖVÙ@ÒærÙ@Ö ÓAÐbÜuÖVßgÛ@ÖVâdçd×@ÏZÐ Û@Ö Ú-nÐ → àRáWÖV∞ â@ÖRÒ 0RÒ]äÏdÖVÙ@ÒærÙ@ÖgÓAÐbÜuÖVß 0RÖÜuÒ]ÒàRÞä]ÖVÏdÖ43OÖ(kÛ@+ ÖVâdçd1)×@ÏrÐZÚ
an = (−1)k/2+1
È ×uÒ]vä 8 Û@ÒâràWÖRÒ£äÜ Ð 3]ÐVÒ]ÓÖRÒ£ÛrârØ VÖ áRÙ@ÖVéÒÙrØu§Ï àWÒÜ@ØrærØrÙ@Ò 1/n Å æ@Ò]Ó (k + 1) ÚAÛ@ÖRäÏZÖÜ8RÏrå äOÐVÓyÏZÖ4G(IØrrØ@ÒÙrá a(k+1) → 0 ÛrârØ n → ∞ ê uÜ Ò×uÖVàYÐáWÒÜ8RÙ@Ö@Ú Ïd4Ö G¾Ø rrØ@ÒÙrá a Ø@ÓÒ]Òá¦Û@ÖVâYçd×uÖVÏÓAÐbÜuÖRäáRØ (k+1)n ÖVáRÙ@ÖRäØráWÒÜ8RÙ@Ö¦àWÒÜ@ØrærØrÙrÞ 1/n ¾ê sÖRäÜuÒ×@Ù@Ò]Ò n à àRØZ×uÒ ÓÖOÑÙ@ÖÝbÐÛrØ@äOÐ?á 8¸ SRB ê MOB 1 an ∼ k+1 , n 3
×uÒ±äØ@ÓàWÖÜ ÖVÛrâ@Ò×uÒÜuÒÙàâuÐÝ× ê ê sÊâ@ÖRäáWÒßréØ@ÒÐVäØ@ÓÛráWÖVáRØræuÒ]äÏrØ@Ò4Ö r@ÒÙrÏr Ø ë@ñ S ó ê E ÙuÐbÜu4Ö 3]Ør∼ ærÙ@Ö×rÜ@ç b Û@ÖÜ@ådæuÐVÒ]Ó n
bn = (−1)k/2
i h δ (k) (x ) cos(πnx /l) l 1 1 (k+1) + b . πnk+1 πnk+1 n
SRB ê MbG?
1
SZMOB BáWÖÖVÝ]ÙuÐOæuÐVÒáÚ@æráWÖ
×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í
RS B ê MOI sÖVàRáWÖVârØràgÐÙuÐbÜuÖ43]ØrærÙrÞÒÍàRÞÏZÜÐb×@ÏrØÔ×rÜ@çÙ@Òæ@ÒáRÙrÞã Ú ÜuÒv3]ÏZÖ å+0RÒ×@Ørá?8WäçÚæráWÖòÖ4r@ÒÙrÏrØ SRB ê MOB Øj SRB ê MOI äÛrâuÐàWÒ×rÜ@ØràRÞ#Ø×rÜ@ç9G]áWÖ43OÖ äÜ@ådæuÐçê2ÐkÏrØ@ÓÉÖ?0VâuÐÝOÖRÓÚÖ4r@ÒÙrÏrØ» SRB ê MOB Ø SRB ê MOI äÛrâuÐàWÒ×rÜ@ØràRÞy×rÜ@ç Üu0VÞã ê}ƱÐÛ@ÖRÓÙrØ@ÓÚ æráWÖhG]áRØÖ4r@ÒÙrÏrØ&ÓÞ(Û@ÖÜ@ådærØZÜ@ØàÍÛrâ@Ò× Û@ÖÜuÖOÑÒÙrØrØ ÚæráWÖ äådÕÒ]äáRàRåWÒá Ü@Ørée8 k ÖO×@ÙuÐáWÖVærÏrÐ âuÐÝ]ârÞàYÐ x êƱÐbÜ@ØrærØ@Òx0RÖÜ 8RévÒ 3OÖ ærØ@äÜÐ / 1 áWÖVæ@ÒÏÍâuÐÝ]ârÞàYÐÙ@Ò±ÓÒÙrçrÒá¸äådÕÒ]äáRàYÐ×uÒÜÐZÚrÛ@ÖRäÏdÖÜ 8RÏrå bn ∼
an = (−1)k/2+1
bn =
h
h
p P
δ (k) (xi ) sin(πnxi /l)
i=1
p P
(−1)k/2 i=1
1 . nk+1
πnk+1
+
δ (k) (xi ) cos(πnxi /l) πnk+1
+
SRB ê MOL
SRB ê SRN
i l (k+1) ; a πnk+1 n
i l (k+1) b , n πnk+1
ÖVáRÏråb×Ð ÓÞäÙ@ÖVàYÐ ÛrârØZãYÖO×@Ø@Ó&Ï£Ö4r@ÒÙrÏZÐVÓj SRB ê MOB Øq SRB ê MOI ê+2 ÐÏ@Ø@Ó&Ö?0VâuÐÝOÖRÓÚdáWÒ]ÖVâ@Ò]ÓAб×ÖVÏZÐ ÝbÐÙuÐZê % ÖbÑÙ@ÖÛ@ÖVÏrÐÝbÐá?8@Ú@æráWÖäÛrâuÐàWÒ×rÜ@ØràYÐ ØÍÖ?0VâuÐáRÙuÐç /
ëìYíuî ìdï ðF;÷ó'&ó*ø ! y SRB ê MOI Û ù?"
) ( "$) !') ´ ) ü SRB ê MOB ü ü# a b ýþÿ f (x) ý}ý ÿ þ ÿ þ ù ü "$# 4% #¾ ù "+ ü # (k)n n !')()v ò%^ù?" ')) ü ) k f (x) [−l, l] ÿ
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ådÙrÏ A ì$ ìYü ù ì@ó t æ@ÒàRØZ×@Ù@ÖrÚdæráWÖ âdçd×S SWG ê 0 ÛrâuÒ×äáRÐàVÜ@çrÒáâuÐÝ]ârÞàRÙrå!u"ÙuÐ ÖVáRâ@ÒÝ]ÏZÒ l]Ø@ÓÖRäá?8 / rrØuÚÛ@ÖRäÏdÖÜ8RÏråÏZÖ4G¾ØrrØ@ÒÙráRÞ àRáWÖVâ@ÖVߨäåZÓÓÞ|Ø@ÓÒvu áòÛ@ÖVâdçd×uÖVÏ
Ä3OÖ[−l, ê ä d ã ] Ö @ × 1/n âYçd×ÐÓÖOÑÙ@ÖåbÜ@ådæréØrá? 8 ä Û@ÖRÓÖVÕe8?u"Û@ÖÜ@ådæ@ÒÙrÙ@Ö43OÖâuÐÙ@Ò]Ò âuÐÝÜuÖbÑÒÙrØrçS MOI ê L êrè±ÐÏäÜuÒ×@åWÒá ØrÝ MOI ê L Úrâdçd×
VÖ Ûrâ@Ò×uÒÜ@çrÒá ÑÒÙrØ@Ò]Ó
∞ l X (−1)n+1 πnx sin 2π n l
Û@ÒârØ@Ö]×@Øræ@Ò]äÏrå!uzådÙrÏrrØu 2l n=1
f1 (x)
f1 (x) =
áê ÒRê 5äãdÖ]×@ÙrÞßÍâdçd×6
ÚÝbÐb×ÐÙrÙrå!u³ÙuÐÛrâ@ÖRÓÒÑådáRÏZÒ
SWG ê 7
àRÞâuÐ
[−l, l]
SWG ê B
SWG ê G?
SWG ê I
x , 4
∞ l X (−1)n+1 πnx f1 (x) = sin . 2π n l
/
ä ådæ@ÒáWÖRÓÌ SWG ê G? ÓÖbÑÙ@ÖÝbÐÛrØ@äOÐá?8 SWG ê 0 Ê n=1
f (x) =
∞ 3l X π(2n + 1)x 3l 1 − 2 cos + f1 (x) 2 8 π (2n + 1) l n=0
ØZÜ@Ø
∞ 3l X π(2n + 1)x 3l 1 − 2 cos . ϕ(x) = f (x) − f1 (x) = 8 π (2n + 1)2 l
sÖÜ@ådæ@ÒÙrÙrÞßâdçd× ÙuÐçx ådÙrÏrr Ørç
ä
ãYÖO×@ØráWä
ç×uÖRäáVÐáWÖVærÙ@Ö¥0VÞäáRâ@Ö Ø Ú@Ïrâ@ÖRÓÒ±áWÖ43OÖrÚ@âuÐàRÙ@ÖRÓÒârÙ@ÖrÚtÐåbÜ@ådæréÒÙ Ù@ÒÛrâ@ÒârÞàRÙuÐÙÐàWä]ÒßÍærØ@äÜuÖVàWÖVß ÖRäØ ê ϕ(x) n=0
:î ùï ìYn î ;ô<ó ;t> ó ½ÐÝÜuÖbÑØr?á 8±àÊâdçd¥ × Ã ådâ 8W´Ò ådÙrÏ rrØ uÚbØrÝO?Ö 0VâuÐÑÒÙrÙr!å u¨ÙuÐ ârØ@äRê SRB ê47&äÜ@ådæuÐVÒ Ù@Ò]Ö?0]ãdÖ]×@Ø@ÓÖRäáRØåbÜ@ådæréØrá?8¸äãdÖ]×@Ø@ÓÖRäá?8¸Û@ÖÜ@ådæ@ÒÙrÙ@Ö43OÖ âuÐÝÜuÖOÑÒÙrØrçê
/
9 e=g=j A $ì ìYü ù ì@ ó ƱÐârØ@äRê SRB ØrÝOÖ?0VâuÐÑÒÙ@Ö Û@ÒârØ@Ö]×@Øræ@Ò]äÏZÖRÒ Ûrâ@ÖO×uÖÜ@ÑÒÙrØ@ÒådÙrÏrrØrØ ×ÐÙrÙ@ÖVßÍÙuÐÛrâ@ÖRÓÒÑådáRÏdÒ [−l, l] àRÞâuÐÑ2lÒ/ÙrØ@Ò]Ó f (x) =
7¨áWÖVærÏZÐbãÍäéØràYÐÙrØrç
Ú
f (x)
x, 0 < x < l; −x/2, −l < x ≤ 0.
xk = (2k + 1)l k = 0, ±∞
ÚtÝbÐ
SWG ê L
ÚådÙrÏrrØ@çØrÝOÓÒÙrçrÒáWäçÍäÏrÐOærÏdÖRÓ
δ(xk ) = f (−l + 0) − f (l − 0) =
l l −l =− . 2 2
SWG ê MON
/
Ë ×Ý,ÓVIÍÐÒÖ ÍTÎ¯Ñ Û0Ö ÞÑQÎ]¥ÖòðÙ0ÖÜCÑ¥Ò Ñ6ÞÍ]ðÙ0ÖÓ=ÍÅÎ ÏÖZ7ÙÚpÛÑ¥Ø&
PÝpÙu¿Í
ZS MOL t æ@ÒàRØZ×@Ù@ÖrÚVæráWÖ âdçdץà ådâ8WÒ-áVÐÏZÖVß*ådÙrÏrrØrØ*0VåO×Òáä]ÖO×uÒâYÑÐá?8ÏZÖ4G¾ØrrØ@ÒÙráRÞgÛ@ÖVâYçd×@ÏrÐ ÚæráWÖÊÛ@ÖVáRâ@Ò(0VåYÒáåbÜ@ådæréÒÙrØrçÒv3OÖäãdÖ]×@Ø@ÓÖRäáRØ ê4.Ü@çÙuÐbãdÖOÑ×uÒÙrØrçâuÐÝÜuÖOÑÒÙrØrçÝbÐb×ÐÙrÙ@ÖVß 1/n ådÙrÏrrØrØ à£âYçd×6à ådâ8WÒàWÖRäÛ@ÖÜ8RÝ]åYÒ]Óä
ç â@ÒÝ]åbÜ8áVÐáWÖRÓÛrârØ@ÓÒâuÐ MOI ê M ÚtÛ@ÖÜuÖbÑØrà Ú b=1 a= 2 Ö43×ÐØrÝ MOI ê 7 äÜÒ×uåWÒá ê −1/2 0/
SWG ê MRM
∞ ∞ 1 3l X π(2k + 1)x l X (−1)n+1 πnx 3l − 2 cos + sin . f (x) = 2 8 π (2k + 1) l 2π n l
n=1
k=0
7-áWÖVâuÐç äåZÓÓAÐZÚbÏrÐÏÓÞ&Ø Ûrâ@Ò×@Û@ÖÜÐ 3]ÐbÜ@Ø ÚVä]ÖO×uÒâYÑØráÊÏZÖ4G¾ØrrØ@ÒÙráRÞÛ@ÖVâdçd×@ÏZÐ ê Ö43
ÜÐVä Ù@Ö¸ÓÒáWÖ]×@åèÊârÞÜuÖVàYÐZÚuåbÜ@ådæréÒÙrØ@ÒäãdÖ]×@Ø@ÓÖRäáRØ âdçd×Ð@ SWG ê MRM ÓÖbÑÙ@Ö¸Ûrâ@ÖVàWÒ]äáRØ 1/n ä±Û@ÖRÓÖVÕe8?u / êB áRåy± ådÙrÏr r Øu ÓÞ¥ÓÖbÑÒ]ÓÔÛ@ÖRäáRâ@ÖVØrá?¸ 8 áRÐÏ ÚræráWÖ?0VÞ"âuÐÝ àWäÛ@ÖRÓÖ43 ÐáWÒÜW8 Ù@ÖVß& ådÙrÏrr ØrØ Ù@ÖRäá?8 f (x) − f (x) ÖVÏZÐÝbÐbÜÐVfäv8 1 (x) Ù@ÒÛrâ@ÒârÞàRÙ@ÖVßµ ådÙrÏr r Ø@Òß&ÙuÐÍàWä]Òß&ærØ@äÜuÖVàWÖVßÖRäØ êoÊ 5 ÝârØ@äR/ ê 1 ådÙrÏr r Ørß Ú@àWÖRÖ?V0 ÕÒO3 ÖVàWÖVâdçÚuäådÕÒ]äáRàRåYÒáR0 Ò]äÏZÖVÙ@ÒærÙ@ÖRÒ SRB Ù@ÒáRâråb×@Ù@Ö åYäáVÐÙuÖVàRØrá?@8 Ú@æráWÖ áVÐÏ@ØZãh ÓÙ@ÖOÑÒ]äáRàWÖrê½ Ð rr Ø@ÖVÙuÐbÜR8 Ù@Ò]ÒàWä]ÒvO3 ÖàWÖRäÛ@ÖÜR8 ÝOÖVàYÐá?W8 äçH ådÙrÏr r ØrçrÓØ ÚÛrâ@Ò×uäáRÐàVÜ@ç+u ÕØ@ÓØ
f1 (x) ÛrâYçrÓÞÒÊÜ@ØrÙrØrØ ê ½ÐVä]ä]ÓÖVáRârØuÓò×@àYÐÛrâ@ÖRäáWÒßréØZã àYÐârØuÐÙráVбåbÜ@ådæréÒÙrØrç ä
ãYÖO×@Ø@ÓÖRäáRظâdçd×ÐH SWG ê MRM äÛ@ÖRÓÖ / Õe8?u ÛrâYçrÓÞãê M ¾ê ½ÐÝ]Ù@ÖRä?á 8 f (x)−f (x) ÓÖbÑÙ@Ö¦ä×uÒÜÐ?á 8¦Ù@ÒÛrâ@ÒârÞàRÙ@ÖRß ÙuÐàWä]Òß ærØ@äÜuÖVàWÖVßÖRäØ ÚOÛ@ÖÜuÖOÑØrà 1 +Ú 3×uÒ Uä]ÓêrârØ@äRê SW?G ê f (x) = x/4 x ∈ [−l, l] 1
9 e=gZ:
ådàWÒÜuØrærØr?á 8ÙuÐ ÚáWÖ . Òß@äáRàRØráWÒÜ8RÙ@Ö@Úä]Ö43ÜÐVäÙ@Ö» SWG ê MON ÚÒ]äÜ@ØÝ]ÙuÐOæ@ÒÙrØ@Ò Ý]ÙuÐ]æ@ÒÙrØ@Ò f (l − 0) äÜuÒ×@åWÒáÙuÐòäáWÖÜ8RÏdÖÑÒÍåZÓÒÙ8RéØrá?8@fê(−l 7ÀG]+ áWÖR0) ÓyäÜ@ådæuÐVÒÍàWÓÒ]äáWÖ l/4 SWG ê MON 0Våb×uÒ]ÓgØ@ÓÒá?8 h li h li δ(xk ) = f (−l + 0) + − f (l − 0) − = 4 4 l l l = f (−l + 0) − f (l − 0) + = − + = 0. 2 2 2
uÜ Ò×uÖVàYÐáWÒÜ8RÙ@ÖrÚZÙuÐVÓ&ÙrådÑÙ@ÖÝbÐÛrØ@äOÐá?8ådâuÐàRÙ@ÒÙrØ@Ò ÛrâdçrÓÖVß ÚdÛrâ@ÖOãYÖO×@çZÕÒßæ@Òâ@ÒÝ ×@àWÒ áWÖVærÏrØ}w Ø ê Î âuÐàRÙ@ÒÙrØ@Ò£ÛrâYçrÓÖVß ÚÛrâ@ÖOãYÖO×@çZÕÒßgæ@Òâ@ÒÝxG]áRØ&×@àWÒáWÖVærÏrØ ÚØÒ]äá?8 A(−l, l/4)ê(½Ø@äRê SWG ÙuÐ 3Ü@çd×@Ù@ÖØZÜ@ÜuÊäáRârØrâråWÒá¦Ù@ÒÛrâ@ÒârÞàRÙ@ÖRäá?8ådÙrÏrrØrØ ådâuÐàRÙ@Ò−l/4) ÙrØ@Ò f (x)B(l, = x/4 f (x)− 1 ê f1 (x) sÖRäÏdÖ Ü 8RÏråâuÐÝÜuÖOÑÒÙrØ@^ Ò ådÙrÏ rrØrØ (x) 0VÞÜuÖÛ@ÖÜ@ådæ@ÒÙ@ÖâuÐÙ@Ò]ÒÊààRØZ×u* Ò SWG ê MRM ÚrÙuÐßZ×uÒ]Ó âuÐÝÜuÖOÑÒÙrØ@ÒàâYçd × Ã ådâ 8W Ò ±ådÙrÏ rrØrØ ff(x) ê BáWÖÙ@ÒáRâråb×@Ù@Ö ä×uÒÜÐ?á 8 äáVÐÙZ×ÐâráRÙrÞÓ äÛ@ÖRä]Ö / o 0RÖRÓÚÛ@ÖRäÏdÖ Ü 8RÏrµ å ådÙrÏ rrØrç f (x) ØrÝ]àWÒ]ä1áRÙuÐZê È ×uÒ]vä 8òÛrâ@ÖVÕÒàWÖRäÛ@ÖÜ 8RÝOÖVàYÐ?á 8WäçÔâ@ÒÝ]åbÜ 8áVÐáWÖRÓ 1 ÛrârØ@ÓÒâuÐ MOI ê M w ∞ SWG ê MOS l X (−1)n+1 πnx f (x) = sin .
1
7-Þærábç9 SWG
ÖVáRÏråb×Ð
ê MOS ØrÝ¥ SWG ê MRM ÚrÛ@ÖÜ@ådærØ@Ó
2π
n=1
n
l
∞ 3l X π(2k + 1)x 3l 1 − 2 cos , f (x) − f1 (x) = 2 8 π (2k + 1) l
SWG ê M¥à
∞ 3l X 1 π(2k + 1)x 3l − 2 cos + f1 (x). 8 π (2k + 1)2 l
SWG ê M 0
k=0
f (x) =
k=0
1 ×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í SRSRN t æ@ÒàRØZ×@Ù@ÖrÚ æráWÖÛuÖÜ@ådæ@ÒÙrÙ@ÖRÒ àØráWÖ43OÒ âuÐÝÜuÖOÑÒÙrØ@Ò& SWG ê M 0 çZàVÜ@çrÒáWäçòåbÜ@ådæréÒÙrÙrÞÓ"âuÐÝÜuÖ / ÑÒÙrØ@Ò]ÓÔâYçd×h Ð SWG ê MRM ê È ÐVÓÒáRØ@ÓÚYæráW Ö ±ÖVâ@ÓåbÜ@h å SWG ê ¥M à ÓÖOÑÙ@ÖÜuvÒ 3]ÏZÖ±Û@ÖÜ@ådærØrá?8@ÚdÒ]äÜ@ظådæ@Ò]äá?8@ÚYæráWÖâuÐÝ]Ù@ÖRäá?8 ♦ u Ù Ð r Û @ â R Ö Ó Ò Ñ d å R á d Ï Ò ÝbÐb×ÐVÒáWäçàRÞâuÐÑÒÙrØ@Ò]Ó f (x) − f (x) [−l, l] 1
f (x) − f1 (x) =
3 |x|, 4
SWG ê MO7
x ∈ [−l, l].
7 äÛrâuÐàWÒ×rÜ@Ø@àWÖRäáRØ@±ÖVâ@ÓåbÜ@Þò SWG ê MO7 áRÐÏrÑÒ ÜuÒv3]ÏdÖå+0RÒ×@Ørá?8Wäç£ØrÝÊârØ@äRê SRB êÆÖâuÐÝÜuÖOÑÒÙrØ@Ò ×rÜ@ç |x| Ú x ∈ [−l, l] 0VÞÜuÖÛ@ÖÜ@ådæ@ÒÙ@ÖâÐÙ@Ò]ÒUä]Óê±ÖVâ@ÓåbÜ@åq MOI ê G? ÛrârØ b = 1 ÚZáVê ÒRê
SWG ê MOB
∞ 4l X π(2k + 1)x l 1 cos . f (x) = − 2 2 π (2k + 1)2 l
∗
Î ÓÙ@ÖOÑØrà SWG ê MOB ÙuÐ à2 0@ÚrÛrârØZãdÖ]×@Ø@ÓÔÏ9 SWG ê M¥à ê Ê ÓÖOÑÙ@Öä
×uÒÜÐá?8Ù@ÒÛrâ@ÒârÞàRÙ@ÖVßòÙuиàWä]Òß ærØ@ä / S êD7ÌG]áWÖRÓäÜ@ådæuÐVÒâÐÝÙuÖRäá?8 f1 (x)Uä]ÓêrârØ@äRê SRI ÜuÖVàWÖVßÖRäØ ÚrÛ@ÖÜuÖOÑØrà¸ÙuÐÛrâ@ÖRÓÒÑfåd(x) áRÏZÒ −[−l, l] k=0
f1 (x) =
. Òß@äáRàRØráWÒÜ8RÙ@ÖrÚä]Ö43
ÜtÐVäÙuÖ SWG ê MON ÚÒ]äÜ@ØÔÝ]ÙuÐ]æ@ÒÙrØ@Ò Ù@Ò£ØrÝOÓÒÙrçZá?8@ÚáWÖ äÜuÒ×@åWÒá ØrÝOÓÒÙrØrá?¸ 8 ÙuÐàWÒÜ@ØrærØrÙrå ê2 Ö43×ÐfàW(−l ÓÒ]ä+ áWÖ@0) SWG ê MON Û@ÖÜ@ådærØ@Ó
f (l − 0)
SWG ê MbG?
x/2, 0 ≤ x ≤ l; 0, −l ≤ x ≤ 0.
Ý]ÙuÐOæ@ÒÙrØ@Ò
l/2
h li l l δ(xk ) = f (−l + 0) − f (l − 0) − = − + = 0. 2 2 2
½Ø@äRê
SRI ÚìíÙuÐ 3
Ü@çd×@Ù@ÖØZÜrÜuÊäáRârØrâråWÒáäÛrâuÐàWÒ×rÜ@ØràWÖRäá8@±ÖVâ@ÓåbÜ@Þ ê ådÙrÏrrØrØ
SWG
ê MbG? Ø Ù@ÒÛrâ@ÒârÞàRÙ@ÖRä]á?8
f (x) − f1 (x)
9 e=g=o
sÖRäÏdÖ Ü 8RÏråâuÐÝÜuÖOÑÒÙrØ@´Ò ådÙrÏrrØ@Ø àÊâYçdץà ådâ8WÒ SWG ê MRM ØrÝ]àWÒäáRÙ@ÖrÚRáWÖ ÙuÐVÓ äÜuÒ×@åYÒá .Ü@çhG]áWÖ43OÖàWÖRäÛ@ÖÜ8RÝ]åYÒ]Óä
çâ@ÒÝ]åbÜ8báRÐáWÖRÓÔÛrârØ@ÓÒâuÐ Û@ÖÜ@ådærØrá?8¸âuÐÝÜuÖOÑÒÙrØ@ÒIådÙrÏrrØrØ f (x)fê!(x) 1 w ß MOI ê I ÛrârØ b = 1/2 MOI ê M Å ±ÖVâ@ÓåbÜuÖV9 f1 (x) =
7-Þærábç9 SWG
ÖVáRÏråb×Ð
SWG ê MOI
∞ l 2l X 1 π(2k + 1)x − 2 cos , 4 π (2k + 1)2 l
SWG ê MOL
∞ 2l X 1 π(2k + 1)x l − 2 cos + f1 (x). 4 π (2k + 1)2 l
SWG ê SRN
∞ ∞ 1 l l X π(2k + 1)x l X (−1)n+1 πnx − 2 cos + sin . 2 8 π (2k + 1) l 2π n l n=1
k=0
ê MOI ØrÝ¥ SWG ê MRM ÚrÛ@ÖÜ@ådærØ@Ó f (x) − f1 (x) =
f (x) =
k=0
k=0
Ë ×Ý,ÓVIÍÐÒÖ ÍTÎ¯Ñ Û0Ö ÞÑQÎ]¥ÖòðÙ0ÖÜCÑ¥Ò Ñ6ÞÍ]ðÙ0ÖÓ=ÍÅÎ ÏÖZ7ÙÚpÛÑ¥Ø&
PÝpÙu¿Í
RS SZM t æ@ÒàRØZ×@Ù@ÖrÚ æráWÖÛuÖÜ@ådæ@ÒÙrÙ@ÖRÒ àØráWÖ43OÒ âuÐÝÜuÖOÑÒÙrØ@Ò& SWG ê SRN çZàVÜ@çrÒáWäçòåbÜ@ådæréÒÙrÙrÞÓ"âuÐÝÜuÖ / ÑÒÙrØ@Ò]ÓÔâYçd×Ðh SWG ê MRM ê âuÐàRÙrØràS±ÖVâ@ÓåbÜ@Þ SWG ê M 0 Øj SWG ê SRN ÚÓÞ#àRØZ×@Ø@ÓÚ æráWÖ âuÐÝ]ÙrÞÓ¿ådÙrÏrrØrçrÓ ä]Ö ÖVáRàWÒáWäáRàRå!u á âuÐÝ]ÙrÞÒåbÜ@ådæréÒÙrÙrÞÒâuÐÝÜuÖOÑÒÙrØrçÚÙ@Öà Ö?0RÖVØZãgäÜ@ådæuÐçdãÚ ÏZÐÏgØgäÜufÒ1×u(x) ÖVàYÐbÜuÖ / ÖOÑØZ×Ðá?8@ÚG]áRØ ådÙrÏrrØrØåYäáRâuÐÙrç+u á¸ØrÝ âuÐÝÜuÖbÑÒÙrØrçf SWG ê MRM äÜÐ 3]ÐVÒ]ÓÖRÒä±ÏZÖ4G¾ØrrØ@ÒÙráVÐ / ÓØ ÚZÛrâ@ÖVÛ@ÖVârrØ@ÖVÙuÐbÜ8RÙrÞ¦ÓØ 1/n ê è±ÐÏ Ø à£äÜ@ådæuÐVÒ M ÚuÝbÐVÓÒáRØ@ÓÚæráWÖy±ÖVâ@ÓåbÜ@å9 SWG ê SRN ÓÖOÑÙ@ÖÛ@ÖÜ@ådærØrá?8@ÚtÒ]äÜ@Ø ådæ@Ò]ä?á 8@Ú æráWÖ ♦àäÜ@ådæuÐVÒ S âuÐÝ]Ù@ÖRäá?8 f (x) − f (x) ÙuÐÛrâ@ÖRÓÒÑådáRÏdÒ [−l, l] ÝbÐb×ÐVÒáWä
çàRÞâuÐÑÒÙrØ@Ò]` Ó ñ äâ ê 1 äH SWG ê MO7 ó SWG ê SZM 1 f (x) − f1 (x) = |x|, x ∈ [−l, l]. 7äÛrâuÐàWÒ×rÜ@ØràWÖRäáR& Ø ±ÖVâ@ÓåbÜ@À Þ SWG ê MO7 áVÐÏÑ2ÒÜuvÒ 3]ÏdÖ +å 0RÒ×@Ør?á 8WäçÍØr Ý 3]âu4Ð ØrÏrÐ SRI ìÚ íVêuè±ÐÏÓÞ ådÑÒ ådÛ@ÖRÓØrÙuÐbÜ@Ø ÚtâuÐÝÜuÖOÑÒÙrØ@Ò×rÜ@ç |x| Ú x ∈ [−l, l] ÝbÐb×ÐVÒáWä
& ç ±ÖVâ@ÓåbÜuÖVµ ß SWG ê MOB êÎÊÓÙ@ÖOÑØ@à SWG ê MOB ÙuÐ M2VS ÚZÛrâ@ØZãYÖO×@Ø@Ó9 Ï SWG ê MOL ê 7(ÝbÐÏZ Ü u æ@ÒÙrØ@Ò£âuÐVä]ä]ÓÖVáRârØ@Ó ÛrârØ@ÓÒâ ÚÛrâ@Ò×rÜuÖOÑÒÙrÙrÞß E ê ÆêèÊârÞÜuÖVàRÞÓÉà âu4Ð 0RÖVáWS Ò êt Ù@ÒÏdÖVáWÖVârÞã&×@Ø ±Òâ@ÒÙ rrØuÐbÜ 8RÙ@Þã&ådâuÐàRÙ@ÒÙrØrçdãÓAÐáWÒ]ÓAÐáRØræ@Ò]äÏZÖV ß ØrÝ]ØrÏrØ Ú Ø@ÓvÒ u ÕØZã&ÛrârØ / ÜuÖOÑÒÙrØrçàáWÒãZÙrØræ@Ò]äÏrØZãÍàWÖVÛrâ@ÖRäOÐbDã ëê :î ùï ìYn î ;ô'ó &óÎÜ@ådæréØr?á 8¸äãdÖ]×@Ø@ÓÖRä?á 8¸âYçd× Ð Ã ådâ 8WÒ 0/
f (x) = −
SWG ê SRS
∞ 2 X n cos(πn/2) sin nx, π n2 − 1
n=1
Ûrâ@Ò×uäáVÐàVÜ@ç+u ÕÒv3OÖÙ@ÒÏZÖVáWÖVârå!u¥Û@ÒârØ@ÖO×@Øræ@Ò]äÏrå!u`ådÙrÏrrØu f (x) ä¦Û@ÒârØ@ÖO×uÖRÓ 2π ÚY×uÖÛrçZáWÖ43OÖ Û@ÖVâYçd×@ÏrÐÖVáRÙ@ÖRäØráWÒÜ8RÙ@Ö 1/n ê A $ì ìYü ù ì@Û ó sÖ]×@æ@ÒârÏrÙ@Ò]ÓÚuæráWÖ àÖVáOÜ@ØrærØ@ÒÖVá¸Ûrâ@Ò×@Þ×@ådÕvÒ 3OÖ¸ÛrârØ@ÓÒâuÐHådÙrÏr@Ørç Ù@ÒØrÝ àWÒ]äáRÙuÐZê 7&G]áWÖRÓÍÖVáRÙ@ÖVéÒÙrØrØ ×ÐÙrÙrÞß ÛrârØ@ÓÒâÐÙuÐbÜu4Ö 3]Øræ@ÒÙÛrârØ@ÓÒârå SWG ê M ¾ê ÆÖ¦Ò]äÜ@رfà(x) Û@ârØ@ÓÒâ@Ò / Ø 0VÞÜuÖ×uÖRäáRÐáWÖVærÙ@ÖÍØ@äÛ@ÖÜ 8RÝOÖVàYÐÙrØrçòâuÐÝÜuÖbÑÒÙrØrç Ü@ØrÙ@Òß / SWG ê M ×rÜ@ç åbÜ@ådæréÒÙrØrç äãdÖ]×@Ø@ÓÖRäáRF Ù@ÖVßÙuÐ [−l, l] ådÙrÏ rrØrØ ÚráWÖà×ÐÙrÙ@ÖRÓÔäÜ@ådæuÐVÒÊ×uÒÜuÖ ?Ö 0RäáWÖVØrá Ù@Ò]äÏZÖÜ 8RÏdÖ¸äÜuÖbÑÙ@Ò]ÒRê 5ÊáVÐÏ ÚrØ@ÓÒ]Ò]ÓÙ@Òæ@ÒáRÙr!å u¿ådÙrÏ rr Ø u f (x) äÏd4Ö G¾Ø rrØ@ÒÙráRÐVÓØ 3ÊáWÖ?0VÞ"Ûrâ@Ò×uäáVÐàRØrá?8
bn = −
bn
2n πn cos . π(n2 − 1) 2
ààRØZ×uÒ SRB ê SRN ÚrÝbÐÛrØréÒ]Óg×@â@Ö?048
n/(n2 − 1)
SWG ê SQà
äÜuÒ×@å!u ÕØ@ÓÖ?0VâuÐÝOÖRÓ´w
n2 n2 − 1 + 1 1 1 n = = = + = n2 − 1 n(n2 − 1) n(n2 − 1) n n(n2 − 1) 1 n2 1 n2 − 1 + 1 1 1 1 = + 3 2 = + 3 2 = + 3+ 3 2 . n n (n − 1) n n (n − 1) n n n (n − 1)
SWG ê S 0
SWG ê SR7
ådæ@ÒáWÖRÓfG]áWÖ43OÖâuÐÝÜuÖbÑÒÙrØrç@±ÖVâ@ÓåbÜ@åq SWG ê SRS ÓÖbÑÙ@ÖÝbÐÛrØ@äOÐá?8
∞ ∞ X 2 hX 1 πn πn 1 f (x) = − cos sin nx + cos sin nx+ 3 π n 2 n 2 n=1 n=1 ∞ i X 1 πn + cos sin nx . n3 (n2 − 1) 2 n=1
2 ÐÏ ÏrÐÏÏZ4Ö G¾Ø rrØ@ÒÙráRÞ Û@ÖRäÜuÒ×@Ù@vÒ 3OÖâdçd×Ðàq SWG ê SR7 Ø@ÓÒvu á£Ö4r@ÒÙrÏrå Ûrâ@ÖRäåZÓÓØrâ@ÖVàYÐá?8 ×@àYÐÛ@ÒâràRÞãâYçd×ÐZê ½ÐVä]ä]ÓÖVáRârØuÓ äÙuÐ]æÐbÜÐÛ@ÒâràRÞßÍâdçd× −
∞ πn 2X1 cos sin nx, π n 2 n=1
O(1/n5 )
Ú@áWÖäÜÒ×uåWÒá
SWG ê SRB
1 ×ß6Ø=ß©=Ë+¢.ÚpÛ0ÔÀ
PÝpÙu¿Í SRSRS Ö?0VåWäÜuÖVàVÜuÒÙrÙrÞß âuÐÝ]ârÞàRÙ@ÖRäá?8?uyäOÐVÓÖVß ådÙrÏrrØrØ f (x) à áWÖVæ@ÏZÐbãÚ@ÏZÖVáWÖVârÞÒ±ÙuÐVÓÔÛrâ@Ò×uäáWÖVØrá ÖVÛrâ@Ò×uÒÜ@Ør?á 8@ê 7ä]ÖRÖVáRàWÒáWäáRàRØrØ ä^±ÖVâ@ÓåbÜuÖVß9 SRB ê SRN Ú@ÛrârØ k = 0 Ø@ÓÒ]Ò]Ó
SWG ê SWG?
n
Ú Ò]äÜ@Ø
SWG ê SRI
p nπ 1X 2 . δ(xi ) cos nxi = − cos π π 2 i=1
BáWÖ ä]ÖRÖVáRÙ@ÖVéÒÙrØ@Òä¸ådæ@ÒáWÖRÓ Ù@Òæ@ÒáRÙ@ÖRäáRØgâdçd×ÐF SWG ê SRB äÛrâÐàWÒ×@Ü@ØràWÖ ÛrârØÜu0RÖRÓ àRÞ10VâuÐá?8 π x1 = − , 2
i = 1, 2;
δ(−π/2) = −1,
x2 =
π , 2
δ(π/2) = −1.
t Ûrâ@Ò×uÒÜ@ØràáWÖVærÏrØâuÐÝ]ârÞàYÐZÚÛ@ÖRäáRâ@ÖVØ@Ó àWäÛ@ÖRÓÖ43ÐáWÒÜ8RÙrå!u ådÙrÏr@Øu ÚÛrârØrÙrçZàÊàWÖ¦àRÙrØ AÓ ÐÙrØ@ÒRÚYæráWÖ f (−π) = f (π) = 0 ê t ×@Ù@ÖVßØrÝ Ûrâ@ÖRäáWÒßréØZã¸Ù@Òæ@ÒáRÙrÞãyådfÙr1Ï(x) rrØrß ÚWÙ@ÒÛrâ@ÒârÞàRÙ@ÖVß / àáWÖVærÏZÐbã ÙuÐ93]âuÐÙrØruÐbãØrÙráWÒâràYÐbÜÐgØØrÝOÓÒÙrç+u ÕÒß@äçäÏrÐ]ærÏZÖRÓ âuÐÝ]ârÞàYÐZÚuçZàVÜ@çrÒáWäç@±ådÙrÏrrØrç f (x) = −f (−x) Ú@ÝbÐb×ÐÙrδ(−π/2) ÙuÐçÙuÐ Û@Ö=Ü@ådδ(π/2) Û@ÒârØ@Ö]×uÒ =[0,1π] ä]ÖRÖVáRÙ@Ö / 1 1 éÒÙrØrçrÓØ SWG ê SRL x/π, 0 ≤ x < π/2; f (x) = ÍâuÐ4ØrÏ 1 (x) ½ÐÝÜuÖbfÑ Ørà
1
(x − π)/π, π/2 < x ≤ π.
ØrÝOÖ?0VâuÐÑÒÙÙuÐârØ@äRê MO7 ìÚ íVÚ3
×uÒÙrådÑÙ@ÖÛ@ÖÜuÖOÑØrá?8 l = π Ú A = 0,5 ê ÏZÐ@ Ï ådÙrÏ rrØ uÉä @× àWÖVßrÙ@ÖVß äØ@ÓÓÒáRârØ@Òßàâdçd×&à ådâ8WÒRÚrÛ@ÖÜ@ådærØ@Ó f (x) 1
SWG ê àRN
∞ 2X1 πn f1 (x) = − cos sin nx. π n 2
n=1
2 ÐÏrØ@Ó Ö?0VâuÐÝOÖRÓÚÓÞ§Û@Ö]×uÖ?0VâuÐbÜ@Ø"àWäÛ@ÖRÓÖ43]ÐáWÒÜ8RÙ@å!u ådÙrÏr@ØuÃáRÐÏ ÚæráWÖÒ]Ò âuÐÝÜuÖOÑÒÙrØ@Ò SWG ê RN ä]ÖVàRÛuÐb×ÐVÒáÔä âuÐÝÜuÖOÑÒÙrØ@Ò]Ó SWG ê SRB ÚáVê ÒVê´ ÐÏráRØræ@Ò]äÏrØ"ÓÞ.Ûrâ@ÖRäåZÓÓØrâ@ÖVàYÐbÜ@Ø"âdçd× SWG ê SRB ê È ×uÒ]äv8 åZÓÒ]äáRÙ@Ö ÖVáWÓÒáRØrá?8@ÚæráWÖ âdçd×Ì SWG ê SRB ÓÖbÑÙ@Ö&0VÞÜuÖ Ûrâ@ÖRäåZÓÓØrâ@ÖVàYÐá?8&ÐÙuÐbÜuÖ 3]ØrærÙ@♦ÖáWÖRÓåWÚrÏZÐÏh]G áWÖ ä×uÒÜÐÙ@Ö àÛrârØ@ÓÒâ@Ò SWG ê M Ú@äÊÛ@ÖRÓÖVÕe8?É u Ûrâ@ÖRäáWÒßréØZãâuÐÝÜuÖbÑÒÙrØrß ê / . Òß@äáRàRØráWÒÜR8 Ù@Ö@Úrä ådæ@ÒáWÖRÓ&áRârØO3 ÖVÙ@ÖRÓÒáRârØræuÒ]äÏrØZãä]ÖRÖVáRÙ@ÖVéÒÙrØrßÍâYçd×9 SWG ê SRB ÓÖbÑÙ@ÖÝbÐ / ÛrØ@äOÐá? 8 ààRØZ×uÒ
à
−
∞ ∞ πn 1 X 1h 2X1 π π i cos cos nx = − sin x + − sin x − = π n 2 π n 2 2 n=1 n=1 ∞ ∞ 1X1 π 1 X 1 π + , =− sin x + sin x − π n 2 π n 2 n=1
SWG ê àZM
SWG ê àRS
n=1
ÖVáRÏråb×ÐÛ@ÖRäÜuÒÙ@ÒÏdÖVáWÖVârÞãÍÛrâ@Ò]Ö?0VâuÐÝOÖVàYÐÙrØrß ØÍäÜuÒ×@åWÒá SWG ê SRL ê ½ÐVä]ä]ÓÖVáRârØuÓÔáWÒÛ@Òâ 8 àRáWÖVâ@ÖRÒ äÜ Ð 3ÐVÒ]ÓÖRÒ äåZÓÓÞ¶ SWG ê SR7
à
−
∞ 2X 1 πn cos sin nx. 3 π n 2 n=1
½çd×F SWG ê RS Ö?0VåWäÜuÖVàVÜuÒÙâuÐÝ]ârÞàRÙ@ÖRäá?8?u àRáWÖVâ@ÖVßÍÛrâ@ÖVØrÝ]àWÖO×@Ù@ÖVß ådÙrÏrrØrØ ê+p±Òv3]ÏdÖ ådàRØ ×uÒá?@8 ÚVæráWÖG]áWÖVáâdçd×Ûrâ@Ò×uäáRÐàVÜ@çrÒá ä]Ö?0RÖVß×@àYÐÑ×@ÞÔÛrâ@ÖVØrÙráWÒv3]ârØrâ@ÖVàYÐÙ@ÙrÞ߸âYfçd(x) × SWG ê SRB êÄAäÜ@Ø / àWÖRäÛ@ÖÜR8 ÝOÖVàYÐá?W8 äçÍâ@ÒÝ]åbÜb8 áRÐáWÖRÓÔÛrârØ@ÓÒâuÐ S ê ÚráWÖ×rÜ@ç Ø@ÓÒ]Ò]Ó
Qà à
f2 (x) = −
f2 (x) = −f2 (−x)
∞ 2X 1 πn cos sin nx = 3 π n 2
n=1 3 x π π + x, 0≤x< ; − 6π 24 2 = 3 − (x − π) + π (x − π), π < x ≤ π. 6π 24 2
SWG ê àQà
Ë0/ ×Ý,ÓVIÍÐÒÖ ÍTÎ¯Ñ Û0Ö ÞÑQÎ]¥ÖòðÙ0ÖÜCÑ¥Ò Ñ6ÞÍ]ðÙ0ÖÓ=ÍÅÎ ÏÖZ7ÙÚpÛÑ¥Ø&
PÝpÙu¿Í 2 ÐÏrØ@ÓÔÖ?0VâuÐÝOÖRÓÚr×rÜ@çØ@äãdÖ]×@Ù@Ö43OÖ âuÐÝÜuÖOÑÒÙrØrçF SWG ê SRS ÓÞ"Û@ÖÜ@ådærØZÜ@ØÍàRÞâuÐÑÒÙ@Ø@Ò f (x) = f1 (x) + f2 (x) −
∞ πn 2X 1 cos sin nx, 3 2 π n (n − 1) 2 n=1
çZàVÜ@ç+u ÕÒ]Ò]ä
çâ@ÒéÒÙrØ@Ò]ÓÛ@ÖRäáRÐàVÜuÒÙrÙ@ÖVßÝbÐb×ÐOærØ ê
SRSQà SWG ê à 0
SRS 0
Á áI½¾§½ 3 65ê 7&ì9æ VìSIê 4
w´±
1
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í
çy+- ! ×#$Ê,"8 '³ + ~
ª0dµ # d ¤ ªt¥~
e6^²p¤ = pcQò
^ e^ , ^p0e^© f 0¤ ¦0
e¥c c7d =¥®¯=¤ ^ ª0d9^¤ c=§¬ ¦þ=c;d9e~
6cpc ,=¤ ¡V7 , C¶d e^f ò^±~¤ ¬ô ¤cC0p
h=cf®¯d ¡ª0[2l¤ ¬º^¤ ¤ ccC¥
ñ6
e6c pd#^=e^ªºpf0 ,c¨ ±©6ª0 f e^ c« d9c 6±yf¬~e6(x) ¥µ ¤ cC
2l X ÷ g=oúq?ø nπx nπx a + + b sin a cos f (x) = , 2 l l A
∞
0
n
n
n=1
1 a0 = l
Zl
f (t)dt,
Zl
f (t) cos
nπt dt, l
Zl
f (t) sin
nπt dt. l
−l
1 an = l
÷ g=o0ùgø
−l
1 bn = l
−l
¥e ©¨ª0 f « ^ ^¤ cC
^e^fp chc,ôd9c?¡6~ò¬w ¤ ¥
e6p=p ^,h¤ 0
cd ¨ª0 f « ~¥
,ª Ve^^Q~ cpµ¸¤,pC cd#ª c===6ò0
,( ,^y ^¤ cp , cCª
ce^ Ce^f 0¦I¨;bª0^ d f0µ « ± b#;=cf ¤ dô¥
c¥ ¤,^p C còdT¦C=, y,=¤,e^^p±M¤ 0
e¥c Mc®¯cª0±¤ c=¬e^P 9] −¤,=∞, yd9^ ^=9ñ6c== ,=¤,pey ^f0c=c¤ Td9ºd9cC
¨( f0=∞[« d9d9cQ¡V cw e^ cp ]0,¬=C∞[cp¬
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^e^f ±º ¤ c« ^e^ey ^¤ cC
0µ f c=^e^f c ¤ dTT c^eC ^6¤ ¡V cC
c d ¤ lwe6p=¤ f ^cd#±c7 ^d9¤,=eÅ« ô f¶ ;^ e^¤ f c ¥^
, ª ~ce6fw Q=d9¹^¤ ¥7
¤¥ 0¬
ôT®¯ª0 ¤ ^¬¤ ¥¦ cC^
f cpµ c¤ Td*
c± Td ^^¤,? cd e^e^f c ^ Td9© ¤ ¥
¥ =d9 C c=h ^^¤,? »§ ,¤ cpp C¬ = º c~^¤6¤0?
®¯cd´ª0¤ ®¯¬ª0¤
¬ ,=ô§c°^º¤,= ^ ¤, =^6 §
,¦ þ ^c6¤,= ^ ò¦þ ^¤ ? c=ªt¡ ª0¤ª[Q ?¤
¥=
,¥ ¬ c¤ ^cc¯d9 6^¡¤ ª ¥¦f cC
Ic=(¤0 µ
ªC
(c®¯p ª0=6e^¤ e^¬d9Tcc=f[¤ ¤ 6 hd»^÷
,^ ,¤,V?f0 ,?cª ^®¯¤ e6¥ª0
¤ ¥^¬ =^ c¹ d©cªe6e6ª0¤ ¬IcøP¨ª ª0e¥T f c¤ « c« ¥d©f
,(x) l¯ ¤ 0¦ =0bcð
(c,Id9cQ¡[6[−l, ¯òl]¬ f0?¤ ¦ô¥
¤,e6ppC=¤ pT ^,[[e6¯ª0d9dPc [f0¤?0¦¶
h^ ÷ g=¤ o^ú¤ q?Tø&¤, =ce6, ôe6ª0d9d9c±ôe^c^^c¤0
[®¯ª0¤ ¬=÷ g=oúq?ø¥¾þc0µ /®¯ª0¤ ¬ôcp ¬f c©,f(x) ^f0c=c¤ c=dKf c ^ cdÀ ^¤ ? =§¾§ ~ñ6c^c» ^¤ ? ©¤0
*
f (x − 0) + f (x + 0) 2
e^¤ ¥
^d#ª=¤ ¨(d96 ^e^f0cd#ªI ^c^c7 ¤,=c^c ¤ ¥
¥ c¨ª0 f « 4ñ6c±7c f0= # 0 !
¯ ® 0 ª
¤ ¬ V ÷ = g o ú ? q & ø
¤ ¥
6 e p = p ,
6 V ¨ 0 ª
f
«
¢ 0
6 e 0 ª 9 d 9 d 6 p
¤ P d = c
^ ^ e
f 0 ¦ f (x) e¥ =¥=^d9§¦ ÷ g=o ø nπx πnx nπx , + b sin = M sin ϕ + a cos n
l
n
l
n
n
l
$¡=Ë}PÎÐÒ Ñ¥ØQÒÔòÍ ¬ Ñ¥ÒÚ¥ÖÚÖ¶ÑO¬=ÙÍ ÛÍÐ× ÍÐÒÖÚ
,=e6c= f0cpc¤ §¦
nπ/l
¤ dP=¢M[
e^f ¤ 6,TIC,Q ^
SRSR7
nπ π 2π , ,..., ,... l l l
0,
Ç
^e^¬=d90 =ªQ
M = pa + b tg ϕ = a /b ϕ ö,?,? ¬ T(¨IpC e^e^d9c=Q?¤
=dT f0c±~=f , d ce^^±~¤,pcCe^c d d9cQ ¡V c~ ¤ c^e6»6=¤ d9c ^e^f ±t=,? y¨ª0 f0µ « f=(x) ¹ªe6¬h¨ª0 f « f (x) c ¤ ¥
¥Ox ^ ,p©,¶^e^f c ^ cd ¤ c=dP¥¡ª f0 ] − ∞, ∞[ ªC
cp 6c¤ 6ye¥ ¥
,ª0¢M7,dþªe¥ c dT q^^T¤,¨? ª0 f « f (x) =e^c? ¢M cþ ^^¤ ¤ª^dPÉ? =Ée6ª 7^e6=ª^ ^e^ce6^ ɱ 2 n
n
2 n
Z∞
n
n
n
n
|f (x)|dx = Q,
¦ c=Qhg0,ô ¢4e6dPòcd¼e¥ ¯f cA ^= ccd¼^c[ C¤ ,cQd9 6^¡ ª 5f0¶¨ª0 f « ªQ
cp 6c¤ 6ªe¥ c d f (x) l¯ ¤ 0¦ = ¤ª6dt» ¤ 0^f
c=®¯cª0¤ ¤ c¬Mw ¤ ,c Ccp ¬ò c¾§ òl e^ cdTe^ òcpf0 =¬=f*^ª^d9Cd9eÅV^ ¤,p6eÅ ²cQ¡ ¤,^ = p^*d²,÷ g=poe6úq?¬ ø ¨¤,=ª0 Ç ^f =« e6¨( f he^f÷ (x) g=oúq?ø¥ ^e¥ ~ ^¤ ^±wf~[−l, ¤ ¥
l]¥ ,ªô ¤ l → ∞ ô cpl( , ,ª0y ñ6d c^c7 cQ
e6p= d¯e^cc= c=·7^ ÷ g=oúq?ø# T±y0
ôf0cñ^¨¨( « ^ch÷ g=oùgø −∞
1 f (x) = 2l
Zl
f (t)dt +
n=1
−l
+
0 1 f (x) = 2l
Zl
∞ h Z X 1
1 l
∞
1X f (t)dt + l n=1
l
l
−l
nπt nπx f (t) cos dt cos + l l
Zl
f (t) sin
Zl
nπt nπx nπt nπx f (t) cos dt. cos + sin sin l l l l
−l
nπt nπx i dt sin l l
¾§ce^ cp ¬=Cc==·7 e^¬[¤ ^c cd96¤ Ce^f dcQ¡[
^e6cd¼÷ÐqV3 ø¥C= ·7^d −l
1 f (x) = 2l
Zl
−l
∞
1X f (t)dt + l n=1
Zl
f (t) cos
nπ (x − t)dt. l
÷ g=o 3 ø
l ^ c~ ¤,c¢4e6=tc^ I÷ e6¨( c~f0e^ d9¤ ^cø¥6 h= d9 ^ce6Ic~ e¥¤ c e^b¥^¦ ^x¤ ¬ªC
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cp µ ë ¹¤ 6¡[
(l →e^^+∞ ^c[Q=d96 dTc ¤ l → +∞ C
−l
1 lim l→+∞ 2l
Zl
−l
−l
f (t)dt = 0.
÷ g=oùnø
× ß6Ø=ß ¼ ËéòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í l¯^± e6¥ ¬ c p=f~f0=f~¨ª0 f « f (x) =e^cp ¢M c ^^¤ ¤ª^dP? = 1
SRSRB
Z∞
c
|f (x)|dx = Q,
−∞
Zl Z∞ 1 Zl 1 1 Q f (t)dt ≤ |f (t)|dt ≤ |f (t)|dt = . 2l 2l 2l 2l
¤,4° =e^^¢§
e6¶ y¤ ÷ g=ol 3 →ø&f~∞ ¤ e¥¥
¥¥
, ,ªªô6~ ¤ e^ ¤,l=→¥
∞ , ccpe6 ,ª0¬» ÷ g=d oùnø¥b#=f d*c¤,pCcdT ^¤ ¥¦ cC
, −l
−∞
−l
Zl ∞ nπ 1Xπ f (t) cos (x − t)dt. f (x) = lim l→∞ π l l n=1
¾§¥
^dþ c=ª0¢¼ ^¤ ^d9^ ª0¢
,A
−l
un = πn/l
∆un = un+1 − un =
n = 1, ∞
÷ g=o jø
,¹ce^f0cp ¬fª
π π(n + 1) πn − = , l l l
e^cc= c=·7^ y÷ g=o jø§d9cQ¡V c[ ^¤ ^ eCp¬V0
l
?^^¤,?
Z ∞ X 1 f (x) = lim ∆un f (t) cos un (x − t)dt. π ∆un →0 n=1
÷ g=o;:ø
cC
hC,=f cdþe6ª0d9d9 p , ¢MeðôC,? ^ d9~¨ª0 f « −l
φ(u, x) =
Zl
f (t) cos u(x − t)dt,
,T=e6 ~e¥ ¨(^c ¤ d#TªQd9 7KÉ÷ g=co ;:fø¥? ¦ ,u= =cd9u , =¤ 6V¯¨( f ^e^^ ¤,¤ ?c ¬= ª0 ¢ cd e6ª0xd9=d#ó&ª©ª0d9÷| dP? e6p ,cQ e^¬[p¯^¢¥M ¤ùø ¤,=c± −l
n
¤e^c¥e6¦pcC=
!p h^÷ g=oª0;:¢KøP
= ,6 ¤ cd96¡ª f
∞ X
φ(un , x)∆un ,
n=1
]0, +∞[
.¹cñ6cd#ª!¨(c¤ dP? ¬ T± ¤ Å
¥ ¬ T±© ¥µ
Z∞ Z∞ ∞ X 1 1 lim φ(un , x)∆un = f (x) = du f (t) cos u(x − t) dt. π ∆un →0 n=1 π
÷ g=o oø
®7c¤ d#ªQ »÷ g=o oø,pCT=6eð» ^^¤,? ¬ c±t¨(c¤ d#ªC c±t®¯ª0¤ ¬=ôe6c?7 ±»
^±~ ¹¤ ^¥^
¤,e6?p h=p´ ^
c7± ¨Tª0dþ f « ^^¤,? cd®¯0
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0
−∞
$¡=Ë}PÎÐÒ Ñ¥ØQÒÔòÍ ¬ Ñ¥ÒÚ¥ÖÚÖ¶ÑO¬=ÙÍ ÛÍÐ× ÍÐÒÖÚ SRSWG ?^^¤,? ®¯ª0¤ ¬÷ g=o oød9c?¡V c ¤ ¥
e6==©0
=§cp ^h cC¦c?¡^dK,¤ 0
¯® ª0¤ ¬V÷ g=oúq?ø¥ ¤ l(^c , Vcd9ñ66c^¤ c7 , ^¤ e^^fcª¢ ¤,p¨(^ªc^¤ dd# ªQ¤, ,=ª =ª0¢,=e6¬7¨(c¤ d#ªQ ÷ g=o oø¥@?e^ cp ¬=^ªVC^e6ª¢ cos u(x − t) = cos ux cos ut + sin ux sin ut,
cp ,ª0 d 0
1 f (x) = π
Z∞h Z∞ 0
f (t) cos ux cos ut dt +
f (x) =
Z∞h 0
°c=C,Q cp ,ª0 d
Z∞
1 π
0
1 A(u) = π
Z∞
i
f (t) sin ux sin ut dt du
−∞
−∞
+
Z∞
1 π
Z∞
−∞
Z∞
÷ g=o ø
f (t) sin ut dt sin ux du.
−∞
f (t) cos ut dt,
−∞
i f (t) cos ut dt cos ux du+
1 B(u) = π
Z∞
÷ g=oúqQiø
f (t) sin ut dt,
−∞
÷ g=oúqq?ø ó§cc= c=·7^ I÷ g=oúqq?ø. ¤ ¥
e6==p , 6( ^ ^¤ cC
^eCfª0¢²¨ª0 f « ¢ ^^¤,?µ c=c7d! ®¯±©ª0¤ 0 ¬&^©M^¤^0c4
=w®¯c¤ ª0c¤ ±¯¬=¨(.c¤ cpd9 =¬?f ¹c~cQ6
T^e^¬w6,^¤=e6? c=¬p, p¯¨ ª0^ f ¤ « ^¤ ¯T ÷ cwg=oúq=Cq?d9fø ^(x) ,= ce^d9¬ , =¤ 6cp µ ^^¥ =cV6h e^6[^¤,C, Q¤ c^ = » ^c=d*~ c ªQ ,;®¯
c~ª0 f ^« e^f c ^ ce6 u cc ñ6¤ c¥d#
¥ª ,e6 ª0¢Md9d9eÅw ¤ ccV=¨ c ¤ [d#QªQ= d9=¥d µ ÷ g=oúqQiø¥.=,? c^ Td¨(c¤ ud#ªC =d C ±0 ^¤,A(u) 4 ®¯ª0¤ B(u) ¬y© ¤ ©Cd9^ ^ c=wi¶
c ª0« f0 p± C T=¢M¶Q=f c!Cd9^ ^ =d90 =ªC
©!,Q,? ¬ §¦!¨Ip7¥=¤ d9c u^e^f 0¦¨ª0 ∞f0µ ^ ¤ ^¤ T ò
p¡w^e¥ ¨ª0 ♦f « ¡fw(x)d9 ¤,p Cc¤ fTp¡,^dTô§e6cº¤ ^¨d#ª0 eðf h« f~ ªQA(u) ¢¼ ¤ B(u) |u| → ∞ # b =
f
d c , ¤ p C c T d . 9 d , ô 0 f ?
^ 6 e ^
c d 0 ª
¤ c
[ ª 6 e = =
c ¤ ¨¬ª0ô f d9« c? ¡V c¤,=e^e6e^dP¤ p^d9¤ eÅp¬»fwf=^e^fºf c ¤ ^¥
¥ ce6¬ T;±*? =e¥ , ª0,=±þ¤p
=©0f ;®¯ ª0¤ c¬=ch& f cA
^^¤,? þ^¤ ®¯ cCª
µ fl¯(x) l÷ g=→ ∞ ∆u = ¤,π∆n/l = P ¤ 0 ª ^
9 d * ¥ e c = 9 d § ( ¨ c
¤ # d C ª ÷ = g o o ¥ ø o ú q ? q ø c 6 e ª 7 ^ 6 e p ,
M ¢ º p 6 Q c ¡ π/l → 0 ^ ¤ ^^¤ T^¤ cycC
d9 ^ ^ eC¢Mf c7±© ¨d9 ª0eð !f « c= ,¶^
e^f c c ^ c[,d9= e6cQ¡[c==^=e6d9 c¶c ¤ ¤ c¥e6
¥§ ,¦ ¥=f0¤ =d9fwc 6 cfcp¥e µµ ¤?b,^^^ gc ^¤ p¬ ^ ^¤ ¤ ^^¤ ±0T
^ d Éf ±he^e6 ^¤ fc^¤cd#u¢? ª =c0ce^ cu== ∞ ¢ ^^¤,? ¬ c±¼¨(c¤ d#ªC ®¯ª0¤ ¬ d9÷ g= o o øcp[ ,ª0e¥ ^¥
, ª0T¢Md97~^±V ¤ ¶É ¤,^=Te^e^¨(d9cc=¤ d#¤ ^ªC ,~÷ ¤g=o0
úq[q?ø¥®¯d iª0¤ ¬c= ºd9c^e^e6 c7cp
¬=cCf0cpQpp¬¥eð [¬¤ e66^ªC Q¬Q==f pc=0?µµ ce6 ¤ ¥
¥ ¬ c^c~ ^¤ ¥¦ cC
©÷ g=o oø4d9
cf0p¡[^dT;c~w÷ g=oúqQiøMe¥ ¥
,ª6÷ g=oúqq?ø Z∞ f (x) = [A(u) cos ux + B(u) sin ux]du. 0
n
× ß6Ø=ß ¼ ËéòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í 0 ¤ ¥
P cp ccQ¡c^ ¡[0ô¦~eCc==d9 cc=e^Pe^ ¥¤, =¬ ¥c
c e6¬»¨(c¤ d#ªC ®¯ª0¤ ¬t÷ g=o oø¯ ¤ ^f c=c¤ §¦ f (x) =e^e^d9c=¤ d* 6^¤? 1
SRSRI
1 I(l) = π
Z l h Z∞
÷ g=oúq?gø
i f (t) cos u(x0 − t)dt du,
A
¤ cCcp ¬ chf c ^ cw e¥ c ò ^f c=c¤ cw¨( f e^ ¤ c= c C,,=Qe6 l ^ > c0±*x Ée6ª0d96d9 ¤ªC
¤ 0cþ
tC=®¯d9ª06¤ ¬ ¬ É& cºc=C cp ,^ ^6¤,»? ëQx=÷ g= oeCúpq?gø¥¬t§ p ,^^ ¤,e^?¬º =®¯,?ª0 ¤ c¬^cd ÷ g=on µ¸o± ø S f0=f~ ¤ ¥
¥ Z Z i ÷ g=oúqQø 1 h lim I(l) = lim f (t) cos u(x − t)dt du −∞
0
0
n
l
l→∞
l→∞
∞
0
π
e¥ ¥
cp¥ ¬ c ; ^¤ ^ ^e6!eA¦ ^d#ªh e^e¥ ¥
cp !,=e6 c±!e6ª0d9d9 ^¤,? I(l) ôªe6p= c¬ªe¥ c~eA¦cQ
d9ce6h ^^¤,? y®¯ª0¤ ¬= S , ¥µ <~ ' ( D X >x =,x4l¯cfpQp¬ pc¨ª0 f « f (x) dPcQ¡6ò¬ ¤ ¥
e6p=p ^ ^^¤,?µ cd®¯ª0¤ ¬I[0
Z ÷ g=oúVq 3 ø f (x) = M (u) sin(ux + ϕ )du. 0
−∞
n
∞
u
0
@9 x ¹cQ
T^^¤,? ¬ª0¢ ¨ª0 f « ¢¼w÷ g=oúqqQø§ ¤ ¥
^dfô¨(c¤ d9==,? c^ 0µ c±©÷ g=o ø¥ l( ,ôñ6c^cV cp cQ¡V d M (u) =
? =
p A2 (u) + B 2 (u),
A(u) = sin ϕu , M (u)
tg ϕu =
bcA
B(u) = cos ϕu , M (u)
A(u) . B(u)
÷ g=oúq?nø
A(u) cos ux + B(u) sin ux = = M (u)[sin ϕu cos ux + cos ϕu sin ux] = = M (u) sin(ux + ϕu ).
bcA
^^¤,? ®¯ª0¤ ¬y÷ g=oúqq?ø& ¤ d96V0
f (x) =
A
M (u)
~,Q,? ¬,pô¨IpQ
Z∞
c ¤ ¥
¥ ^ e^cc= c=·7^ d9÷ g=oúq?nø¥ 0
ϕu
M (u) sin(ux + ϕu ) du,
÷ g=oúqQjø
!§pËZ&Ù0Ö@®ðÒß6Ï¥ÞÖÒÖ¶Î> Ñ Û0Ö ÞÑQÎ]¥Ö[ÖÒtQÍÐÜAÙ,ß6×ß@
PÝpÙu¿Í w ©± xyµ !Ñ ´ ß# * %& ++- ! ! # $Ê,
SRSRL
ó§¨(c¤ d#ªQ ¤ª^dK
,* ^^¤,? º®¯ª0¤ ¬hª ^¤¡[
^ T=,? c^ cwce^ c c± ^d9d9(¥=¤ d9c ^e^f0c6c¶p,? Q Qh ''¨ D ` >x =,x ò] N[O D _ N uN f (x) :=ap]^> ºTZD>MN,D,Z[Y @N=@ Y
:RM`=@>(Y¥B³Z_0Y ]a, ∞[G Z[> Z ÷ g=úq?ø lim f (x)e dx = 0. U -+
xk4^¤¡[
^ ¶÷ g=úq?ø&ñ^f ? ^ c[
=ª0dþ¤,=^ e6=d Z Z ÷ g=ùgø f (x) sin λx dx = 0, f (x) cos λx dx = lim lim
qV3 cùf0g7pQepª0 ¥ 6¬ce6dþcc^f c c=c¤ c§¦ =,^?^ ¤,?c ^ ë ÷ g=cú
q?cøPf0öpQp ^¥e^ c=¬e6e6=^ª 4, Te^ =cp ¬p^ª^d9cd#ªt ^d9d9 l7? ^=, cp =¥p f (t) =e^cp ¢M c ^^¤ ¤ª^d9c±w, R ,¤,=e^e^d9c=¤ d ^^¤,? Z ÷ g= ø f (t) cos u(x − t) dt. ^cd9cQ¡V c~÷ ¦ c=Q~ ~[e^d9Te¥ IA = c^c[C,Q ^ øT¤,=e^e^dPp¤ p¬Vf0=f~ ¤ ¥
¥ / Z Z ÷ g= 3 ø lim f (t) cos u(x − t) dt = f (t) cos u(x − t) dt. P k ¥ e c
ô = ^ e p c M ¢
c º ±
^ ^
¤
¤ ª ^ 9 d c 6 e þ ¨ 0 ª
f
«
= c C p c ,
6
c ¥ e ¥
c ? µ f (x) ¥ ¬ c7ª ^¤¡[
p¬ ,c cpµ¸ ^¤ §¦; ^^¤,? ÷ g= øPeA¦cC
eÅô¤,= cd9^¤ cc= cpµ e^¥ ¬ c u , ce^f cp ¬fªV
,~e^¥¦ u côdPp¡c¤ ¤ª6eðheA¦cC
,7 d9eÅh ^^¤,? cd ∞
iλx
λ→∞
a
∞
∞
λ→∞
λ→∞
a
a
∞
0
−∞
L
∞
0
L→∞ −L
0
−∞
Z∞ Z∞ Z∞ f (t) cos u(x0 − t) dt ≤ |f (t)| dt = Q < ∞. |f (t)| | cos u(x0 − t)| dt <
e6§¾ cp¤ µ¸^°=d9 c ¤ ¤,§eÅph¦; §f~e^¬þd9e^cQ,¡Vc^ d#ñ6cª¶ ª ¤ ª ¥^
¤ ¥^¡[ ,¤ª~
¡[p¤,
=^¬ § c= dPct6¤ e¥ c ¥ ^
,^ª=¤, ? eð¦^d9 ©^ c=e^±*e¥ ,¥=
e6c²= ¤,= =^ µ¸e6,=e6 ÷ g= c3 ± ø nd#ª!0
,ªw 0µ e6ª0^^d9¤,d9? ¤l¯0
ô¤ 0®¯¦ ª0 ¤ V¬÷Ð=qV.3 úq?¤ ø¥^0cl(¤, ,p~^ªñ6^dc6cV ^^¤ ^¤,c?, Q,I(l) f 0 ,
ª . = , ? c ^
c ? ¬ c ¤ ¥
e6p= d^^c0
−∞
−∞
−∞
1 I(l) = π
Zl h
lim
ZL
L→∞ −L
i f (t) cos u(x0 − t) dt du.
÷ g=ùnø
¹c¤ cC±ô¥
¥^c ¤ ¤ ¬f 0
c^f0^dTc[ ^¤ c~¥^¦^¤,cQ¤ =
,7¤ cQc7d9=^C ¤ ,,=hpf¶h eð÷ ¦g=cC^
ù^n ¤,ø d9? c[e6 ¬tc[÷ gp^¤ 3 ^ø4d9 ^ c= Cccp± ,u 6eh cTe¥ ¥^
,e6ª0©¢M7c C^±ô¤,Q==« d9 ¥¢ µ Z i hZ ÷ g=0 jø 1 cos u(x − t) du dt. I(l) = lim f (t) 0
l
L
L→∞
0
π
−L
0
SQàRN
× ß6Ø=ß ¼ ËéòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í ¾§cT^ e¥ c^dþ 4^^ª ¤,?¤ ,^ª¶ l¯C ^c7¤ 0 ¦ y^^÷Ф,qV?3 úq?¯ø¥y÷ g= jø c=Ccp , 6IQ= eCp¬¯^^c¯I0
=0=,?µ 1
1 I(l) = π
Z∞
f (t)
÷ g=;:ø
sin l(t − x0 ) dt t − x0
^¤,=¢M76d =ª»¡y¤ cp ¬ ¤ ¤,=e^e^d9c=¤ ^ ,ª ¡[¶ ^^¤,? ©®¯ª0¤ ¬=Pm(=ft ¤ ¤,=e^e^d9c=¤ ^ h¤ 0
®¯ª0¤ ¬=¥
^dþ¨ª0 f « ¢ ÷ g= oø ψ(τ ) = f (x + τ ) + f (x − τ ) − [f (x + 0) + f (x − 0)], ¤ ¥
cp cQ¡V [e6ª 7^e^c= ¯cC
ce6c¤ c 0¦~ ¤ ¥
¥ c f (x − 0) f (x + 0) @ V: ¶Ô^ @ YO D,_ N,N,N GÉp:=D,ap:=]^\> Y6D,ºòN Z[º D,>ôN,D,Z[Y @N @ %&
Y ( >Q Fô'D, :7D `=` G§@>x > Y^=(]¶Y¥N!U³B `=@Z[N! _ ·?DY x Y6]_ − >QZ[D,∞,Z[>A@Y ∞[ # G I R [ Z Q > 0 \ _ Y ] >Q ]>Q
0
0
f (x0 − 0)]/2
0
0
δ
Zδ
0
0
÷ g= ø
|ψ(τ )| dτ τ
G§>C`=@Y6
d9xcQ?¡V ^c[^ ¤,¤ ? ²¥
e6÷ g=p=;:øÉC¬V=d9[^ 0c
± τ = t − x !¶e^0 ,ª! 6 ce6dP c?µ (sin lτ )/τ 0
0
1h I(l) = π =
Z∞
sin lτ f (x0 + τ ) dτ + τ
0
Z∞ 1h
π
Z0
f (x0 + τ )
−∞
f (x0 + τ )
sin lτ dτ + τ
Z∞
f (x0 − τ )
sin lτ i dτ = τ
sin lτ i dτ = τ
÷ g=úqQiø l7? ^Mce^ cp ¬=^ª^d9eÅy ¤ ¥
e6p=p ^, ^d¥
« ²C^e6 Td ^e^ce6^ Éd ¥µ ^¤,? cdT Z ÷ g=úqq?ø sin lτ 2 1= dτ. π τ ¾§T ^~d»,=7±0÷
g=^d úqQiøe^cc= c=·7^ ÷ g=úqq?ø¥=ª0d9 cQ¡^ cM, e¥ c [f (x + 0) + f (x − 0
=
1 π
Z∞ 0
0
f (x0 + τ ) + f (x0 − τ )
sin lτ dτ. τ
∞
0
0
0)]/2
f (x0 + 0) + f (x0 − 0) I(l) − = 2 Z∞ sin lτ n 1 f (x0 + τ ) + f (x0 − τ )− = π τ 0
− f (x0 + 0) + f (x0 − 0)
o
1 dτ = π
Z∞ 0
ψ(τ )
sin lτ dτ. τ
0
÷ g=úq?gø
!§pËZ&Ù0Ö@®ðÒß6Ï¥ÞÖÒÖ¶Î> Ñ Û0Ö ÞÑQÎ]¥Ö[ÖÒtQÍÐÜAÙ,ß6×ß@
PÝpÙu¿Í
¹¤ ¥
¥ ¬ T±h ^¤ ¥¦cC
!w÷ g=úq?gø& ¤ 1 π
Z∞ Z∞
l→∞
e^cð =e^ cw÷ g=oúqQø¥0
=¥
f (t) cos u(x0 − t) dt −
0 −∞
1 = lim l→∞ π
Z∞
SQàZM
f (x0 + 0) + f (x0 − 0) = 2
÷ g=úqQø
ψ(τ ) sin lτ dτ. τ
¥ V4° ^(^¤ e^ ¤,¢§¤ =
c=7c=±[ ^,^=hf e6c[,wª0¯÷
0g=
6úqQ¬ ø,¤,=c^[^cªC¤ ^¢¯dP[ ¾§=ªQT
6 e¥
cf0d©pñ6Q=c=,¯ ^¤ e¥¥
~¥ c¤,f0ppCQp (¬ c^¤ ¤ ?¥ µ 0
lim
l→∞
Z∞
Z∞
Zδ
ψ(τ ) sin lτ dτ + τ
Zδ
ψ(τ ) sin lτ dτ = 0. τ
h ψ(τ ) sin lτ dτ = lim l→∞ τ
÷ g=úqV3 ø
i ψ(τ ) sin lτ dτ . τ
ó*ª0 6cdþªe¥ c»÷ g= ø&^c¤ ^d9 c ^d9d9[qV3 ùg7 d9^^d 0
lim
l→∞
0
δ
l7? ^&ª0e¥^ dT¥
,ª67c=Ée^=cp e^¢Mcp ,¢M? c ± 6^¤,^ ^¤¤ ª ^¤d9ªc^e6d9c¬Ie6¨[ª0¨ f ª0« f « f (x) ,4 ¤ cd96¡ª f ]−∞, , ¤ c∞[d96¡ª f ]δ, ∞[ ,bcA
¯
,~ce6p==·7^^ceðhw÷ g=0úVq 3 ø§ ¤ [f¥(x
¥ +τ )+f (x −τ )]/τ 0
0
0
Z∞ Z∞ ψ(τ ) f (x0 + τ ) + f (x0 − τ ) lim sin lτ dτ = lim sin lτ dτ − l→∞ l→∞ τ τ δ
δ
−[f (x0 + 0) + f (x0 − 0)] lim
[e^cc=6e6 he ^d9d9c±wg=úq( d9^^d lim
l→∞
Z∞
l→∞
Z∞
sin lτ dτ, τ
δ
f (x0 + τ ) + f (x0 − τ ) sin lτ dτ = 0. τ
¢¼¹ ¤ ^ce^c¥
,e6h=^cV, =cc^cV¤ c d* ^¤ ^¥¤,
?¥ ô ÷ cpg= ,ª0úq? n øòd Q=d9^ª~ ^¤ ^d9^ c± δ
lim
l→∞
Z∞
sin lτ dτ = lim l→∞ τ
÷ g=úq?nø
Z∞
sin y dy = 0. y
lτ = y
cyc ¤ ¥
¥ ¥µ
b#=f d¼c¤,pCcdT ¤ ¥
¥ þ! ¤,=c±t,=e6 ÷ g=úqQø
^± e66 ¬ c¤,=^tªQ ¢¯ c¶
cf0pCT=6ª ^¤¡[
^ (^c¤ ^d9? = Z Z ÷ g=úqQjø f (x + 0) + f (x − 0) 1 f (t) cos u(x − t) dt = du . δ
∞
lδ
∞
0
0
π
0
−∞
0
2
SQàRS ♦
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í 1
¹ce^f cp ¬fª¶c f?¦ô ^ ¤ ^¤ T ce6 f (x0 + 0) + f (x0 − 0) = f (x0 ), 2
c¯C,? ^ ^^¤,? ®¯ª0¤ ¬M
=67ñ6c±yc f04C,Q ^ eC=d9c±y¨ª0 f « f (x ) m(=f¯
,¤ 0
(®¯ª0¤ ¬==ªe¥ c I÷ g= ød9cQ¡V c4Q=d9^ ¬(ªe¥ c ^de6ª 7^e6cpµ = ♦7c f x f0c ^ §¦cQ
ce6c¤ c 0¦ ¤ cCcQ
§¦ f (x +0) f (x −0) ÷|e^dT ¤ ¤ ^ e=« , o¤ø¥ªe^^cd9c,e6?
= ¢M70¦;,^e¥ ôc f0 x p , 6eð¶c f0c=±~ ^ ¤ ^¤ T ce6~y
¨(¨(¥µ 0
0
0
0
0
0
0
°1³±
a%&!#$Ê,
=e^e^d9c=¤ dþ67(cC
ô ¤ C,=fôeA¦ cC
d9ce6~ ^^¤,? V®¯ª0¤ ¬=l( ,¶ñ6c^c[,=d c=¤ ^=ª6eðô
ce6=pc cV ¤ ce6cIccp7^ ^d9d9 qW:0úq Qh ''¨ Hh.>x =,x ò] N!O D,_ N,uN f (x) D,:[`=@]> (Y¥³B Z_ Y [0, L]G L > 0
ZL
f (x)
÷ iúq?ø
sin λx dx = f (+0). x
U +-
x;áI^d9d9ÀqW:0úqIwiúq¯¤,p6 ,=¢Meðwªe¥ c d9 ¤ ¥
,³. p , ^d9§µ d9, f¨ª0= f c« ^ d9fdP(x) ú/ qòe¥ ¤ ^= ª^6d9¯dP!T Wq :0cp úq ^ ¤ ^=[ªª6e¥! d9c=c cpl¯c ¤ 0ce6¦ »= cc(^¤,É= eC =d9 ^c ±[ ¨(ce6c¤0 µ ( i df #cªC [0,^ ¤ L] ccf [ ªe¥eÅ c¶c,=e6l¯^ ± ¤ .0y¦ f pÉ¡[
òc±w^f=6?f c=ccI¤ § ¤ ¦c¨d96ª0¡ f ª « cf [0, L] =d9ªQcQ
¡V6ô cd9¤,cp Cc=c ¬I c,± ôc^¤,= ^ c± ;ó*ª0 6cdñ6c^cV ^^¤,? º÷ iúq?ø§d9cQ¡V cfQ=(x) eCp¬ 0
ZL1
2 lim λ→∞ π
ZL
2 sin λx dx = lim f (x) λ→∞ π x
2 + lim λ→∞ π
ZL2
sin λx 2 f (x) dx + · · · + lim λ→∞ π x
0
f (x)
sin λx dx + x
0
ZL
f (x)
sin λx dx. x
m ^¤ cd#ªe¥ =6p^d9cd#ªV7 ¤,==c±y,=e6¶ ¤ d9^ dP( ^d9dPVWq :0úq0¯fVce6=? ¬ Td© ^d9c»dPºqV¤ 3 ^ùg0cóP? ¥c
e^c¬p
c¥f0 p¬Q pc ¬² ^÷|e^¤ dTcP4p=e¥ f¡=¥h=^¤,d9pc64
¤,I=¬òl¯ c ¥ f¬Q(+0) ª0I f ce6« p?º ¬l¯ T¤, =4fªQþ ¢¯^ p ? ¸ µ ¨ e^c± e6 ®Kû g?üýø¥ %&Y
:~ D, :~'R= ]^Y6HF»h.>x>?=]6N©¶ N¨
1 π
Z∞
du
Z∞
Ln
f (t) cos u(x − t) dt =
f (x + 0) + f (x − 0) . 2
2Z[>Q\0_:R!DY6`=@Y¸@EPRpD>?]6ZN!N,D,Z[Y @ :Vy@ :R=Y6DOD,_N,N,N 0
−∞
f (x)
3
÷ iùgø
©$²?Ë}X ÍÅÑðÙÍ Þßh
PÝpÙu¿Í U +-
x4¾§ce^ cp ¬=^ª^d9eÅ
SQàQà
¦cC
¶
cf0pQp¥ ¬e6^c¤ ^d9 g=úq9 b#^=f f c=#c,¤ =T d9¤ d9¤ ^6¤ ^#ªQ
,¬Qpºp= =d9^ ^M¤, ?cp ,tª0®¯ ^ª0 ¤ ¬T~d9 ceÅ µ cp ¬=^ª^d9eðh ¤ ¥
e6p=p ^, ^d ÷ g=úqQiø¥ Z∞
1 π
du
l→∞
h 1 Z∞ π
f (t) cos u(x − t) dt = lim I(l) = l→∞
−∞
0
lim
Z∞
1 sin lτ dτ + f (x + τ ) τ π
Z∞
f (x − τ )
sin lτ i dτ . τ
÷ i ø
^f c==ce^¤ e^d9Tc=d¤ dõe¥ c d ^¤ LT>±0 ,d9cQ^¡V^¤, ?c ¼Q= ¤,eCp=¬Vc±²[,0=e6
K÷ i ø¥òf c=c¤ T± TQ?
==·7 e^¬ 0
0
1 lim l→∞ π
Z∞
f (x + τ )
sin lτ dτ = τ
0
1 = lim l→∞ π
ZL
1 sin lτ dτ + lim f (x + τ ) l→∞ π τ
Z∞
f (x + τ )
sin lτ dτ. τ
¾e^0 , ª¶,e¥ ^¥d9
d9cp¥i úqI¬ c ^¤ c¯e¥ =6=6dPc=7V ¤,=c±h,=e6!e6cc= c=·7^ !¤,= c 0
L
ϕ(x +
0)/2
1 lim l→∞ π
Z∞
#b =fþf0=f ¤ c« ^ f0
0
ϕ(x + 0) 1 sin lτ dτ = + lim f (x + τ ) l→∞ π τ 2
d9 c?¡V¥ ¬ τ > 0
Z∞
f (x + τ )
sin lτ dτ. τ
÷ i 3ø
¤ º ¢4cd l Pc»e^ ¤,=¥
|(sin lτ )/τ | < 1 L
÷ iùnø ¤ c!cd9¨6ª0¡ ª f « f ]0,f (x∞[+;óPτ ) #¥
e^ccA p=e^¥ c!¬ ªc e¥ d9ccQ¡V ¢ c[^QcQ
¤ ^pd9¬p&==f e^ cp ε¢M> 0c L>^^¤ 1 ,¤ª^c[dP
, e^¥¦ l ,e^cA =e^ cw÷ iùnø¥ =ªQ
6VT cp ¬eðh ^¤,=^ e6c Z∞ 1 Z∞ sin lτ 1 f (x + τ ) |f (x + τ )| dτ. dτ ≤ π τ π L
L
1 Z∞ sin lτ f (x + τ ) dτ < ε π τ
,
,ª0¢M¶7 ¢4^(cC,^cVQ e^^f cp =¬ ª06cQ
cVdP? c^c ε 6 =^cC
=¤ ~ñ6cd#ª~ ¤ ¥
¥ þ÷ i 3 ø§ ¤ dP¥¶eÅ ¥µ Z 1 ÷ i jø sin lτ f (x + 0) f (x + τ ) lim dτ = . π τ 2 ½4,? c^ cd9cQ¡V c[ cf0pQp¬ ,c Z f (x − 0) 1 ÷ i; :ø sin lτ dτ = . f (x − τ ) lim L
∞
l→∞
0
∞
l→∞
π
τ
0
2
SQà 0
× ß6Ø=ß ¼ ËéòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í Q¹=cC,
e6 p= cf0ô÷ i jø9÷ i;:øP¶÷ i øP ¤ cC
f÷ iùgø¥ b^d»eC=d9Tdt^c¤ ^dPI
cfQµ = ^ e ^ e 9 d = c
¤
© d 6 7 T C c
¯ ª ( ¨ c
¤ # d ¯ ª
^ ^ , ¤ ? ¯ ¯ ® 0 ª
¤ ¬ = = 7 ·
¤ c f c
^ e
p c = ¬ ^ ª ^ # d ª * ¢
¤ 0 µ cQ¡^ 0¦; °Éÿ г% ! Ñy%&! +- ! !Ò}$Ê, °¤,p d9eðIf( ^^¤,? ¬ c±I¨(c¤ d#ªQ §®¯ª0¤ ¬M÷ g=oúqq?ø¥QÇ =d96 dTQcMª ¤ 6 , ± Q = ^e^^¤,ô? ¤ c=¥ô
e6eCp==d9pc ,± 6 !^¤ e^^cd9^c ± c±^f0c= cy¤c=ª0ô¢ö^¨[ª0f cf e^« ª¢ëeCc= ucô.cb#,=f
fcp= ,f¡Vñ6,py~¨òª0 ¬ôf « 6p µ ^^¤?u t [Cd9^ eÅ ¹cñ6cd#ª! c~e^c± e6=ª! 6 c± ¨cª0± f « ¤ ~Qd9=c?d9¡^ ^dº u ^,¤ ^ −u eCp¬V¨(c¤ d#ªQ ,ª¶e¥ ¥
,ª0¢M7^d*0
= 1
1 π
C
Z∞
du
Z∞
1 f (t) cos u(x − t) dt = 2π
Z∞
du
c c=Ccp , 6V¨(c¤ d#ªC ,ªh®¯ª0¤ ¬Q= eCp¬V[0
0
−∞
−∞
Z∞
1 f (x) = 2π
du
Z∞
Z∞
f (t) cos u(x − t) dt.
−∞
f (t) cos u(x − t) dt.
¹¤ d9^ dþf~ d9^¢M7^d#ªeðhñ6c±~¨(c¤ d#ªQ (f0c=e^ ªe6ªô¨(c¤ d#ªQ ,ª C 0± ^¤, −∞
−∞
¹cp ,ª0 d 0
1 cos u(x − t) = [eiu(x−t) + e−iu(x−t) ]. 2
1 f (x) = 4π
Z∞ Z∞
−∞
f (t)[e
iu(x−t)
+e
−iu(x−t)
]dt du
−∞
1 f (x) = 4π
Z∞ Z∞
f (t)e
iu(x−t)
dt du +
−∞ −∞
1 + 4π
Z∞ Z∞
f (t)e
−iu(x−t)
dt du.
ó[ ¤,c=d9c=c7±ô¬,¢À=e6 cC
e6¤,p== cf ¹czñ6=cd#−uª 6¤ªQ
côª0¥
¬eð .c¶ ^^¤,? (.e6cQ7 −∞ −∞
1 f (x) = 2π
Z∞ Z∞
f (t)e
iu(x−t)
dt du.
®7c¤ d#ªC ~÷ qúq?øP,pCT=6eÅô¤,p6 c?¡^ ^dº¨ª0 f « [f cd90 ^f e^ c±h¨(c¤ d9= −∞ −∞
÷ qúq?ø
f (x)
^^¤,? ®¯ª0¤ ¬=
©pÊ?Ë+Ï4Ñ6Þ¬× ÍÐÏ0ÎÐÒß6Úq£ÉÑðÙÞßMÖÒtQÍÐÜAÙ,ß6×ß@
PÝpÙu¿Í
SQàR7
¾§T ^eðhd9 cQ¡V¥ ¬ , Q= eÅ7 ±~c= t cp ,ª0 d
0
1 f (x) = 2π
Z∞ Z∞
f (t)e
−iut
−∞ −∞
1 f (x) = 2π
Z∞
eiux du
÷ q= ø
f (t)e−iut dt.
e¥ h6e6hf0c=dP0 6f0e^ c=C,Q ª0¢ ¨ª0 f « ¢
−∞
−∞
/
Z∞
÷ qùgø
dt eiux du
1 ϕ(u) = √ 2π
Z∞
f (t)e−iut dt,
Z∞
ϕ(u)eiux du.
÷ q= 3 ø
c ^^¤,? ¬ª0¢ ¨(c¤ d#ªC ,ªh®¯ª0¤ ¬V÷ q ø§d9cQ¡V cQ= eCp¬0
1 f (x) = √ 2π
−∞
®¯ª0 f « ¢ ,pCTp¢M¯e^ ^f^h¤,?d9 cC¬
, ªQc ±y¬ ¨ª0 f « ,^p±VCT0 =¢Me^t ^=fd9¤,0? ¬= ªQc
±V T0dT c=Tp µ =¤ c6e6ª0¬d9¢£^¨ª0 f « ϕ(u) f (x)¨Ip&C¹c¤ Tdþñ6e^ cdK |ϕ(u)| ^ f , ¤ = 9 d h ¨ 0 ª
f « b#=f0parg^¤ d9ϕ(u) cp 4 c^ôc=ªe¥ cp ^,¯^dTc7 ^^f¤,(x)? !®¯ª0¤ ¬¤,=e^e^dPp¤ =6eð f0¨=ª0fy f e^ « ^ f ¤,? ¬ c¤,p6 cQ¡[^ ^ 6¤ cQ
^e^f0c± Q?
= c±y,¯e^^±¶ e¥ cc±¶ce^
d9ce6¬=f=(x) f c^c[,[¤,p¥6= ¤ c?d9¡c^ f ô~ce= ªe¥^ c¤ p^ ¤ ^T, cVdP d9c=^^c ¢M 7e¥ ^d9 ÉeÅhd9,h=¨(e6c=Cp = d9^ e6f u dPô^Qc?C
¦cp?µµ ,=d9 °^¤,= d9eðcC
c± 0¦; c=C f=¢M76±©ôe^ e¥CdP?¦÷|ñ¥ ^f¤ c §¦ c=0µ C¤ ^ ^e^f 0Q¦hp=¶f??¯
e^ù ø¥e6 ^¤ dP^c4ñ^¤,f p^ ª0¢M?7 ^0¦h, M^f cc= ^c¤,¤ pTc¯¤e^ª ^,QQc e6ª 7 ó*^e6d&p? , ^¢MdP7pC d# ªICe^ f0¤ c^±~c¤,cp Cf cp µ = &¨ª0 f « f ?d9cQ
¥ ¤ª0¢M7^±p¦ cC
c±¯e^ ^,?R =,?CT=^d9T±67§c¤ ^ ,? cdT ¹¤ *cT^± ª0 4 ^ § ¦ cC
¤,= cc=± 4 e^p =^f ,c?±² e^, pe6^T^d9=£^d9ªQT
c±w=6 7c¯e¥ c=cQf0¡V Tf ±²cd¼p¦÷|ccQ
¤, pc=C±²cd&e^ øò^ ,eA?¦ cC
¤ c¥^
c µ eb6pc=A
¬¯T ^e¥± cV±ycf c¤,d9pCK ,÷|=c=« f0^ ±y f ñ¥ ^øTd9ñ¥^ ^=d9=^¤ =§=¦¶¤ e^§ ¦w^,e^? ^c, ?e c÷|d90 sincp ,ª0ux dcoscux¤,pø¥ ÷|c=f0 føT eð¦cC
c^cye¥ cQ¡V c^ce^ ^,? m¤ cd9(c^c ^e¥ w¨ª0 f « p , 6pµ eðe^=f Q=?
¤,pe^cc= c=c~·7 ^ ^^^d ¤ ¤ªCd9c±,¶e^^± e¥ cc±ce^ cô¨ª0 f « ff(x)(x) ϕ(u) −∞
iux
Z∞
2
f (x) dx =
Z∞
|ϕ(u)|2 du,
=¹4, ?= c·7^ ^ ¤ ¥ ,T d* ¤,=^ e6=ªh¹(=¤ e^^? ,ô
,h¤ 0
y®¯ª0¤ ¬¯h,pCT=^d9Td*¤,=^ e6cd ¾ ¤,=d9f?¦w ¤ =¥
^ c±¨(C ^e^f0c± ^¤ ¤ 6==« ^^¤,? ~®¯ª0¤ ¬7¤,=^0µ e6p¦cCc
¹4c^ c=e^ ·7^^,¤ ?¥ ,[ ¤,^=e6,¬dPe6ª0pd9d9^dPIpñ^ ^ ¤ ^^eC f ±hcþ^^cT6¤,=p¤ ¡d9c^ t ^e^f c^0c¦~f0¨Ic=d9f pc ^Qc ñ6 ^¤ ^ <~ ' ( H=,>x =,xI®¯ª0 f « ¢ f (x) = e ¤ ¥
e6==¬y 6^¤? c=d®¯ª0¤ ¬= −∞
−∞
−|x|
1 ×ß6Ø=ß ¼ ËéòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í SQàRB @9 x4Ç ?
= ,p¨ª0 f « ªC
cp 6c¤ 6e^^dKªe¥ c dK^c¤ ^d9 ®¯ª0¤ ¬=
ce^f cp ¬fª
Z∞
|e
|−x|
Z∞
| dx =
e
|−x|
dx = 2
bcA
e^cA =e^ cw÷ q ø¥ d9c?¡ cQ= eCp¬ −∞
−∞
1 = 2π
e|−x|
¾§T e¥ dª ¤ ^ ±w ^^¤,? Z∞
e
−|t|−iut
Z∞
0
eiux du
dt =
∞ e−x dx = −2e−x 0 = 2.
Z∞
e−|t| e−iut dt.
÷ q=ùnø
−∞
−∞
Z0
Z∞
e
(1−iu)t
dt +
Z∞
e−(1+iu)t dt =
÷ q jø ¹cC
e6p= cf0~÷ q jøP~÷ qùnø#
=6[¤,p6 cQ¡[^ ¨ª0 f « ~7 ^^¤,? ®¯ª0¤ ¬f cdµ 0 ^f e^ c±h¨(c¤ d9 −∞
−∞
0
e−(1+iu)t ∞ 1 1 2 e(1−iu)t 0 − . + = = = 1 − iu −∞ 1 + iu 0 1−iu 1+iu 1+u2 e−|x|
1 = 2π
Z∞
2 eiux du. 1 + u2
ó² cd9c=7¬¢¼¨(c¤ d#ªC C 0± ^¤,[^^cd9cQ¡V c[ ¤ ¥
e6p=¬V[0
−∞
e−|x|
1 = 2π
Z∞
2 (cos ux + i sin ux) du = 1 + u2
−∞
1 = π
h Z∞
cos ux du + i 1 + u2
Z∞
sin ux i du , 1 + u2
f ¨c=ª0 cf ¤ « T ±º±~eyd9c?ª0¡V 6 c7cd Q= 6eC pc¬Ve6f0þ=f ^ 6 ce6ºe^cc=6e6=ª0¢470¦º cQ
T^^¤,? ¬ ò¦ −∞
e
−|x|
−∞
2 = π
Z∞
cos ux du. 1 + u2
÷ q;:ø
®7c¤ d#ªC ,ªô÷ q; :ø.d9cQ¡V c4¤,=e^e^dPp¤ p¬(f0=f7 ^^¤,? ¬ª0¢*¨(c¤ d#ªQ ,ª®¯ª0¤ ¬ò¨(c¤0µ d9÷ q;÷ :g=ø&oúd9qc?q?¡Vø9e cA(u) cp ,ª0= 2/[π(1 ¬V~ +^ uce^)]¤ ¥
B(u) e6^ = cV0I¨(6c¤¤ ªQd#
ªQ c7Kª0÷ g=¥
oúqQi¬ø¥eð c¤ 6^ªC ¬?pp 0
2
°¦w
çy+- ! ×#$Ê,
+z +z($Êed
´ 6,pô¨ª0 f « 0ªQ
cp 6c¤ ¢Mp¶ªe¥ c dº ¤ ¥
e6p= d9ce6 ¹^^ª¤,e6? ¬ cfd²(x)®¯ª0¤ ¬=
©Ë+éòÒtQÍÐÜAÙ,ß6×f
PÝpÙu¿Í4Ó=Í]¥Ò Ñ¥Õ[Ö[Ò Í6Ó=Í]¥Ò Ñ¥Õq£PÝÒÏ[ÖÖ
SQàWG
^ k6 §¦hT¨pª0 f e^« c ± ± e6Ic!¤, =^^ ^e6¤,?! c÷ g=o úqQc!ø§ e^ cpd9 ,ª0d9 6 d¤ ,cd#ªt ^¤ ? ,ª»c= 6 §¦ Z∞
2 A(u) = π
f (t) cos ut dt,
B(u) = 0.
b#=f dºc¤,pCc=dÉ ^^¤,? ®¯ª0¤ ¬I 6 c±h¨ª0 f « ôQ= ·76eðhp=f 0
f (x) =
0
Z∞
÷ g0úq?ø
A(u) cos ux du
0
÷ g0ùgø ½4e6,~? c^ ^ ^¤, ?cy
c ,d!®¯ ª0^¤ ¬6= ,c ±cp ,¨ª0 ª0 df « ªQ
cp 6c¤ ¢M7^±ªe¥ c d ¤ ¥
e6p= d9cpµ 2 f (x) = π
Z∞hZ∞
i
f (t) cos ut dt cos ux du.
0
0
2 B(u) = π
A(u) = 0,
Z∞
f (t) sin ut dt,
e¥ ¥
cp¥ ¬ c ^^¤,? ©®¯ª0¤ ¬( ^ 6 c±w¨ª0 f « h d9^6V0
0
f (x) =
0
Z∞
÷ g0 ø
B(u) sin ux du
0
2 f (x) = π
Z∞hZ∞
i
<~ ' (H D x>=,x(#p6 cQ¡V¬[ ^^¤,? ©®¯ª0¤ ¬( 6ª0¢ 0
0
¨ª0 f « ¢ö÷ e6dÉ ¤ e=;g=ø
^e¥ |x| ≤ l; ^e¥ |x| > l. f (x) = ^ @e^9f c ^ cxdbc C¤ cd9cM6¨¡ª0ª f f « ¯ô ªCf
(x)cp =6e^c?c ¤¢M 6 yc4ª e¥ c^^¤ d¤ªl¯C dP¤ M0 µ ¦ óP &¥
Tc ¢4p¥c dh¬=f0 cc .^¤, p 6c dwc?¡ ¤ ^ c d96¡[¶ª f0=6p^¤0?
cM®¯ ª0^¤ ¬ce^¤ e6ª ¥
7e6^e6^= ª ^c ? b#=f¶f0=fôñ6p[¨ª0 f « ¶ë 6,p , ^^¤,? ®¯ª0¤ ¬( d9^60
f (x) =
A
1, 0,
Z∞
A(u) cos ux du,
0
2 A(u) = π
Z∞ 0
÷ g0 3 ø
f (t) sin ut dt sin ux du.
f (t) cos ut dt,
B(u) = 0.
9 e=g=
SQàRI
¾§T e¥ df cñ^¨(¨( « ^ 2 A(u) = π
Z∞
A(u)
1
Zl
2 f (t) cos ut dt = π
1 · cos ut dt =
b#=f dºc¤,pCc=dÉ e^f cd9Td¤,p6 cQ¡^ ^dº p , 6eÅ 0
0
Z∞
2 f (x) = π
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í
l 2 21 sin ut = sin ul. πu πu 0
sin ul cos ux du. u
¤ §¦~¨(¨cª0¤ d#f ªC« 7we^ d9¤,^=6y¥
¤, p C¤ ¯TV
, ¶^¤ e^c¥¦V^cVC,¤ ?cC
^ ¾± ñ6x0Q¦ô7
=e^f0ª¦~ ¢4 c^ f0 ?^¦~d x=^^±l¤Q »7®¯f0ª0c=¤ ¬cp µ p ¤,=^ 1/2 ~c ¤ ¥
¥ , 6V¨ª0 f « ¢ ^e¥ |x| < l; ( Z 1, ^e¥ 2 sin ul 1/2, ^e¥ |x| = ±l; cos ux du = π u |x| > l. 0, ¾*,=e6 ce6. ¤ l = 1 =ªC
^d d96¬ ^e¥ ( Z π/2, ^e¥ |x| < 1; sin u π/4, ^e¥ |x| = ±1; cos ux du = u 0, |x| > 1. ¹cp cQ¡V 76
^e^¬ x = 0 ,,=±0
^dºC,Q ^ I ^^¤,? 0
C
∞
0
∞
0
Z∞
π sin u du = . u 2
¾*ñ6l(c ,d[e¥ñ6 ,ª0c,±=¡(T ¤ ¨ ª0d9 ^f « ^Vd,¨(=±0c
¤ ^d#dªQ ,ª¶^ ^¤ ¬¯^^ ¤,? ^y^¤,®¯? hª0¤ ®¯¬ª0I¤ [¬Éf0c(d9f00c d9^0f e^ ^f ce^±h ¨(c±Vc¤ ¨(d9c=¤ d9= 0
1 f (x) = 2π
¾§T e¥ dª ¤ ^ ±w ^^¤,? Z∞
−∞
=−
f (t)e
iu(x−t)
Z∞ h Z∞
−∞ −∞
dt =
Zl
e
i f (t)eiu(x−t) dt du.
iu(x−t)
−l
1 iu(x−t) l dt = − e = iu −l
2eiux eiul − e−iul 2eiux 1 iu(x−l) [e − eiu(x+l) ] = = sin ul. iu u 2i u
óP¤ ¥c
c¤ 6pp=¥6 y¬= e¥c .¥
,
ª0 ,¢M!7Q?
±w= 0
c±¨ª0 f « © ^^¤,? º®¯ª0¤ ¬[ôf0cd90 ^f e^ c±©¨(c¤ d9 1 f (x) = π
Z∞
−∞
sin ul iux e du. u
R© ©=Ë+éòÒtQÍÐÜAÙ,ß6×f
PÝpÙu¿Í£PÝÒÏ[ÖÖDâu®6ßÅÛ,ß6ÒÒ Ñ¥ÕÒß ¬,Ñ¥×ÝÑQÎÐÖ °°± xy +! ³+- ! % #$Ê, ($Êe dx ãµ!- !z !. $¦
SQàRL
¹ªe6¬h¨ª0 f « Q?
=,ô,¶ cp ,ªce^ .=e^cp ¢M c~ ^^¤ ¤ªCdPô, ^±~~,¯ ¢4cdºf0c f^ (x) cdþ ¤ cd96¡ª f (ªQ
cp [0,6∞[ c¤ 6Vªe¥ c dl¯ ¤ 0¦ = b#=¶f² ¡¤ c=d9T6f¡=ª fcñ6f cº
¥ ? ce^ ¬þ¹¤ c=7^c¤ e^ ^^¤ c¶0ñ6
ccôd9®¯cQ¡Vª0¤ ¬cy=TeA
¥c cp ¤ ¬ ¥.
¥¤ cC
dõcp ,¨¡Vª0 yf « ^ ¢ f ¤
(x) cd96¡ª cf ] − ∞, 0[] −I∞, ¤ 0[cd96¡ª f [0, ∞[ 6 òd*0 ~ ^ 6 Édc¤,pCcdT ¹¤ ! 6 cd ¤ cC
cp ,¡^ !¨ª0 f « d9¼ cp ¬=^ª^d9eð ^^¤,? cd®¯ª0¤ ¬ ÷ cg0±wùg¨øª0
,f 7« 6 c±= ¨cª0d9 f « d² 6=7 ¤ 7¤,p ^; 6 côcVd!f (x) c f0?¦!^^¤,¤,p?C ¤ cTd©y®¯ ª0¤ ¬^(^÷ ¤,?g0 » 3 ¤,ø;=
,^7! ^cp ,6ªpµµ e6ª0d9d9( ¤ ¥
¥ cV¨ª0 f « he^ ¤,=[~e¥ ^ <~ ' (H H x>=,x(#p6 cQ¡V¬[ ^^¤,? ©®¯ª0¤ ¬(¨ª0 f « ¢
f (x) =
x, 0,
^^e¥e¥
0 ≤ x ≤ l; x > l,
ªC¤ cC
^d¼cp ,÷|¡Ve^dT ,¯¤ ^ e=,¯ic=ø¥ ¤ «,p¥ ¬ª¢ cp ,ªce^¬ ^ 6 Tdºy 6 Tdºc¤,pCcdTIp=f¡[
9 e=i ªe¥ @ 9c dþ xɤ ¹¥¤
ºe6p= ¢4 d9ccdÀe6wh ¤ ¥^¦*^¤, ?¤ cCc
d²cp ,®¯¡ª0^¤ ¬ =±º
= ,p¨ª0 f « þªQ
c? 6c¤ 6 Ç =
7 · ^ » d
^ ^ , ¤ ? w ¯ ® 0 ª
¤ ¬ = 0
¤ C c
p c , V ¡
I ¨ 0 ª
f
«
¢
^
6 , T d c , ¤ p C c * d | ÷ ^ e T d
¤
= e iúa^ø¥ 2 f (x) = π
Z∞hZ∞
f (t) sin ut dt sin ux du.
¾§T e¥ dª ¤ ^ ±w ^^¤,? ! c,=e6Q dT 0
0
Z∞ 0
¹cñ6cd#ª
i
Zl
Zl l t 1 f (t) sin ut dt = t sin ut dt = − cos ut + t cos ut dt = u u 0 0 0 l l 1 sin ul l . = − cos ul + 2 sin ut = − cos ul + u u u u2 0 2 f (x) = − π
Z∞
ul cos ul − sin ul sin ux du. u2
p¯¨(c¤ d#ªQ ¯e^ ¤,=¥
(
,ye^¥¦C,Q ^ ± x Q¯ e^f0 ¢4 ^ ^dc ^fy¤,pC¤ T
, x = l C,Q ^ M ¤,==c±,=e6=ªQ
6Ip
cTd9^ ¬=·7ÉC,Q ^ ¨ª0 f « x = ±l 5 C
0
S0N
ô¤,= c
l/2
1
?p=f f (x) =
0
(
x, l/2, 0,
^e¥ ^^e¥e¥
0 ≤ x < l; x = l; x>l
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í
f (−x) = −f (x)
÷ úq?ø
^e¥ |x| < l; ^e¥ x = ±l; ul cos ul − sin ul 2 sin ux du = − ^e¥ |x| > l. π u ¤ e=Ç = iú:Q·7ø¥ ^d^ ^¤ ¬¯ ^^¤,? ô®¯ª0¤ ¬= ¤ cC
cp ,¡V (¨ª0 f « ¢² 6 Td©c¤,pCcd÷|e^dT Z∞
(
2
0
2 f (x) = π
Z∞
cos ux du
Zl
t cos ut dt =
¾§T e¥ ^ ¯ª ¤ ^ ^^cy ^^¤,?
=6 0
Z∞
¹cñ6cd#ª
f (t) cos ut dt =
0
Z∞
x, ±l/2, 0,
f (t) cos ut dt.
0
cos ul − 1 l sin ul . + u2 u
0
2 f (x) = π
Z∞
cos ul − 1 l sin ul + cos ux du. u2 u
Q(m [= fte^tf0 !¢M ^¤ ¥,
^§d
,ª 7c C^d fhe¥¤, ,pª0C,¤ =T=# ñ6=x!=¨(±lc¤ d#
ªC , ~!f e6 c=¤,c=¤ §¥
¦~ C ,Qh
^ , I e^¥¦»^C^¤,,?Q ^V p
± cx d9^ ¬=·7C,? ^ ~¨ª0 f « h~¤,= c l/2 ?p=f ^e¥ 0 ≤ x < l; ( x, ^e¥ f (−x) = f (x) ÷ ùgø f (x) = l/2, ^e¥ x = l; x>l 0, 0 ^e¥ |x| < l; ( Z |x|, ^e¥ 2 cos ul − 1 l sin ul l/2, ^e¥ |x| = l; cos ux du = + π u u 0, |x| > l. Ç =
7 · ^ T d # , = f c
^ « 9
^ ^ , ¤ ? ² ¯ ® 0 ª
¤ ¬ y
, t ¨ 0 ª
f
«
9
¤ C c
p c , «,p¥ ¬ª0¢ cp ,ªce^¬wªQ ^dT f (x) = 0
, x < 0 ÷|e^dT¤ e=#i RCø¥¡^¾§ c e^c ±cp ,¬=!^ªc=^d9¤ eð0 µ ¨(c¤ d#ªC c±»÷ g=oúqq?ø¥ [f c=c¤ c± 0
∞
2
0
1 A(u) = π
Z∞ Z∞ Zl 1 1 f (t) cos ut dt = f (t) cos ut dt = t cos ut dt; π π
−∞
1 B(u) = π
Z∞
0
0
Z∞
0
Zl 1 1 f (t) sin ut dt = f (t) sin ut dt = t sin ut dt. π π
p y¥ ¬ c ^^¤,? *ª ¡ÉT e¥ ^ ² ¤ V 6 cdtV ^ 6 cdt ¤ cC
cp ,¡[^ 0¦ e¥ ¥
cpµ −∞
C
A(u) =
1 sin ul − ul cos ul , π u2
0
B(u) =
1 cos ul − 1 + ul sin ul . π u2
© ¼ ZË &Ù0Ö ÞÍ Ù0ÔtÙ,ß®ð× Ñ¾(ÍÐÒÖÚq£PÝÒÏ[ÖÕØ(ÖÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í
°4e^¢§
d9^^dþ¤,p6 cQ¡^ 1 f (x) = π
Z∞
S0M
[cos ul − 1 + ul sin ul] cos ux + [sin ul − ul cos ul] sin ux du, u2
f e6c=c¬V¤ cf~7e70
,cª d9c=7¬¢ÀC^e6 §¦!¤ ^c cd96¤ Ce^f 0¦e^cc= c=·7^ ±d9cQ¡V cVª0 ¤ cpµ 0
1 f (x) = π
Z∞
cos u(x − l) − cos ux + ul sin u(l − x) du. u2
÷ ø
0
~ Q c V ¡
c
¤ c ^
¤
¬ & © c 6 ñ c [ ¡ ô T , ¤ p ¡ ^
~
p c , 0 ª
ð e § ^ ¥ e þ 9 d ^ 6 e » c ( ¨ c
¤ 9 d ÷ g=oúqq?ø&ce^ cp ¬=Ccp¬eð~¨(c¤ d9c±©÷ g=o oø¥ ¢4e6T¤ ¤, p¤ ¡ª^^ h ^dë c=Cd9¨cQª0¡V f c« e6 ¬¶ò Q0?µµ ^^¤,°4? d9þ6®¯ ª0dT¤ .¬côc , =e^e6Tc?p7¬ ±¥
¤ Td9dõ^¤=,,?= A ,0
6côe^f,0 d
= T,y¤,pC §¦! ¤ cd96¡ª f0?¦¤,p6 Td9¨(c¤ d#ªC =d9 .¹¤ !ñ6cd¨ª0 f « Q?
= ,p¶,7 cp ,ªce^ d9c?¡6?f0=fô¯
= cde¥ ,ª0,== ò¬[c eC=,=T¤,p¡[^ d9 ,÷ V úq?cpø¥ ,.ª÷ ce^ ùg=ø&x 0> ©0 ÷ c ¤ ¥ø¥
¥m , ¤ ¢Mcd9~cC
c^ªwc hñ6=ªh ,¡c7c¨=7ª0 f ^« c ¢Kc¤ ¤,e¥ p¥6
c p T¥ I¬ c .^e6^c¤,?, ? µ
=¢M
¤ª0Ie
¤ª0^c=dÉ,f cð
x > 0 ° À
xy%&' !µ AÂ Ñ($Êedz .+- ! ×#$Ê,
<~ ' (HtJx>=,x(¹¤ ¥
e6==¬V ^^¤,? cd²®¯ª0¤ ¬I¨ª0 f « ¢ f (x) =
(
0 1 0
¤ ¤ ¤
x < 0; 0 ≤ x ≤ 1; x > 1.
@¢ 9 e6ª 7 ^e6xb#=cf=f0 =f[²ñ6= ¨^ª0^ ¤,f ?« þ[®¯ªQ
ª0¤ cp¬ Qø¥6§cc¤ ^!6Id9ªcQe¥¡V c ct d¤ ¥
^e¥c=¤ =^d9²¬º®¯ ª0¤ ¬^I^¤,÷ýª?e¥ ccpd µ ®¯ª0¤ ¬ A
Z∞ f (x) = [A(u) cos ux + B(u) sin ux]du, 0
1 A(u) = π B(u) =
Z∞
−∞ Z∞
1 π
−∞
f (t) cos ut dt;
f (t) sin ut dt.
S0S
¾§T e¥ d
A(u)
B(u) Z0
1 A(u) = π
1
1 0 · cos ut dt + π
−∞
1 π
Z1
cos ut dt =
0
Z0
1 0 · sin ut dt + π
−∞
=
b#=f¶f=fô[c f0?¦
1 π
1 cos ut dt + π
0
= 1 B(u) = π
Z1
0
x=0
sin ut dt = − x=1
1 sin u 1 sin ut 0 = ; πu πu
Z1
1 sin ut dt + π
Z∞
0 · sin ut dt =
1
1 1 cos u 1 cos ut 0 = − + . πu πu πu
¨ª0 f « ~¤,pC¤ T, c
f (0 − 0) + f (0 + 0) 1 = , 2 2
°f c ,p¥ ¬ c[Q= ·7^d
0 · cos ut dt =
1
0
Z1
Z∞
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í
f (1 − 0) + f (1 + 0) 1+0 1 = = . 2 2 2
Z∞h
i cos u sin u 1 sin ux du = cos ux + − + πu πu πu 0 x < 0; 0 Z∞ 1 0 < x < 1; sin xu − sin(x − 1)u 1 x > 1; 0 = du = π u 1/2 x = 0; 0 x = 1. 1/2
¤ ¤ ¤ ¤ ¤ ¹¤ ¥
e6p= dñ6c¤,p6 cQ¡[^ ò¨ª0 f « ^^¤,? cd»®¯ª0¤ ¬Tf cd90 ^f e^ c± ¨(c¤ d9=0l( ,~ñ6c^cT e¥ de^ ^f¤,? ¬ª0f¢ (x)0 c= ce6¬ Z∞ Z1 1 1 −iut ϕ(u) = √ f (t)e dt = √ e−iut dt = 2π 2π −∞ 0 −iut 1 −iu 1 1 e e 1 1 = √ (1 − e−iu ). =√ − − + =√ iu 0 iu iu 2π 2π iu 2π
óP ¥
cp¥ ¬= c e^f cd9c( ¤ ¥
e6p=p C ¯ d9^6V0
1 f (x) = 2π
Z∞
1 iux e−iu 1 + − e du = iu iu 2πi
Z∞
1 − e−iu iux e du. u
<~ ' (HtJxED x(¹¤ ¥
e6==¬V ^^¤,? cd²®¯ª0¤ ¬I¨ª0 f « ¢ −∞
f (x) =
(
0 2−x 0
¤ ¤ ¤
−∞
x < 0; 0 ≤ x ≤ 2; x > 2.
© ¼ ËZ&Ù0Ö ÞÍ Ù0ÔtÙ,ß®ð× Ñ¾(ÍÐÒÖÚq£PÝÒÏ[ÖÕØ(ÖÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í S0à @9 xb#=ff0=f[ñ6=¨ª0 f « [ªQ
cp 6c¤ 6Iªe¥ c d^c¤ ^d9²®¯ª0¤ ¬I÷ýªe¥ cpµ
¢õe6ª 7^e6c= º ^^¤,? »®¯ª0¤ ¬Qø¥#c^=9f0=f ¤ ¥
§
,ª 0¦þ ¤ d9^¤,?¦; d9cQ¡V c ¤ ¥
e6==¬V ^^¤? cd²®¯ª0¤ ¬(¨c¤ d9V÷ g=oúqq?ø¥ ¾§T e¥ d
A(u)
Z∞ f (x) = [A(u) cos ux + B(u) sin ux]du.
B(u)
1 A(u) = π
Z∞
0
1 f (t) cos ut dt = π
Z2
(2 − t) cos ut dt =
1 cos 2u − ; πu2 πu2
1 π
Z2
(2 − t) sin ut dt =
sin 2u 2 − . πu πu2
−∞
1 π
B(u) =
b#=f¶f=f
0
Z∞
f (t) sin ut dt =
ëc f0¤,pC¤ T[¨ª0 f « c −∞
x=0
0
0+2 f (0 − 0) + f (0 + 0) = = 1. 2 2
°f c ,p¥ ¬ c[Q= ·7^d Z∞h 0
2 i 1 cos 2u sin 2u cos ux + sin ux du = − − πu2 πu2 πu πu2 1 = π
Z∞ 0
2u sin ux + cos ux − cos u(x − 2) du = u2 x < 0; 0 1 x = 0; = 2 − x 0 < x ≤ 2; 0 x > 2.
¤ ¤ ¤ ¤
Ç ^^=¤,d9?6 ¬ dTc±h¨(cyc¤ ñ6d#cªQ ¡[c(±w ®¯¤ ª0¥¤
¬=e6Ip=[p ^0,
V7÷ d9g=ocQ ¡Voø¥ c cp ,ª0 ¬ ^e¥ wc=e^ cp ¬=Ccp¬eÅ 1 f (x) = π
Z∞
du
Z∞
f (t) cos u(x − t) dt.
l¯ ^^± dþe6 c[,¥= e6¬? c dT ª ¤ ^ ±© ^^¤,? tT e¥ , 6eÅcQ
cf ¤,? Td ^^¤ ¤ c?µ 0
Z∞
−∞
f (t) cos u(x − t) dt =
−∞
Z2
(2 − t) cos u(x − t) dt =
0
=
2u sin ux + cos ux − cos u(x − 2) . u2
× ß6Ø=ß ¼ ËéòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í c±ô¹¨(¤ c¥¤
d9e6p==l( ,d!ô==ñ6fc¡^§cV¤,pT6 cQe¥¡ ^ d òe^ ¨^ª0f f ¤,« ? ¬fª0(x) ¢ 0 c=^ ^c¤,e6? ¬ cd©®¯ª0¤ ¬ò4f0cd90 ^f e¥µ S 0R0
1
1 ϕ(u) = √ 2π
Z∞
f (t)e
−iut
−∞
1 dt = √ 2π
Z2
(2 − t)e−iut dt =
0
1 2 1 e−iut 2 e−2iu 1 2 − 2 =√ − 2 + 2 . =√ u 0 u u 2π iu 2π iu
óP ¥
cp¥ ¬= c e^f cd9c( ¤ ¥
e6p=p C ¯[f cd90 ^f e^ c±h¨(c¤ d9( d9^6V0
1 f (x) = 2π
Z∞
2 1 − e−2iu iux e du. + iu u2
<~ ' (HtJxEH x(¹¤ ¥
e6==¬ ^^¤,? cd®¯ª0¤ ¬òMf0cd90 ^f e^ c±7¨(c¤ d9§¨ª0 f « ¢ −∞
f (x) =
@9 xl( ,~e^ ^f¤,? ¬ c±h0 c= ce6 1 ϕ(u) = √ 2π
Z∞
cos x x ∈ [0, π/2[; 0 x∈ / [0, π/2].
1 f (t)e−iut dt = √ 2π
T e¥ dK ^6¤? ² c©,=e6Q dT& cp cQ¡V cos t dt V = sin t −∞
Zπ/2 cos te−iut dt 0
Zπ/2 Zπ/2 π/2 cos te−iut dt = e−iut sin t 0 + iue−iut sin t dt.
¹c=c¤ c( 6^¤, ¤ c= I c,=e6Q dº
p6 0
°4e^¢§
0
Zπ/2 Zπ/2 −iut iπu/2 2 e cos t dt = e + iu + u e−iut cos t dt. 0
0
Zπ/2 eiuπ/2 + iu −iut e cos t dt = 1 − u2 0
1 e−iuπ/2 + iu . ϕ(u) = √ 2π 1 − u2
óP ¥
cp¥ ¬= c e^f cd9c( ¤ ¥
e6p=p C ¯ d9^6V0
1 f (x) = 2π
Z∞
−∞
U = e−iut dU = (−iu)e−iut dt dV =
e−iuπ/2 + iu iux e du. 1 − u2
© ¼ ZË &Ù0Ö ÞÍ Ù0ÔtÙ,ß®ð× Ñ¾(ÍÐÒÖÚq£PÝÒÏ[ÖÕØ(ÖÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í <~ ' (HtJ xKJ x(¹¤ ¥
e6==¬V ^^¤,? cd²®¯ª0¤ ¬I¨ª0 f « ¢
S07
¤ −2 ≤ x < 0; ¤ 0 ≤ x ≤ 2; f (x) = ¤ |x| ≥ 2. ¤ @¥9
e6 p= x ¬yC pt¨^^ª0¤, ?f « cd²*®¯ªQª0
¤ c¬p 6c¤ 6ªe¥ c d´^c¤ ^d9䮯ª0¤ ¬= / !d9cQ¡V c (
x+2 −x + 2 0
Z∞ f (x) = [A(u) cos ux + B(u) sin ux]du.
®¯ª0 f « ~ 6,p cñ6cd#ª 2 A(u) = π
Z∞ 0
0
B(u) = 0
¾§T e¥ d
A(u)
Z2 Z0 2 2 f (t) cos ut dt = (−t + 2) cos ut dt + 0 · cos ut dt = π π 0 −∞ 1 2 1 − cos 2u 2 cos 2u − 2 + 2 = . = π u u π u2
b#=f¶f=fô¨ª0 f « h ^ ¤ ^¤ T,,e^^±~ e¥ cc±~ce^ c 2 f (x) = π
Z∞
Z∞
1 − cos 2u 4 cos ux du = 2 u π
0
sin2 u cos ux du. u2
0
<~ ' (HtJxEP x(¹¤ ¥
e6==¬V ^^¤,? cd²®¯ª0¤ ¬I¨ª0 f « ¢
¤ ¤ x > 0; f (x) = ¤ xx =< 0;0. @9 x C py¨ª0 f « ! d9^6ô¤,pC¤ Ty ^¤ c^c¶¤ cC
VVc f ;¹¤ c^¤ dT p , 6eð¶ ~¨ª0 f « f (x) =e^cp ¢M c ^^¤ ¤ªCd9c±h,e^^±hx =e¥ 0c=c±~ce^ (
Z∞
|f (x)|dx =
Z0
x
| − e |dx +
Z∞
0 |e−x |dx = ex
−∞
?cd^®¯^¤,ª0?¤ y¬ eA¦ cC
eð óP ¥
cp¥ ¬ c ¨ª0 f « ¢ −∞
0
−∞
e−x 0 −ex
f (x)
∞ − e−x = 2. 0
d9cQ¡V c ¤ ¥
e6p=¬I ^^¤,?µ
Z∞ f (x) = [A(u) cos ux + B(u) sin ux]du.
b#=f¶f=fôñ6=[¨ª0 f « ~ ^ 6,p cñ6c=d9ª 0
2 B(u) = π
Z∞ 0
A(u) = 0
2 f (t) sin ut dt = π
Z∞ 0
¾§T e¥ d
e−t sin ut dt.
B(u)
S0B
1
¹¤ c ^^¤ ¤ c= c,=e6Q dT,=±0
^d
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í
h e−b cos ub 1 e−t sin ut b 2 + − B(u) = lim − + 2 b→∞ π u u u 0 Zb i 2 h e−b cos ub sin ut −t (−e )dt = lim + − + u2 π b→∞ u 0
1 e−b sin ub i 2 1 + − − lim 2 2 u u π b→∞ u
0 ce^f cp ¬fª
2 lim π b→∞
Zb
e
−t
Zb
e−t sin ut dt
0
2 2 1 sin ut dt = − lim 2 πu π b→∞ u
Zb
e−t sin ut dt,
0
0
2 Zb 1 21 , lim 2 + 1 e−t sin ut dt = b→∞ u π πu
c
0
ôe^cc=6e6^ ,c
2 π
e−t sin ut dt =
2u2 πu(1 + u2 )
0
B(u) =
óP ¥
cp¥ ¬= c ¹cp cQ¡V
Z∞
2 f (x) = π
cp ,ª0 d x=1
Z∞
2u . π(1 + u2 ) u sin ux du. 1 + u2
0
e−1
c=fªC
2 = π
Z∞
u sin u du, 1 + u2
0
Z∞
x sin x π dx = , 2 1+x 2e
?^¤, =?9 c? ,ª0 d e¥ ccyC,Q ^ ~67ôcC
c^c 6¤ Ce^ c^c ^e^ce6^ ,c=Cc ¥µ 0
<~ ' (HtJxEQ x(¹¤ ¥
e6==¬V ^^¤,? cd²®¯ª0¤ ¬I¨ª0 f « ¢ f (x) =
|x| + 1 0
¤ ¤
|x| ≤ 1; |x| > 1.
© ¼ ËZ&Ù0Ö ÞÍ Ù0ÔtÙ,ß®ð× Ñ¾(ÍÐÒÖÚq£PÝÒÏ[ÖÕØ(ÖÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í S0G @9 x C p¨ª0 f « 7ªQ
c=p 6c¤ 6ªeÅ c d!^c¤ ^d9*®¯ª0¤ ¬== óP ¥
cp¥ ¬pµ
c ^(d9c?¡V c ¤ ¥
e6=p¬V ^^¤,? cd²®¯ª0¤ ¬
Z∞ f (x) = [A(u) cos ux + B(u) sin ux]du.
®¯ª0 f « ~ 6,p cñ6cd#ª ?^^¤,?
0
B(u) = 0
2 A(u) = π
Z∞
¾§T e¥ d
2 f (t) cos ut dt = π
A(u) Z1
(t + 1) cos ut dt.
,=e6Q d
=6 p=f c^c~0
¶d9¼ª ¡[ ^cC
cf ¤,p ch=T e¥ ,0 ?^^¤ ¤ c= c 0
0
2 2 sin u cos u 1 2 2 sin u 1 − cos u . + 2 − 2 = − π u u u π u u2
óP ¥
cp¥ ¬= c c f0?¦~ ^ ¤ ^¤ T ce6 A(u) =
2 f (x) = π
Z∞
2 sin u 1 − cos u − cos ux du. u u2
<~ ' (HtJxUT;x(¹¤ ¥
e6==¬V ^^¤,? cd²®¯ª0¤ ¬I¨ª0 f « ¢ 0
f (x) =
sin x 0
¤ ¤
0 ≤ x ≤ π; π < x,
¤ cC
cp ,¡V [^( ^ 6 Édc¤,pCcdT c @ 9^( d9c? ¡Vx C cp ¤ ¨¥
ª0e6 =f p« 7¬VªQ
c=p ^6^¤,?c ¤ c d²6®¯ªª0eÅ ¤ ¬c d!^c¤ ^d9*®¯ª0¤ ¬== óP ¥
cp¥ ¬pµ Z∞ f (x) = [A(u) cos ux + B(u) sin ux]du.
®¯ª0 f « ~ ¤ cQ
cp ,¡^, ^ 6 Édc¤,pCcdT, c=ñ6cd#ª 0
2 B(u) = π
Z∞ 0
=
2 π
Zπ 0
Z∞
¾§T e¥ d
sin t sin ut dt =
0
1 cos(t − ut) − cos(t + ut) dt = 2
Zπ 1 cos t(1 − u) dt − cos t(1 + u) dt = π 0 0 1 sin t(1 − u) π 1 sin t(1 + u) π = − = π 1−u π 1+u 0 0 h i 1 sin π(1 − u) sin π(1 + u) = − = π 1−u 1+u 1 sin πu sin πu 2 sin πu = = + . π 1−u 1+u π(1 − u2 )
1 = π
Zπ
2 f (t) sin ut dt = π
A(u) = 0
B(u)
S0I
b#=f¶f=f
f (x)
ö ^ ¤ ^¤ T,ph¨ª0 f « c 2 f (x) = π
Z∞ 0
°Y\
1
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í
sin πu sin ux du. 1 − u2
xy!µ !z#$Ê,
=e^e^d9c=¤ dw ^f c=c¤ T±If0 =e^eP¨ª0 f « ±
cp ,ª0^e^f c ^ cd b = ∞ 0 w^e^f c ^ cd fa(x)= ^∞c ¤ b¥
=¥ ∞^ ¤ òc¦Id9,6ò¡f0ª c f0^7 w c¨(d ]a,f e^0b[µ ¤ c= ª0¢
,¶ñ6c^c[f0 =e^eC7¨ª0 f « ±ô¨ª0 f « ¢ K(x, u) c ¤ ¥
¥ ^ ª0¢¼ ¤ d9cpµ ª0^cp ¬ ? f ^]a, b[×]c, d[ ¤ ^c¤,pCc= ^d ÷|0 * ^^¤,? ¬ Édöc¤,pCcd&ø¨ª0 f « 6 , ¤ ? ¬
É ö d e0
¤ cd ,pCT=6eðh¨ª0 f « f (x) K(x, u) ϕ(u) =
Zb
÷ n0úq?ø
f (x)K(x, u) dx.
7e^ ^c4e^ó§ce^ccyA Qc=?±e^
=c© c^ ^¤ ¤,¨¥?ª0
Å0 f Q^« = c ¢7,eð?P 7 ¬ ±y§^c=¦!^¯¤,Q?,= =¬¤, =e^cd9 hd96 c¤,e6¤ ^0^c±.d9.¤,cQ¡V p^e^ c cIcp= ¤, ¬== ^e^»ªe^^dPd9÷ pòn0¤ ¦úq? ø¥¶pdP ¤ p¬¥f0
^=dPe6=fVp= p c ,f T¢É± µ ^^~¤,? ¤ ¬0 ccQ^¡ch^ ¤ 60c¦;¤,&pCc,==e6 ce6©dPpû 0^ dP©pf0 =^e^e^e^f0 c±¨(ü C d9f0pcQ¡V ? côCd9 ^cp ,ª0º,p0
¬w¤ c©c ¤,pC¥ µ cp ?
=¢M7 Ic= ¤ ¥
¥ ^K(x, Éd9u)we6c± e6=d9 f (x) ϕ(u) e¥ ¶ ^¤ 6 cc=C,Q ¬ ^6¤? ¬ T±ôc ^¤,pc¤
^± e¥=ª¢M7,±¶ c7 ¤,=0 ,ª ÷ n0ú/ q?ø¥c[ ^^K¤? ¬ T±wc=¤,p ϕ(u) ¨ª0 f « f (x) d9c?¡V c7Q= eCp¬Vf0=f a
ϕ(u) = K[f ](u) = Kx→u f (x) =
bcA
¶ eA¦ cC
ª0¢ö¨ª0 f « ¢ c¤,pCcp
Zb
f (x)K(x, u) dx.
Zb
ϕ(u)K −1 (x, u) du.
÷ n0ùgø
d9c?¡V c¶ ¤ ¥
e6==¬wô0
^^¤,? ¬ c^ch ¤ ¥µ a
f (x)
−1 f (x) = K−1 [ϕ](x) = Ku→x ϕ(u) =
÷ n0 ø
¤ ^c^^¤,¤,??p Cc¬^ ^=c¤, d#?ªy »¬ ÷ ¤ c^n0wc ¤,ø¥¤ p^Ccc¤,=p C c¢ë=÷ n0tùg÷ø¥n0¯ ¨øª0, pf C« T =6Keð²(x,cu)¤,p4 0T
dK¤ cf¼dt÷|c ¤ ¤, pd9c d#c^ª c ø * ó 0 ª
6 c * d = ¥
^
§ h ¦ c
¤ ¥
¥ ^
w ± ¨ 0 ª
f
«
¢ 9 d ? c V ¡
c
¤ ¥
6 e = = y ¬ ¥ e ¥ ,
ª µ f (x) ¢M7 d
c± Tdþ ^^¤,? cdT ÷ n0 3 ø f (x) = K K[f ] (x). ¾§^¤ ^d9eðhf~ ^^¤,? ,ªw®¯ª0¤ ¬(¨ª0 f « f (x) [f0cd90 ^f e^ c±h¨(c¤ d9= a
−1
−1
1 f (x) = 2π
Z∞ h Z∞
−∞ −∞
i f (t)eiut dt eiux du.
÷ n0ùnø
©?T=ËZ&ÙÍÅÑR«ðÙ,ß®^Ñ¥Ø=ß^ÒÖ ÍH
PÝpÙu¿Í
S0L
^7f0=fI¨cª0 cf =« 7 ^¢ & c;ó*
¤¯ª0¤ ^0c
±!4e¥®¯cª0¤ ¤ c¬ &,É
^c ± ^ ¤ c ±hcQ
^^e^^f¤,ª?¢ , c= ¤ cd96^¡^¤,ª ? [f ]cp− ,ª0∞, ÷ n0ùnø¥?c ¤ ¥
¥ , ¢M7 ±[ ∞[^^¤,? ¶®¯ª0¤ ¬fT(x)¨ª0 f « f (x) pd9cQ¡V c4¤,=e^e^dPp¤ p¬(f=f ,=e6 T±he¥ ,ª0,=±ô
c± c^c[ ^^¤,? w÷ n0 3 ø§e( ¤ d9Td C
1 ϕ(u) = √ 2π
Z∞
f (t)e−iux dx
Z∞
ϕ(u)eiux du,
÷ n0 jø
ôc¤,p Td ^^¤,? ¬ Td* ¤ ^c¤,?Cc= ^d 1 f (x) = √ 2π
−∞
÷ n0;:ø
? =, ^^¤,? ¬ Éd9h ¤ ^c¤,pCc= d9»÷ n0ùgø¥÷ n0 ø§e0
¤,=d9 −∞
e−iux K(x, u) = √ , 2π
eiux K −1 (x, u) = √ . 2π
« ®¯ ª0¤ ϕ(u) c² ¤ c¥
¤,¥p ,C c^dëdPp®¯tª0 ¤ ¤ ¬^!c=¨¤,ª0p Cf c« = ^dë÷ n0÷ © jø¥p=f,¡pC=Tòf0==6f*eðt* ¤,¤ =^ c^p=µ ,e^¤ p^Cfc¤,=®¯ ?ª0 ¬^fdö ¬ w 0 c±w¨ª0 f « ^±!0 he^ ^f¤,? ¬ c±!0 c= cfe6(x)¬¢øÉhe¯ cd9c=7¬¢ c ^¤,pcpµ ¤, F cc=C,Q,=6eÅ~f0=f ÷ n0 oø ϕ(u) = F [f ](u) = F f (x). ó§chc=®¯6ª0¤ e6¬IC0 ~c c ¨(¤,cp¤ d# TªC dþ! ÷ ¤ n0^c; :øɤ,,pCpcCT= = 6^d*eð!®¯¨(ª0c¤ ¤ ¬d#IªQ~ cd9±wcQ¡c6¤,pò c^¬[c¶Q= ¤ ^ceC=,¤,[p^f0c=f ?µ ÷ n0 ø f (x) = F [ϕ](x) = F ϕ(u). e6 c ± c ¨ª0 b#f =« f d .yc ¤,c¶pC¨(ccdT¤ d# ªC c²~¨(÷ cn0¤ ;d#:ªQøÉ ce^÷ e6pn0= jcø~d9c?¬¡V pc*c¤ , =^ ±,?A c¢Q= ¤,?p = ¨®¯ª0ª0 ¤ f ¬« ¢ Cf^(x) ^,=¤0C
,ª^e6eV cd#ª¶^^c¤,?¤, pc^d ªô®¯®¯ª0ª0¤ ¤ ¬¬=!,÷#=n0e^ù nø¤ c=ce6 ¤,¤,==p
C= , ¶ce6^¬dT ¤ ^ccw¤,pTC ccp = ^ ~ ®¯yª0 ¤ ^f ¬Vc=÷ c¤ n0§ j¦ ø c ^¤,=« ±[,?
y®¯ª0¤ ¬¥µ¸c¤,pCcd ϕ(u) cf0pCT=6eð cA
¤ c=7= ^d,?
V eA¦cQ
c± ¨ª0 f « ^± f (x) ,¹¤ ~ñ6cdþ cA
[^e^¬dP[ cp 6C Tdcf0pCT=6eð~c?¡[
^e¥c ÷ n0úqQiø F [f (x)] = F [f (−x)], ò^f0=¢M7^¯y÷ n0 jø§»÷ n0; :ø¥ ¤ ¬=,c^ f ¤ c=Å
c¥¤ ,T T^d9==T¯c¤ e^tcc=d9 c=^e6·7^c[ ÷ n0d9 j ø.¶÷ n0; :ø ¤,=e^e^dPp¤ =¢M( ¤ ^c=¤,pCc= [®¯ªµ x→u
−1
−1 u→x
−1
ϕ(u) =
Z∞
f (x)e
−iux
0 ~e^ d9d96¤ Cc= Td9hQ=d9^ c±
dx,
1 f (x) = 2π
−∞
ϕ(v) =
Z∞
−∞
ϕ(u)eiux du
÷ n0úqq?ø
ϕ(v)ei2πvx dv,
÷ n0úq?gø
Z∞
−∞
u = 2πv
f (x)e−i2πvx dx,
f (x) =
Z∞
−∞
1
SR7RN
7p=f¡ ϕ(u) =
Z∞
f (x)e
iux
1 f (x) = 2π
dx,
Z∞
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í
ϕ(u)e−iux du
÷ n0úqQø
y
¤ dPp¤ e6pcQ¬t7f0^±t=f cC
»C^,^=¤,f ?c d ¬ cw^ª0^¤,¤,=? ^ ¤ w6·7c=^ cp µ e^f c=¥c ¤ =¬c ^^c»cw e6
^=6cCeð÷ ^e6n0¨( j cøc¤ ±td9d#cQ¨ªC¡V ª0c ± c©f « ¤,÷ =n0e^;e^:ϕ(u) øw÷|e^dT§ð ==ª d?^^¤,? ¬ Tª0¤,== ^ ¢ û 3=üýø¥ ¾§ ¤ c ^dT,¤,=^ e6h÷ n0 jø§»÷ n0; :ø§d9cQ¡V c cd9^¬Vd9^e6p=d9 ¹c=6
^Vd9K¤,=e^e^d9c=¤ d¼ ¤ ^c¤,pCc= ©e[
¤ª0^ d9» ^^¤,? ¬ Éd9tc ^¤,?µ c¤,=d9÷ýá¯=0 =eC ~¥ ,
¤ ùø9
,ye^cc=6e6=ª0¢M7¦¶f0 =e^e^c¨ª0 f « ± f ϕ =¤ 6ª0d9^f c=c¤ §¦~d9c6ª Vò¬~f cd90 ^f e^ Td9w ^¤ ^d9^ Td9 =e^e^d9c=¤ d6 ^¤ ¬y,=e6 ò¯e¥ ,ª0,=h ¤ ^c¤,pCc= h®¯ª0¤ ¬= ¹ ª 6 e ¬
6 , p þ ¨ 0 ª
f
« & ( =
c 9 d
d ( ¨ c
¤ # d C ª , ª
^ ^ , ¤ ? t ¯ ® 0 ª
¤ ¬ ¶
, f (x) 6 c±h¨ª0 f « −∞
−∞
¹^¤ ^ ·7^d^(0
ô cp c?¡V d bcA
cp ,ª0 d
Z∞hZ∞
2 f (x) = π
0
0
i f (t) cos ut dt cos ux du.
r Z∞hr Z∞ i 2 2 f (x) = f (t) cos ut dt cos ux du π π 0
0
r Z∞ 2 ϕc (u) = Fc [f ](u) = f (t) cos ut dt. π
÷ n0úqV3 ø
0
r Z∞ 2 f (x) = Fc−1 [ϕ](u) = ϕc (u) cos ux du. π
÷ n0úqQnø
¾§0
dT,c¨ª0 f « e^c=^¤·7^ cVcQ
,=f cc[T¤,p¡[=¢Með¶
¤ª0I ¥µ ¤ ®¯6ª0#¤
¬¤=ª0 ¥É ^^¤,? ¬ cf±¯(x)¨(c¤ d#ϕªC (u)c± Cf0c=c¤,p(,pCò=6eÅIf ce^ ªe¥µ¸ ¤ ^c¤,pCc== ^d ®7c¤ d#ªQ »÷ n0úVq 3 øc ¤ ¥
¥ , 6! ¤ d9cVf ce^ ªe¥µ¸ ¤ ^c¤,pCc= ô®¯ª0¤ ¬y ¥ c± ¨f cª0e^ f « ª e¥µ¸ ¤ f^(x) c¤,Tp C¤ c = cC
, 77®¯^ª0¤ f²¬=¨ª0 f « ϕ (u) T¨(c¤ d#ªC ÷ n0úq?nø c¤,p c ¹ªe6¬ f (x) ^ 6,p¨ª0 f « òÇ = ·7^dÀ ^^¤,? ®¯ª0¤ ¬ô
,þ ^ 6 c± ¨ª0 f « 0
c
c
2 f (x) = π
Z∞hZ∞
i
f (t) sin ut dt sin ux du
ô cV=,? c^ ~d9cQ¡^dºc ¤ ¥
¥ ¬y ¤ d9c(e^ ªe¥µ¸ ¤ ^c¤,pCc= 7®¯ª0¤ ¬ 0
0
r Z∞ 2 f (t) sin ut dt ϕs (u) = Fs [f ](u) = π 0
÷ n0úqQjø
©?T=ËZ&ÙÍÅÑR«ðÙ,ß®^Ñ¥Ø=ß^ÒÖ ÍH
PÝpÙu¿Í
SR7ZM
ôc¤,p c(e^ ªe¥µ¸ ¤ ^c¤,pCc==
÷ n0úqW:ø
r Z∞ 2 f (x) = Fs−1 [ϕ](u) = ϕs (u) sin ux du. π
¹ce^f cp ¬ f6ª~ c=¢4±ô=¨ª0¢Kª0 ¨f « ª0 f « ¢ f (x) d9cQ¡V c¶ ¤ ¥
e6p=¶ ^ ¬~6¶ c0±¶
¨ª0e6ª0 d9f « d9 f (x) = p(x) + q(x) p(x) = ( [f®¯ (x) + f (−x)]/2 q(x) =¬ 7 c
¤ ^ c , ¤ p C c =
0 ª
¤ ¬ É 4 ¢ c y ± ¨ 0 ª
f
«
¶ 9 d ? c V ¡
¯ c
¤ ¥
6 e p = [f [(x)0
− f (−x)]/2 0
1 ϕ(u) = √ 2π 1 =√ 2π
Z∞
Z∞
f (x)e−iux dx =
−∞
f (x)[cos ux − i sin ux] dx = ϕc (u) − iϕs (u),
e^A
cc= ϕ6e6 ϕ^, 4 c f ce^ ªe¥µ¶e^ ªe¥µ¸ ¤ ^c¤,pCc= ¼®¯ª0¤ ¬¨ª0 f « ± <~ ' (H P x>=,x(¹ªe6¬¨ª0 f « c ¤ ¥
¥ ^,¤,=^ e6cd f (x) ^e¥ 0 ≤ x < a; ( 1, ^e¥ 1/2, ^e¥ x = a; f (x) = (=±~^(f ce^ ªe¥µ§~e^ ªe¥µ¸ ¤ ^c¤,0,pCc= !®¯xª0¤ >¬=a. @ 9 x mce^ ªe¥µ¸ ¤ ^c=¤,pCc= Iñ6c±h¨ª0 f « h^e6¬ c
−∞
s
q(x)
p(x)
r Za r Z∞ r Z∞ 2 2 2 f (t) cos ut dt = 1 · cos ut dt + 0 · cos ut dt = ϕc (u) = π π π 0
0
e^ ªe¥µ¸ ¤ ^c¤,pCc=
a
r r Za 2 2 sin au = cos ut dt = ; π π u 0
r Za r Z∞ r Z∞ 2 2 2 ϕs (u) = f (t) sin ut dt = 1 · sin ut dt + 0 · sin ut dt = π π π 0
0
a
r r Za 2 2 1 − cos au = . sin ut dt = π π u
( =±0
,f0ce^ ªe¥µ[e^ ªe¥µ¸ ¤ ^c=¤,pCc= V¨ª0 f « ± cp ,ª0 dTe^cA =e¥µ ch÷ n0úqQjø§©÷ n0úqW:ø¥ ^^¤,? ¬ cI ¤ ¥
e6==p ^ ¯¨ϕª0(u) f « ϕ± (u) ¤ |x| < a; ( Z 1, ¤ ÷ n0úqQoø sin au 2 1/2, ¤ |x| = a; cos xu du = π u 0, |x| > a;
¤ 0, ¤ x0 <= x0;< a; Z 1, ¤ ÷ n0úqQø 2 1 − cos au sin ux du = ±1/2, ¤ x = ±a; π u 0, x > a; ¤ −a < x < 0. −1, 0
c
∞
0
∞
0
s
× ß6Ø=ß ¼ ËéòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í Ç
^e^¬ x = 0 ëc f[¤,pC¤ Tl( , x > 0 ^^¤,? ë÷ n0úqQoø&»÷ n0úqQø&¤,= <~ ' (H P xED x((=±h¤ 6·7^ (ª0¤,= ^ 1
SR7RS
r Z∞ 2 y(u) cos ux du = e−x , π
A
y(u) ö e^f0cdPpô¨ª0 f « @ 9 0 l¯'=ª0ñ6f ¤,c^pcÉ ª0c¤,I= ^ ^^(¤ =¤ ªQc
6=M c ¯¤,p c[ ,c=9e6f0?c e^d
ª=e¥6µ¸ ¤ ^c¤,pCc= §®¯ª0¤ ¬P¨ª0 f0µ « e 0
−x
0
r r Z∞ 2 2 1 −t e cos ut dt = y(u) = π π 1 + u2 0
y(x) =
r
2 1 , π 1 + x2
^e¥ ~ ^¤ ^cc=C,Q ¬ u ^¤ 6 x <~6 c'±y V( H ^P Ex 6H x( c¹±¶cf¨pª0Q pf « ¬ fcô(x)f0ce^eÉ cª e¥ µ4ce6e^¬ ¢²ª
e¥c¯µ¸ d9¤ ^c?c¡V¤,pCÅc ,y= e^c»,®¯?
ª0=¤ ¢M¬7ϕeMf c ñ^¨ϕµ ¨( « ^p=d9 A(u) B(u) ^I 6^¤? ¬= c±w¨(c¤ d#ªC ¼®¯ª0¤ ¬(¨(c¤ d9÷ g=oúqq?ø¥ @ 9 x;b#=fôf0=f c
1 A(u) = π
Z∞
f (t) cos ut dt,
−∞
r Z∞ 2 f (t) cos ut dt, ϕc (u) = π 0
1 ϕc (u) = √ 2π
Z∞
Z∞
f (t) sin ut dt,
−∞
c , ¤ ¥
e6p= ¶®¯ª0¤ ¬¥µ¸c¤,pC 0
,?¦cC
d
1 B(u) = π
f (t) cos ut dt,
−∞
A(u) =
r
2 ϕc (u), π
r Z∞ 2 ϕs (u) = f (t) sin ut dt, π 0
1 ϕs (u) = √ 2π
B(u) =
r
Z∞
f (t) sin ut dt,
−∞
2 ϕs (u). π
cd96¤,¹pcCC
¤, c=§hf¦¶Q p f0^¤e^ªCc,
? e6¦c?¡[^6
¬^ò c ¦~#² ¾ ¨ñ6ª0^^¤ 0¤,¬¦»?¥ µ¸e¥cc , ª0¤,,p
pC ,0c=ô¦»f0d9e6c=pcQ?¡Vc ¤ f §c~ ¦ôc= 6¤,e^ peðcp e!¬=¬CeðctT= f© e¥d9( 6^e6 pcQ =
^0=d
d=¤,¤¨pª0C òf0cp¦ µµ
« ¤ª ±¨(¹¨(f ¤ c^ d9¤ ^0¥
« ^^ f d¤ e^ c
c^pc~ ¤ w^ ¤ hd9^d9^ ¤,^ ^^0c¤ ^ c ¤ .¢4c,e6== ¤ ¤ ¤wd9ª¢M^ ¤!^7e^c^¦~ce6¤ ñ6 c^ Q =T§d9 ¦w^6, =c ^=^V ¤,p?= fc¡y cf,d9=6¤,=cCd9
6=pd µ
s
©?T=ËZ&ÙÍÅÑR«ðÙ,ß®^Ñ¥Ø=ß^ÒÖ ÍH
PÝpÙu¿Í <~ ' (H P xKJ x4l( ,~¨ª0 f «
SR7Qà
f (x) =
1 , a2 + x 2
a > 0,
, =±~ ¤ ^c¤,pCc= w®¯ª0¤ ¬(ôQ= eCp¬V^( ¤ ¥
e6p=p ^ ,I ^^¤,? cd®¯ª0¤ ¬= @9 xi6,p¨ª0 f « p , 6eð[
¨(¨(^¤ ^ « ¤ªCd9c±VV=e^cp ¢M c¯ 0µ ^^¤ ¤ªCd9c±h, R ce^f cp ¬fª f (x) Z∞
1 dx = a2 + x 2
¾ 0
e^0 ,ª 6 ce6þ¨ª0 f « K −∞
Z∞
dx =2 2 a + x2
Z∞
dx 2 x ∞ π = arctg = . a2 + x 2 a a 0 a
f ce^ ªe¥µ¸ ¤ ^c¤,?Cc= w®¯ª0¤ ¬»÷ u > 0ø¯ d9^6 f (x) 0
−∞
1 i ϕc (u) = Fc 2 (u) = a + x2 1 =√ 2π
bcA
h
Z∞
r Z∞ 2 cos ut dt = π a2 + t 2 0
h cos ut 1 eiut i √ dt = Re 2πi Res = t=ia a2 + t2 a2 + t 2 2π −∞ r −au h πe 1 e−au i = . = √ Re 2πi 2ia 2 a 2π
r Z∞ r Z∞ 2 π e−au 1 e−au cos ux du. f (x) = cos ux du = π 2 a a
<~ ' ( H P Ex P x4l( ,¶¨ª0 f « ,=±ô ¤ ^c¤,pCc= ~®¯ª0¤ ¬yQ= 0µ eCp¬^( ¤ ¥
e6p=p ^, ¯ ^6¤?f c(x)d²®¯=ª0e¤ ¬= @ 9 @x i6,p7¨ª0 f « ¯ p , 6eðI
¨(¨(^¤ ^ « ¤ª^d9c±7=e^cp ¢M c ^^¤ 0µ ¤ª^d9c± ce6f0cp ¬fª Z Z Z ÷ n0ùg=iø e dx = 2 e dx < 2 e dx = 2e. 0
0
−x2
∞
∞
∞
−x2
−x2
1−x
(ôñ6cd ce^ c= f0ce^ ªe¥µ¸ ¤ ^c¤,pCc= ¶®¯ª0¤ ¬¨ª0 f « 0
−∞
0
0
f (x)
Q= ·76eÅ»
÷ n0ùg0q?ø (=¤0
,ªweeC=d9c±¨ª0 f « ^±!
¨(¨(^¤ ^ « ¤ª^d9c±©=e^cp ¢M c~ ^^¤ ¤ª^d9c±©, p , 6eðh¨ª0 f « xe R Z Z ÷ n0ùggø xe dx = 2 xe dx = −e = 1.
ϕc (u) = Fc e
−x2
r Z∞ 2 2 (u) = e−t cos ut dt. π 0
−x2
∞
∞
−x2
−x2
−x2
∞ 0
−∞
0
SR7 0
× ß6Ø=ß ¼ ËéòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í [÷ n0ùg0q?ø^^c ¤ cCcQ
,p cos ut 1
¾»e^0 ,ª7ñ6c^c( cC
T^C¤,? ¬ cÉT¤,p¡^
e−t
2
d −t2 2 e cos ut = −te−t sin ut du
c ^ [¤ dP^¤ p¡[Tc ¤ ä ¢M=eðe^cp ¢M^ ^cþ¤,? =d9^^¤ ÷ ,¤n0ªùg=Cid9øäw÷ºn0f ùg?g
ø¥¤,bpcA
]0,,∞[×]0, T c¤ e^f cpC ,¬=ff0ª ∞[
¤ 0 ª c ^ e
c =
¾§^± ^¤·¯¤,=e^eQ¯ ^^¤,? ©÷ n0ùg0q?ø#eA¦cQ
eðy¤,= cd9^¤ c¯^^c¯d9c?¡V c
¨(¨(^¤ ^ « 0µ ¤ cp¬[ c,=¤,=d96¤ª u ÷|e^dT p=f¡(e^c± e6ôqI~o[ ¤ ^c=¤,pCc= ±w®¯ª0¤ ¬Qø¥ l¯ ¨(¨(^¤ ^ « ¤ c= y÷ n0ùg0q?ø& c u
=6T¤,p¡[6 r Z∞ 2 2 te−t sin ut dt, ϕ0c (u) = π
¤ c ^^¤ ¤ c=Vf c=c¤ c( c,=e6Q dT,Qp ·7^d 0
r ∞ u Z∞ u 2 1 −t2 0 −t2 ϕc (u) = e cos ut dt = − ϕc (u). e sin ut − π 2 2 2 0
¹ ¤ c ^^¤ ¤ c=( cp ,ª0 ^ c§
¨(¨(^¤ ^ « ,? ¬ cTª0¤,= ^ TeT¤,p6
¥ , ¢M7 d9 eð ^¤ ^d9^ Td9 ,=±0
^d 0
u2 + ln |c| 4
0 A
c ë ¤ cC Ccp ¬,pô ce¥c? ,pϕ (u)/ =d9c?ce¡V c7c , ¤ ¥
¥ ¬ ^e¥ ôce^ cp ¬=Ccp¬eÅ ^^¤,? cd ±0 ^¤, 4 ¹ª=e^e^c, ln |ϕc (u)| = − c
−u2 /4
r √ r Z∞ 2 2 π 1 −t2 e dt = ϕc (0) = = √ = c. π π 2 2
bcA
0
1 2 ϕc (u) = √ e−u /4 2
r Z∞ Z∞ 2 1 2 ϕc (u) cos ux dx = √ e−u /4 cos ux dx. f (x) = π π 0
°1~±
0
) z +- !.!µ ! # $Ê,
O c ^( ¤ ^cº¤, wpC1.c = x ¯/ ®¯e¥ ª0¤ ¬¨ ª0 f « = e^cp ¢M c =,x(z~^^ ¤ ) ¤ª ^ dPu[V , ] t−~∞,) ∞[ 1 ϕ(u) = F [f ](u) = √ 2π
c^¤,= ^ cþ ^ ¤ ^¤ T cþº ¤ cd96¡ª f
|u| → ∞
Z∞
f (x)e−iux dx
−∞
] − ∞, ∞[
f (x)
÷ júq?ø
e6¤ ^d9eÅ fªQ ¢í ¤
©${?Ë}|ØÑ¥Õ Î]¥Ø=߬=ÙÍÅÑR«ðÙ,ß®^Ñ¥Ø=ß6Ò ÖÚ6
PÝpÙu¿Í U +-
x ? (=e^cp ¢M c±ô ^^¤ ¤ ªCd9ce6~¨ª0 f «
,ô ^^¤,? !÷ júq?ø&e6ª 7^e6=ª6ydPp¡[c¤ ¤ª0¢É7 ±h ^^¤,?
SR7R7
e¥ ¥
,ª6p,c
f (x)
Z∞ Z∞ 1 1 −iux |ϕ(u)| = √ f (x)e dx ≤ √ |f (x)| dx = Q < ∞. 2π 2π
Pó c~ ¥
c~,cM,p=¤,e^¥^= ±¯d9¬=6 c ¤e¥ ª ¨cuª0 cf ±¯°4« ce^e^ ¢§ pϕ(u)
¹côÅc
e^^^¤,c =¢´ § c ¨ ^ª0^,¤ ¤ (¥¬
[¥µ¸¬ ce¥¤, p^¥^Q
,¤, ª?6 w÷ j^, úÉq?¤ ø^eð¤ ^¦e^TcCf
c c^e6eÅ ¬wc¤,e6¨=ª0I c f d9^« ^^ f0¤0c µ ϕ(u) ªe6p=,=ó§p c ± e6=6ceÅhq7e(Qp dPc6d9,c=p7¥¬ ¢ ¬ cô^d9d9^dT Vq 3 ùcyg0
p¡(
,!¤,ϕ(u) pC¤ T c±¨ª0 f « f (x) ^ ¨ª0¤ ♦¬¥µ¸c¤,p ϕ(u) =ªC
6V ^ ¤ ^¤ T c±w¨ª0 f « ^± cyô
,~f0ce^ ÷|e^dTªe¥,µò h¤ e^d9 ^¤wªe¥µ¸n0 ¤ ^ø¥c ¤,pCc= ±®¯ª0¤ ¬=; p=f♦¡ó§
,c~± e6f cñ^ch¨(q¯¨(e^ « ¤, =^=¥
c A(u) O
D *x Qh t öÀ )¤ c CB(u) ¬ T I c=e6 c? w 1. T= ,x c / e¥ ϕ(u) = F [f ](u) p c ψ(u) = F [g](u) C C ÷ jùgø F [C f + C g](u) = C F [f ](u) + C F [g](u). U -+
wc ^0
c¶wò^f0=6ôe6c± e6 ^± ce6c ¤ ¥
¥ ^ c^c ^^¤,? w÷ júq?ø¥ A<~
' ( H Q >x =,x((=±t ¤ ^c¤,pCc= ¶®¯ª0¤ ¬y¨ª0 f « f (x) = 2e + g(x)/3 −∞
1
−∞
2
1
2
1
2
−|x|
g(x) =
(
0, x < 0; 1, 0 < x < 1; 0, x > 0.
@9 x ¾§ce^ cp ¬=Cc==·7 e^¬¤ 6^ªQ ¬Qpp==d9w ¤ d9^¤ cVqúq(~d3 úq d9^^d F e−|x| (u) = √
2 , 2π(1 + u2 )
°4e^¢§
e^cA =e^ cw÷ jùgø¥ ,=±0
^d
i(1 − eiu ) F [g](u) = − √ . 2πu
√ i 2 g(x) 2 i(1 − eiu ) √ . F [f ](u) = F 2e−|x| + (u) = √ − 3 π(1 + u2 ) 3 2πu H / ϕ(u) = F [f (x)](u) c ∈ R c 6= 0 u 1 1 u F [f (cx)](u) = F [f (x)] = ϕ |c| c |c| c h
,
|÷ ¤ c = −1 d9^^d e6cIc=¤,p¡^ Qø¥ U +-
~d9cQ¡V c[ cp ,ª0 ¬ ¤ c¥
,~h÷ júq?ø&Q=d9^ª
cx = y
O
x(%§ ') "#9x e¥
Z∞ 1 f (cx)e−iux dx = F [f (cx)](u) = √ 2π −∞ ∞ Z 1 f (y)e−i(u/c)y dy, c > 0; c −∞ = Z∞ 1 − f (y)e−i(u/c)y dy, c < 0, c
c=fªC
~e¥ ¥
,ª6÷ j ø¥
−∞
c ÷ j ø
,bcA
SR7RB
<~ ' (H Q xED x((=±h ¤ ^c¤,p^c= 7®¯ª0¤ ¬(¨ª0 f « g(x) =
(
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í 1
0, x > 0; 2 + x/3, −6 < x < 0; 0, x < −6.
@9 x?I¤ 6·7^ h ¤ d9^¤,Vd3ùg7C^e6 cy ¤ ^c¤,pCc== 7®¯ª0¤ ¬ ÷ j 3 ø
1 h 2i 1 − e−2iu i F [f (x)](u) = ϕ(u) = √ − + u u2 2π
¨ª0 f «
÷ jùnø ¹ce^f cp ¬fªô¨ª0 f « g(x) Q= e^T=6eÅwf=f g(x) = f (−x/3) ,cy^ICc¤,p¡[^ d9cQ¡V c,=± ce^ cp ¬=Cc==·7 e^¬¨(c¤ d#ªC c±»÷ j ø¥ f (x) =
(
0, x < 0; 2 − x, 0 < x < 2; 0, x > 2.
u 1 F [f (x)] = | − 1/3| −1/3 1 h 2i 1 − e6iu i 3 h 2i 1 − e6iu i √ = . = 3ϕ(−3u) = √ + + 9u2 3u2 2π 3u 2π u F [g(x)](u) = F [f (−x/3)](u) =
¾ ^ e
, ¤ = = ¥
c 6 e 7
p c , 0 ª
^
c ¯ ± ( ¨ c
¤ # d Q ª © 9 d Q c V ¡
T c 0 ª ¥
¬ ð e ¯
^
c ^ e
¤ ¥
6 e ^
, T d T e¥ ^ ^d¨ª0¤ ¬¥µ¸c¤,pQ[¨ª0 f « g(x) 0l¯^± e66 ¬ c 1 F [g(x)] = √ 2π
Z∞
g(x)e
−iux
1 dx = √ 2π
¾§T e¥ d ^^¤,? h c,=e6Q dT cp cQ¡V V = −e /(iu) −iux
Z0
−6
bcA
−∞
Z0
−6
2+
x −iux e dx. 3
U = 2 + 3/x dV = e−iux dx dU = dx/3
Z0 1 2 + x/3 0 x −iux e dx = − 2+ e−iux dx = + 3 iu 3iu −6 −6 1 −iux 0 1 2 2i + 2 (1 − e6iu . = − + 2e = iu 3u u 3u −6 i 1 h 2i 1 F [g(x)] = ϕ(u) = √ + 2 (1 − e6iu . 3u 2π u
Pó ¥
,ª6»c=d96¬ Pc ¦c=?¨ª0 f « ÷ jùnø( d9^6©¤,pC¤ Twc f0 ¨ª0 f « ϕ(u) ^ ¤ ^¤ T=,Vc f u = 0 , ce^f cp ¬fª 1 h 2i 2iu i 1 h 2i 1 − e−2iu i √ = − + − + 2 = 0, lim ϕ(u) = lim √ u→0 u→0 u u2 u u 2π 2π
c cC
,^¤¡[
=6Ve^c± e6chq,m¤ cd9c^c
lim ϕ(u) = 0
u→∞
x = 0
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PÝpÙu¿Í O
J x(%§ '¼ )"#u[ 9x%§ 'À' ¦ 9x
c F [f (x)](u)
U +-
~d9cQ¡V c[ cp ,ª0 ¬ ¤ c¥
,~h÷ júq?ø&Q=d9^ª
/
F [f (x + x0 )](u) = eiux0 F [f (x)](u) = eiux0 ϕ(u).
1 F [f (x + x0 )](u) = √ 2π 1 =√ 2π
Z∞
f (y)e
−iu(y−x0 )
Z∞
x + x0
e¥
SR7WG ϕ(u) =
÷ j jø ,bcA
=y
f (x + x0 )e−iux dx =
−∞
dy = e
iux0
1 √ 2π
Z∞
f (y)e−iuy dy,
c=fªC
~e¥ ¥
,ª6÷ j jø¥ <~ ' ( H Q Ex H x((=±h ¤ ^c¤,p^c= 7®¯ª0¤ ¬(¨ª0 f « −∞
g(x) =
(
−∞
0, x < −1; 1 − x, −1 < x < 1; 0, x > 1.
¨ @ª09 f « » ÷ x£j?ùnø¥ ¤ ¹6·7ce^^f0 cp ¬f ªy¤ ¨d9ª0^ ¤,f ²« d¢ 3 ùgCd9^e6cQ¡V cc7 Q¤ =^ c eC¤,ppC¬cf== f þ®¯ª0¤ ¬ ÷ j 3 ø c^(Cc¤,p¡^ (d9cQ¡V c[,=± ,ce^ g(x) cp ¬=Cc==·7 e^¬¨(c¤ d#ªQ c±»÷ g(x) j0 jø¥= f (x + 1) F [g(x)](u) = F [f (x + 1)](u) = eiu F [f (x)](u) = 1 iu h 2i 1 − e−2iu i iu = e ϕ(u) = √ e − + . u u2 2π
b#=f c±t¡ô¤ 6^ªC ¬?pptd9cQ¡V c cp ,ª0 ¬© ^ ce^¤ ¥
e6^ ÉdKT e¥ ^ ^dK¨ª0¤ ¬¥µ c¤,pQ¨ª0 f « g(x) 0l¯^± e6¥ ¬ c 1 F [g(x)] = √ 2π
Z∞
−∞
g(x)e
−iux
1 dx = √ 2π
Z1
(1 − x)e−iux dx =
−1
i 1 1 h 1 1 =√ − (1 − x)e−iux + 2 e−iux = iu u −1 2π i 1 h 1 1 =√ − (−2)e−iu + 2 (e−iu − eiu ) = iu u 2π i h 1 1 2i = √ e−iu − + 2 (1 − e−2iu ) . u u 2π ¦
O c
P x(%§ '' V"+w w1. I x / e¥ − u ) = ϕ(u − u ). U +-
~F [fcp(x)e ,ª0,=6eð](u) hQ==d9^F [fc± (x)](u h÷ júq?ø¥? = u−u iu0 x
0
ϕ(u) = F [f (x)](u)
0
0
1 F [f (x)eiu0 x ](u) = √ 2π 1 =√ 2π
Z∞
−∞
Z∞
f (x)eiu0 x e−iux dx =
−∞
f (x)e−i(u−u0 )x dx = F [f (x)](u − u0 ) = ϕ(u − u0 ).
÷ j;:ø
SR7RI
<~ ' (H Q xKJx((=±h ¤ ^c¤,p^c= 7®¯ª0¤ ¬(¨ª0 f « g(x) =
(
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í 1
0, x < 0; 3x (2 − x)e , 0 < x < 2; 0, x > 2.
@9 c ¤ ¥
x§¥¹ ^c,e^Vf0c?¤, =¬f^ª» e6¨ª0c d¼f « ÷ ¢jùng(x) d9 cQ¤ ¡V^c ch¤,pQC=c =eC p 7¬®¯f0=ª0f ¤ ¬g(x) #A
¬ = fc[(x)e ¥ ø
V c I 9 d ? c V ¡
p c , 0 ª
fc(x) ¨(ce^¤ d#cp ªC ¬=cCc=±»÷==j·7;: ø¥e^ ¬ºbcAC
^e6 Éd ¤ ^c¤,pCc= ^dõ®¯ª0¤ ¬÷ j 3 ø¨ª0 f « f (x) 3x
F [g(x)](u) = F [f (x)e3x ](u) = F [f (x)](u + 3i) = 2i 1 h 1 − e−2i(u+3i) i − = ϕ(u + 3i) = √ . + u + 3i (u + 3i)2 2π
<~ ' (H Q xEP x(¹ªe6¬
ϕ(u) = F [f (x)](u)
¹cf0pQp¬ ,c
F f (x) cos u0 x (u) = 1 = F [f (x)](u − u0 ) + F [f (x)](u + u0 ) , 2 F f (x) sin u0 x (u) = 1 F [f (x)](u − u0 ) + F [f (x)](u + u0 ) . = 2i
÷ j oø ÷ j ø
@9 x.ó§cð =e^ cc ¤ ¥
¥ ^ ¢÷ n0 jø¥ 1 F f (x) cos u0 x (u) = √ 2π
1 =√ 2π 1 = √ 2 2π
Z∞ −∞
Z∞
f (x)
Z∞
f (x)e−iux cos u0 x dx =
−∞
eiu0 x + e−iu0 x −iux e dx = 2
−∞
f (x)e
iu0 x −iux
e
dx +
Z∞
f (x)e
−iu0 x −iux
e
dx =
−∞
o 1n −iu0 x iu0 x (u) = (u) + F f (x)e = F f (x)e 2 n o 1 = F f (x) (u − u0 ) + F f (x) (u + u0 ) , 2
=,cV? ~c^ ¤ ^ c c? ce^¬V cf0pQp¬ ;ó§ ¤,=¥
ce6¬¶e^cc= c=·7^ ÷ j ø&
cf0pCT=6eð <~ ' (H Q xEQ x((=±h ¤ ^c¤,p^c= 7®¯ª0¤ ¬(¨ª0 f «
h i cos x, x ∈ − π , π ; 2h 2 π π i g(x) = 0, x∈R∈ / − , . 2 2
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PÝpÙu¿Í @9 x®¯ª0 f « ¢ d9cQ¡V cT¤,=e^e^dPp¤ p¬Mf0=f( ¤ cC¥
^
A
(¨ª0 f «
=^g(x) p[I ¤ d9^¤,Vg0úq(e f (x) F [f (x)](u) =
e¥ ¥
cp¥ ¬ c e^cð =e^ cw÷ j oø¥
O
,bcA
l = π/2
r
SR7RL g(x) = f (x) cos x
2 sin(nπ/2) π u
F [g(x)](u) = F f (x) cos x (u) = r h 2 1 sin(u − 1)π/2 sin(u + 1)π/2 i + = = π2 u−1 u+1 r h r cos πu/2 cos πu/2 i 1 2 2 cos πu/2 = − = + . 2 π u−1 u+1 π 1 − u2
Q c x(%§ '!©) 'K) w1. x / e¥
F [f (x)](u)
F F [f ] (x) = F [ϕ(u)] = f (−x).
U +-
x,l¯^± e6=ª=w,7cQ¡[
^e6cyn0úqQi
ϕ(u) =
÷ júqQiø
F [f (x)] = F −1 [f (−x)]
c ^¤,pc¤ cd F ¤ 0¦cQ
dþf»÷ júqQiø¥ O d9 ^6
= e^T;c? x(¢M%§ cV 'y^^"#¤ # }~¤ª}~^d9 ò ¦h $¤ c C cC
§9¦;x ,/ e¥c
ϕ(u) = F [f (x)](u)
n
F [f (n) (x)](u) = (iu)n F [f (x)](u) = (iu)n ϕ(u).
U +-
¶
ce6ppc cV ¤ c^e6ô
, 1 F [f (x)](u) = √ 2π 0
?^^¤ ¤ª=w c[,=e6Q,dT, cp ,ª0 d
Z∞
n=1
,bcA
f (x)
÷ júqq?ø
f 0 (x)e−iux dx.
−∞
Z∞ ∞ 1 −iux 0 −iux f (x)e dx = iuF [f (x)](u), + iu F [f (x)](u) = √ f (x)e −∞ 2π
ce^f ^cp^¤ ¬f¤ªôª^ d9^c¤ e6cwI¨e¥ ª0= ¥f =« ^ d9 cIc¤, p=6eðw[ªQ ¬ f (±∞) = 0 e^0 ,ªh=e^cp ¢M c=± −∞
f (x)
<~ ' (H Q xUT;x((=±h ¤ ^c¤,p^c= 7®¯ª0¤ ¬(¨ª0 f « g(x) = −
(a2
2x , + x2 )2
a > 0.
× ß6Ø=ß ¼ ËéòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í =e^cp ¢M cw ^6¤, ¤ª^dPw, R g(x),A
1
SRBRN @9 x#áI^^f chª00
¥¬ cw¨ª0 f «
d9cQ¡6ò¬ ¤ ¥
e6p=p C,[0
g(x) = f 0 (x) f (x) =
¾* ¤ d9^¤ ¯n0 3[ cp ,ª0 ^,¨(c¤ d#ªC
a2
1 . + x2
h
1 i F 2 (u) = a + x2
c=fªC
e^cA =e^ cw÷ júqq?ø¥ d9cQ¡V cQ= eCp¬ h F −
r
π e−ua , 2 a
i h d 1 i 2x (u) = F (u) = (a2 + x2 )2 dx a2 + x2 r −ua h 1 i πe = iuF 2 (u) = iu . 2 a +x 2 a
<~ ' (H Q xEX x(¹cfpQp¬ cV^e¥ ~[¨(c¤ d#ªQ V÷ júqq?ø
A
f (x) =
g(x)
¨(¨(^¤ ^ « ¤ª^dP
]a, b[
c
g(x), x ∈ [a, b], 0, x∈ / [a, b],
1 g(b − 0)e−ibu + g(a + 0)e−iau + iuF [g(x)](u). F [f 0 (x)](u) = √ 2π
@9 x.ó§cð =e^ cc ¤ ¥
¥ ^ ¢¯
1 F [f (x)](u) = √ 2π 0
¹ce¥ ( ^^¤ ¤ c= h c[,=e6? dþ cp ,ª0 d
Zb
g 0 (t)e−iut dt.
a
Zb b 1 −iut −iut dt = F [f (x)](u) = √ g(x)e + iu g(t)e a 2π 0
a
1 =√ g(b − 0)e−ibu + g(a + 0)e−iau + iuF [g(x)](u), 2π
cô¤ ^c=? ce^¬ cfpQp¬ O
d9^e6ôe¶¨ª0X x(f %§« ^± f'(x) "#=#e^}~cp }~¢M c $ ^^¤
¤ ªCõd9T"d9+þä! w 1.¤ cd9 6¡ª f ]− x ∞,/ e¥∞[ e^py , ¢M¤ ceðCcC
¨ ª0T Vf «
c xfµ¸^(x) cx¤0f
(x) úf0úú ¢4x f(x) p¬ cc ¨9f ª0c= f c« ¤ T ¶ϕ(u) ^d9 ^¤ 6¥ µ = F [f ](u) w c
f h ¥ c ^ , ¤ = ^
#
¤ T ~e6¤ ^d#eÅhf~nªC ¢ ¤ |u| → ∞ , ¤ ^d ÷ júq?gø F [(−ix) f (x)](u) = F [f (x)](u) = ϕ (u). 2
n
n
(n)
(n)
©${?Ë}|ØÑ¥Õ Î]¥Ø=߬=ÙÍÅÑR«ðÙ,ß®^Ñ¥Ø=ß6Ò ÖÚ6
PÝpÙu¿Í SRBZM U« +-c^¤,=
~
^ ce6,=pp[ c ^ ¤ c¶^ ¤ ¤ Tc,^pe6h¨
ª0 , f « n =e61 ¤ ó§^d#cA =e^p c¶eðe^fc± ªQe6 ¢*=ª© q¤ ¨ª0 f0µ ®7ϕ(u) c¤ dP4? ¬ Tdþ
¨(¨(^¤ ^ « ¤ c= ^d ^^¤,? ÷ júq?øT cy,=¤,=d96¤ª |u|cp ,→ªµ
dþ ^6¤?
∞
u
1 √ 2π
Z∞
(−ix)f (x)e−iux dx,
feA¦c=cQ
c¤ Teð± ©#¤,e^=cA c=d9e^ ^cw¤ ªc~eÅ ccô ,¢¯=¤,= d9p6 , ¤6ª eð = ¤ e^ cp ^¢Md cd9^ c~^^f ¤ ¤ª^d9Él¯dTc#f0 pQchp¥c ð
¬we6cc u ϕ (u)
, n > 1 =,? c^ c <~ ' ( H Q Ex ` x((=±h ¤ ^c¤,p^c= 7®¯ª0¤ ¬(¨ª0 f « f (x) = xe @ 9 x m(=f[ cf0pQ= c(( ¤ d9^¤ 4n ùn0¨ª0 f « xe =e^cp ¢M cI ^^¤ ¤ª¥µ dCPcIp R¬eð ~óP¨( ¥c
¤ cd#ªCp c¥±» ¬÷ c júq?gø¥c,¹²¤ , =± [y^c[ ¤ ^dPc= ¤, pC=c= c ®¯ª0¤ ¬=d9cQ¡V c¯ce^ c? ¬pµ −∞
0
−x2
−x2
1 2 2 F [e−x ](u) = √ e−u /4 , 2
,?¦cC
d 2
F [xe−x ](u) = i
d h 1 −u2 /4 i iu 2 √ e = √ e−u /4 . du 2 2 2
§¾ ¥
^dþe¥ ¥
,ª0¢M7C¯c ¤ ¥
¥ ^ = ó§^¤f c±y
=ª¦V¨ª0 f « ± f (x) ¨ª0 f « h(x) ,¤,=,p
g(x)
0cc=C,Q,=^d9c±
1 h(x) = f (x) ∗ g(x) = √ 2π
Z∞
f (x) ∗ g(x)
0,pCT=6eð ÷ júqQø
f (y)g(x − y)dy.
e¥ ¨ª0 f « ^ ¤ ^ ¤ ^T ¤ ,^¤ Tc^ ¤,= 0 c ^¤,^,= ¶ ^= þe^cpV ¢M=e^ cpc¶ ¢M ^cI^¤ ¤ ª^^C¤ dP ¶¤,ª^d9R ,/ R c h(x) pf=f(x)¡7 g(x) ¤ ^d ÷ júqV3 ø f (x) ∗ g(x) = g(x) ∗ f (x) Z Z Z ÷ júq?nø h(x) dx = f (x) dx · g(x) dx. ∞
∞
` x(%§ '1.'0s 9x / e¥ −∞
O
∞
−∞
−∞
−∞
ϕ(u) = F [f ]
ψ(u) = F [g]
c
÷ júqQjø ?c=¤,;p C¤ c^cp ¤, p±wCc®¯ª0= ¤ ¬([f®¯pª0¡[¤
¬c±ôc=¶Ie^ ^0¤¦; f w
=ª¦!¨ª0 f « ±!¤,= cy ¤ cC¥
^ ¢´ ¤ ¥µ U +-
~e¥ ¥
,ª6¶I« ^ c f ~ ¤ ^c¤,pCc= ± F [f ∗ g] = F [f ]F [g] = ϕ(u)ψ(u),
1 f (x) ∗ g(x) = √ 2π
Z∞
−∞
f (y)g(x − y) dy =
1
SRBRS 1 =√ 2π 1 =√ 2π
Z∞
−∞ Z∞
−∞
1 f (y) dy √ 2π
−∞
1 ψ(u)eiux du √ 2π
1 =√ 2π
óP ¥
cp¥ ¬= c
Z∞
Z∞
×ß6Ø=ß ¼ Ë éòÒtQÍÐÜAÙ,ß6×6
PÝpÙu¿Í
ψ(u)eiu(x−y) du = Z∞
f (y)e−iuy dy =
−∞
÷ júqW:ø
ψ(u)ϕ(u)eiux du.
−∞
÷ júqQoø ¹cC
^=± e6^ e6c=c!÷ ,~jú÷ qW:jø§úqQd9ocQø§¡Vc c^¤,Qp= c ¤ eCcpd ¬VF, cpcp ,^ª0I d¼¤ ÷ òj úqQ jcø6±w ¨c¤ d9=^e¥ hce^ cp ¬pµ Ccp¬eð~c ¤ ¥
¥ ^ ^d ÷ júqCø¥ f (x) ∗ g(x) = F −1 [ϕ(u)ψ(u)] = F −1 F [f ]F [g] (x).
Z∞
f (y)g(x − y)dy =
−∞
°
#$Ê,ÛØY!µ'
Z∞
ψ(u)ϕ(u)eiux du.
−∞
~ ,A+!lØ$($Êed Òß#!!
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çÍÖ?0RÖ?0VÕÒÙrÙ@ÖVß ådÙrÏrrØ@Òß ê sÖ]×uâ@?Ö 0VÙ@ÖRÒ ØrÝÜuÖbÑÒÙrØ@ÒÊáWÒ]ÖVârØrØÍÖ?0RÖ?0VÕÒÙ@ÙrÞãyådÙrÏrrØrßÛrâ@ÖVàWÒ×uÒÙ@Öàyñ Só ê.Ü@çÏdÖVÙrÏrâ@Òá Ù@ÖVß ÝbÐb×Ð]ærØ Å ÙuÐbãYÖbÑ×uÒÙrØr ç ådâ 8WÒ / ?Ö 0VâuÐÝOÖVàIådÙrÏrrØrß Å ÙuÐVÓS0Våb×uÒá×uÖRäáVÐáWÖVærÙ@ÖäÜuÒ×@å!u ÕØZã / ±ÖVâ@ÓAÐb Ü 8RÙrÞãÛ@ÖÜuÖOÑÒÙrØrß ê sÖRäÏdÖÜ 8RÏrå
áWÖÛrâ@Ò]Ö?0VâuÐÝOÖVàYÐÙ@Ø@Ò>Ã ådâ8WÒ
Z∞
−∞
δ(x − x0 )f (x) dx = f (x0 ),
δ(x − x0 )
1 F [δ(x − x0 )] = √ 2π
Ø@ÓÒ]Òá àRØZ× Z∞
−∞
1 δ(x − x0 )e−iux dx = √ e−iux0 . 2π
è â@ÖRÓÒáWÖ43OÖrÚ ×@ØI±Òâ@ÒÙrrØrâ@ÖVàYÐÙrØ@ÒâuÐÝ]ârÞàRÙ@ÖVßà áWÖVærÏdÒ Ê ÛrâuÐàRØZÜ@å 3
×uÒ
x0
ådÙrÏrrØrØgÛrâ@ÖVàWÖ]×@ØráWäçgÛ@Ö
f 0 (x) = f 0 (x) + [f (x + 0) − f (x − 0)]δ(x − x0 ),
àWG ê M
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