P
r
e
f
The pur-pose of this handbook is to supply a collection of mathematical formulas and tables which will prove to be valuable to students and research workers in the fields of mathematics, physics, engineering and other sciences. TO accomplish this, tare has been taken to include those formulas and tables which are most likely to be needed in practice rather than highly specialized results which are rarely used. Every effort has been made to present results concisely as well as precisely SOthat they may be referred to with a maximum of ease as well as confidence. Topics covered range from elementary to advanced. Elementary topics include those from algebra, geometry, trigonometry, analytic geometry and calculus. Advanced topics include those from differential equations, vector analysis, Fourier series, gamma and beta functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions and various other special functions of importance. This wide coverage of topics has been adopted SOas to provide within a single volume most of the important mathematical results needed by the student or research worker regardless of his particular field of interest or level of attainment. The book is divided into two main parts. Part 1 presents mathematical formulas together with other material, such as definitions, theorems, graphs, diagrams, etc., essential for proper understanding and application of the formulas. Included in this first part are extensive tables of integrals and Laplace transforms which should be extremely useful to the student and research worker. Part II presents numerical tables such as the values of elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as advanced functions (Bessel, Legendre, elliptic, etc.). In order to eliminate confusion, especially to the beginner in mathematics, the numerical tables for each function are separated, Thus, for example, the sine and cosine functions for angles in degrees and minutes are given in separate tables rather than in one table SOthat there is no need to be concerned about the possibility of errer due to looking in the wrong column or row. 1 wish to thank the various authors and publishers who gave me permission to adapt data from their books for use in several tables of this handbook. Appropriate references to such sources are given next to the corresponding tables. In particular 1 am indebted to the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S., and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their book S T tf B a Aao i b a gtMy o R l n ir e l e e d si d 1 also wish to express my gratitude to Nicola Menti, Henry Hayden and Jack Margolin for their excellent editorial cooperation. M. R. SPIEGEL Rensselaer Polytechnic Institute September, 1968
o
s s
tc i
CONTENTS
Page 1.
Special
Constants..
.............................................................
1
2. Special Products and Factors ....................................................
2
3. The Binomial Formula and Binomial Coefficients .................................
3
4. Geometric Formulas ............................................................
5
5. Trigonometric Functions ........................................................
11
6. Complex Numbers ...............................................................
21
7. Exponential and Logarithmic Functions .........................................
23
8. Hyperbolic Functions ...........................................................
26
9. Solutions of Algebraic Equations ................................................
32
10. Formulas from Plane Analytic Geometry ........................................ ...................................................
34 40
11.
Special Plane Curves........~
12.
Formulas from Solid Analytic Geometry ........................................
46
13.
Derivatives .....................................................................
53
14.
Indefinite Integrals ..............................................................
57
15.
Definite Integrals ................................................................
94
16.
The Gamma
Function .........................................................
..10 1
17.
The Beta Function ............................................................
18.
Basic Differential Equations and Solutions .....................................
19.
Series of Constants..............................................................lO
20.
Taylor Series...................................................................ll
21.
Bernoulliand
22.
Formulas from Vector Analysis..
23.
Fourier Series ................................................................
..~3 1
24.
Bessel Functions..
..13 6
2s.
Legendre Functions.............................................................l4
26.
Associated Legendre Functions .................................................
.149
27. 28.
Hermite Polynomials............................................................l5 Laguerre Polynomials ..........................................................
1 .153
29.
Associated Laguerre Polynomials ................................................
30.
Chebyshev Polynomials..........................................................l5
Euler Numbers ................................................. .............................................
............................................................
..lO 3 .104
7 0 ..114 ..116
6
KG
7
Part
I
FORMULAS
THE
GREEK
Greek
name
G&W
ALPHABET
Greek name
Greek Lower case
tter Capital
Alpha
A
Nu
N
Beta
B
Xi
sz
Gamma
l?
Omicron
0
Delta
A
Pi
IT
Epsilon
E
Rho
P
Zeta
Z
Sigma
2
Eta
H
Tau
T
Theta
(3
Upsilon
k
Iota
1
Phi
@
Kappa
K
Chi
X
Lambda
A
Psi
*
MU
M
Omega
n
1.1 1.2
= natural
base of logarithms
1.3
fi
=
1.41421
35623 73095 04889..
1.4
fi
=
1.73205
08075 68877 2935.
1.5
fi
=
2.23606
79774
1.6
h
=
1.25992
1050..
.
1.7
&
=
1.44224
9570..
.
1.8
fi
=
1.14869
8355..
.
1.9
b
=
1.24573
0940..
.
1.10
eT = 23.14069
26327 79269 006..
.
1.11
re = 22.45915
77183 61045 47342
715..
1.12
ee =
22414
.
1.13
logI,, 2
=
0.30102
99956 63981 19521
37389.
..
1.14
logI,, 3
=
0.47712
12547
19662 43729
50279..
.
1.15
logIO e =
0.43429
44819
03251 82765..
1.16
logul ?r =
0.49714
98726
94133 85435 12683.
1.17
loge 10
In 10
1.18
loge 2 =
ln 2
=
0.69314
71805
59945 30941
1.19
loge 3 =
ln 3 =
1.09861
22886
68109
1.20
y =
1.21
ey =
1.22
fi
=
1.23
6
=
15.15426
=
0.57721
56649
1.78107
r(&)
=
79264
2.30258
190..
12707
6512.
9852..
00128 1468..
1.77245
2.67893
85347 07748..
.
1.25
r(i)
3.62560
99082 21908..
.
1-26
1 radian
1.27
1”
=
~/180
radians
.
= =
.. .
57.29577 0.01745
..
7232.
.
69139 5245..
.. = Eukr's co%stu~t
[see 1.201
.
38509 05516
II’(&) =
180°/7r
.
02729
~ZLYLC~~OTZ [sec pages
1.24
=
.
50929 94045 68401 7991..
01532 86060
F is the gummu
=
.
99789 6964..
24179 90197
1.64872
where
=
..
8167..
.O
95130 8232.. 32925
.
101-102).
19943 29576 92.
1
..
radians
THE
4
BINOMIAL
FORMULA
PROPERTIES
OF
AND
BINOMIAL
BINOMIAL
COElFI?ICIFJNTS
COEFFiClEblTS
3.6 This
leads
to Paseal’s
[sec page 2361.
triangk
3.7
(1)
+
(y)
+
(;)
+
...
3.8
(1)
-
(y)
+
(;)
-
..+-w(;)
3.10
(;)
+
(;)
+
(7)
+
.*.
=
2n-1
3.11
(y)
+
(;)
+
(i)
+
..*
=
2n-1
+
(1)
=
27l
=
0
3.9
3.12
3.13
-d
3.14
MUlTlNOMlAk
3.16
(zI+%~+...+zp)~ where
q+n2+
the
mm,
...
denoted
+np =
72..
by
2,
=
FORfvlUlA
~~~!~~~~~..~~!~~1~~2...~~~
is taken over
a11 nonnegative
integers
% %,
. . , np fox- whkh
1
4
GEUMElRlC
FORMULAS &
RECTANGLE
4.1
Area
4.2
Perimeter
OF LENGTH
b AND
WIDTH
a
= ab = 2a + 2b b
Fig. 4-1
PARAllELOGRAM
4.3
Area
=
4.4
Perimeter
bh =
OF ALTITUDE
h AND
BASE b
ab sin e
= 2a + 2b 1 Fig. 4-2
‘fRlAMf3i.E
Area
4.5
=
+bh
OF ALTITUDE
h AND
BASE b
= +ab sine
*
ZZZI/S(S - a)(s - b)(s - c) where s = &(a + b + c) = semiperimeter
b Perimeter
4.6
n_
L,“Z
.,
.,,
= u+ b+ c
Fig. 4-3
:
‘fRAPB%XD
4.7
Area
4.8
Perimeter
C?F At.TlTUDE
fz AND
PARAl.lEL
SlDES u AND
b
= 3h(a + b) = =
/c-
a + b + h
Y&+2 sin 4 C a + b + h(csc e + csc $)
1 Fig. 4-4
5 / -
GEOMETRIC
6
REGUkAR
4.9
Area
= $nb?- cet c
4.10
Perimeter
=
POLYGON
inbz-
FORMULAS
OF n SIDES EACH CJf 1ENGTH
b
COS(AL)
sin (~4%)
= nb
7,’ 0.’ 0 Fig. 4-5
CIRÇLE OF RADIUS
4.11
Area
4.12
Perimeter
r
= & =
277r
Fig. 4-6
SEClOR
4.13 4.14
Area
=
&r%
OF CIRCLE OF RAD+US Y
[e in radians]
T
Arc length s = ~6 A
8
0 T Fig. 4-7
RADIUS
4.15
OF C1RCJ.E INSCRWED
r=
where
&$.s-
tN A TRtANGlE *
OF SIDES a,b,c
U)(S Y b)(s -.q) s
s = +(u + b + c) = semiperimeter
Fig. 4-6
RADIUS- OF CtRClE
4.16
R=
where
CIRCUMSCRIBING
A TRIANGLE
OF SIDES a,b,c
abc 4ds(s - a)@ -
b)(s - c)
e = -&(a.+ b + c) = semiperimeter
Fig. 4-9
G
4
A
=.
4
P
.
&
sr s =
2e
s
1=
n +
1
=
FE
3 ise n
7
r n
OO
6
ni a
2 nr s i y 8
2r
RM
0
n
n ri i n
M7E
UT
°
2
r mn z
e
t
e
!
?
Fig. 4-10
4
A
=.
4
P
.
= 1 n r t a eL T n
t rZ n
n =
2e
2
t
9 r 2 a n a! 0
2 nr t a
=
2
n
n ri a n
T
!
I : e?
r m nk
T
t
e
0 F
SRdMMHW W
4
o .s
A
f=2 h +
pr
( -ae s
C%Ct&
e) 1 a r
e
OF RADWS
ra i
d2
4
i
-
g
1
T
tn
e T
e
d r
tz!? Fig. 4-12
4
A
=.
4
P
.
r
r
2
a
e
2
2 4 1 - kz rs
e c3
b
a
7r/2
=
e 5 4a
ii
m +
l
e
@
t
e
0 =
w
k = ~/=/a.h
4
A
4
A
l
[
27r@sTq See
p
e254 f
=.
$ab
r
2
.
ABC
r = e -&2dw
a
n a
e
r
to
4
c +n E5
p
u g
e
ar
p
m e
b F
r
4e
l
i
-r
o e g
a 4
gl 1
a )
tn
+
h
AOC
@
T
b Fig. 4-14
- f
1i
GEOMETRIC
8
RECTANGULAR
4.26
Volume
=
4.27
Surface
area
PARALLELEPIPED
FORMULAS
OF
LENGTH
u, HEIGHT
r?, WIDTH
c
ubc Z(ab + CLC + bc)
=
a Fig. 4-15
PARALLELEPIPED
4.28
Volume
=
Ah
=
OF CROSS-SECTIONAL
AREA
A AND
HEIGHT
h
abcsine
Fig. 4-16
SPHERE
4.29
Volume
=
OF RADIUS
,r
+
1 ---x
,-------
4.30
Surface
area
=
4wz
@ Fig. 4-17
RIGHT
4.31
Volume
4.32
Lateral
=
CIRCULAR
CYLINDER
OF RADIUS
T AND
HEIGHT
h
77&2
surface
area
=
h
25dz
Fig. 4-18
CIRCULAR
4.33
Volume
4.34
Lateral
=
m2h
surface
area
CYLINDER
=
OF RADIUS
r AND
SLANT
HEIGHT
2
~41 sine =
2777-1 =
2wh
z
=
2wh csc e Fig. 4-19
.
GEOMETRIC
CYLINDER
=
OF CROSS-SECTIONAL
4.35
Volume
4.36
Lateral surface area
Ah
FORMULAS
9
A AND
AREA
SLANT
HEIGHT
I
Alsine
=
=
pZ =
GPh
--
ph csc t
Note that formulas 4.31 to 4.34 are special cases. Fig. 4-20 RIGHT
=
CIRCULAR
4.37
Volume
4.38
Lateral surface area
CONE
OF RADIUS
,r AND
HEIGHT
h
jîw2/z =
77rd77-D
=
~-7-1
Fig. 4-21 PYRAMID
4.39
Volume
=
OF
BASE
AREA
A AND
HEIGHT
h
+Ah
Fig. 4-22 SPHERICAL
4.40
Volume (shaded in figure)
4.41
Surface area
=
CAP
=
OF RADIUS
,r AND
HEIGHT
h
&rIt2(3v - h)
2wh
Fig. 4-23 FRUSTRUM
=
OF RIGHT
4.42
Volume
4.43
Lateral surface area
+h(d
CIRCULAR
CONE
OF RADII
u,h
AND
HEIGHT
h
+ ab + b2) =
T(U + b) dF
=
n(a+b)l
+ (b - CL)~ Fig. 4-24
10
SPHEMCAt hiiWW
4.44
Area of triangle ABC
=
GEOMETRIC
FORMULAS
OF ANG%ES
A,&C
Ubl SPHERE OF RADIUS
(A + B + C - z-)+
Fig. 4-25
TOW$
&F
lNN8R
4.45
Volume
4.46
w Surface area = 7r2(b2- u2)
4.47
Volume
=
RADlU5 a
AND
OUTER RADIUS
b
&z-~(u+ b)(b - u)~
= $abc
Fig. 4-27
T.
4.4a
Volume
=
PARAWlO~D
aF REVOllJTlON
&bza
Fig. 4-28
Y
5
TRtGOhiOAMTRiC
D
OE T
FF R
WNCTIONS
F
l I FU
A R N G T ON
Triangle ABC bas a right angle (9Oo) at C and sides of length u, b, c. angle A are defined as follows. sintz . of A
5 5 5 5
5
sin A
1=
:
=
opposite hypotenuse
i
=
adjacent hypotenuse
cosine . of
A
=
~OSA
2=
. of
A
=
tanA
3= f = -~
. of
A
=
of A
tangent
c
5.5
=
secant
cosecant
. of
A
4=
k
=
adjacent t opposite
=
sec A
=
t
=
-~
=
csc A
6=
z
=
hypotenuse opposite
E
l O R RC
functions
G T
N I T
of
B
opposite adjacent
A
o cet
The trigonometric
I
TX A
c
z
A
n
g
hypotenuse adjacent
W OT
Fig. 5-1
N M
3 HG E
G A
TE I R N9L Y
H C E S0 E
A H A I ’
Consider an rg coordinate system [see Fig. 5-2 and 5-3 belowl. A point P in the ry plane has coordinates (%,y) where x is eonsidered as positive along OX and negative along OX’ while y is positive along OY and negative along OY’. The distance from origin 0 to point P is positive and denoted by r = dm. If it is described dockhse from The angle A described cozmtwcZockwLse from OX is considered pos&ve. OX it is considered negathe. We cal1 X’OX and Y’OY the x and y axis respectively. The various quadrants are denoted by 1, II, III and IV called the first, second, third and fourth quadrants respectively. In Fig. 5-2, for example, angle A is in the second quadrant while in Fig. 5-3 angle A is in the third quadrant.
Y
Y
II
1
II
1
III
IV
III
IV
Y’
Y’ Fig. 5-3
Fig. 5-2
11 f
TRIGONOMETRIC
12
FUNCTIONS
For an angle A in any quadrant the trigonometric
functions of A are defined as follows.
5.7
sin A
=
ylr
5.8
COSA
=
xl?.
5.9
tan A
=
ylx
5.10
cet A
=
xly
5.11
sec A
=
v-lx
5.12
csc A
=
riy
RELAT!ONSHiP BETWEEN DEGREES AN0
RAnIANS N
A radian is that angle e subtended at tenter 0 of a eircle by an arc MN equal to the radius r. Since 2~ radians = 360° we have 5.13
1 radian
= 180°/~
5.14
10 = ~/180 radians
=
1
r
e 0
57.29577 95130 8232. . . o
r
B
= 0.01745 32925 19943 29576 92.. .radians
Fig. 5-4
REkATlONSHlPS 5.15
tanA
= 5
5.16
&A
~II ~ 1
5.17
sec A
=
~
5.18
cscA
=
-
tan A
AMONG
COSA sin A
zz -
1
COS A
TRtGONOMETRK
5.19
sine A +
~OS~A
5.20
sec2A
-
tane
5.21
csceA
- cots A
II
III IV
1
A = 1 =
1
1 sin A
SIaNS AND VARIATIONS
1
=
FUNCTItB4S
+ 0 to 1
+ 1 to 0
+ 1 to 0
0 to -1
0 to -1 -1 to 0
OF TRl@ONOMETRK
+ 0 to m -mtoo + 0 to d
-1 to 0 + 0 to 1
+ CCto 0 oto-m + Ccto 0 -
--
too
oto-m
FUNCTIONS
+ 1 to uz
+ m to 1
-cc to -1
+ 1 to ca
-1to-m + uz to 1
--COto-1 -1 to --
M
TRIGONOMETRIC
E
Angle A in degrees
00
X
F
Angle A in radians
A T
A
O
RL
FC
R
1
IU
O UT
O S
sec A
csc A
0
1
0
w
1
cc
ii/6
1
+ti
450
zl4
J-fi
$fi
60°
VI3
Jti
750
5~112
900
z.12
105O
7~112
*(fi+&)
-&(&-Y%
-(2+fi)
-(2-&)
120°
2~13
*fi
-*
-fi
-$fi
1350
3714
+fi
-*fi
150°
5~16
4
-+ti
#-fi)
2-fi
&(&+fi)
fi
1
0
fi)
-&(G+
0
-*fi
-fi
-(2-fi)
-(2+fi)
180°
?r
-1
1950
13~112
210°
7716
225O
5z-14
-Jfi
240°
4%J3
-#
255O
17~112
270°
3712
-1
285O
19?rll2
-&(&+fi)
3000
5ïrl3
-*fi
2
315O
7?rl4
-4fi
*fi
-1
330°
117rl6
*fi
-+ti
345O
237112
360°
2r
-$(fi-fi)
-*(&+fi)
2-fi
-
1
4
-*fi
-i(fi-
2+fi 0
-(2+6)
&(&+
-ti
fi) 1
0
see pages
206-211
-(2
- fi) 0
++
-fi
\h
-+fi
2
-(fi-fi)
f
-(&-fi)
-2 -(&+?cz)
-@-fi)
&+fi
-(2+6) T-J
i
-36 -(fi-fi) -1 -(fi-fi)
2
-1
f
-fi
Tm
-*fi
-ti
-2
g
-fi
0
*ca -(&+fi)
i -
&fi 2-6
Vz+V-c? -1
3
1
km
*(&-fi)
6)
angles
ti
-&(&-fi)
1
l
1
-4
-&&+&Q
6
fi-fi
-2
2 + ti
&
1
-(&+fi)
Tm
0
fi-fi
km
-1
-1
TG
;G
&+fi 0
N
fi
2
2-&
*CU
fi)
fi
.+fi
2+&
R
2
$fi
1
C N
3
&+fi
fi-fi
fi
1
@-fi)
$(fi-
2+*
*fi
r1
i(fi+m
other
A
cet A
300
involving
FN A
tan A
rIIl2
tables
GE
COSA
0
llrll2
V
sin A
15O
165O
For
V
FUNCTIONS
fi $fi fi-fi
-$fi -fi -2 -(&+fi)
1
?m
and 212-215.
f
I
TRIGONOMETRIC
5.89
y = cet-1%
5.90
y
=
FUNCTIONS
19
sec-l%
5.91
_--/
y
=
csc-lx
Y
I T
---
,
/A--
/’
/ -77 -//
,
Fig. 5-14
Fig. 5-15
RElAilONSHfPS
BETWEEN
The following results hold for sides a, b, c and angles A, B, C.
5.92
ANGtGS
any plane triangle
ABC
OY A PkAtM
with
TRlAF4GlG
’
A
Law of Sines a -=Y=sin A
5.93
SIDES AND
Fig. 5-16
1
b
c
sin B
sin C C
Law of Cosines
/A
cs = a2 +
bz -
Zab
COS
f
C
with similar relations involving the other sides and angles. 5.94
Law of Tangents
tan $(A + B) a+b -a-b = tan i(A -B) with similar relations involving the other sides and angles.
5.95
sinA where
s = &a + b + c)
=
:ds(s
is the semiperimeter
- a)(s - b)(s - c) of the triangle.
B and C cari be obtained. See also formulas 4.5, page 5; 4.15 and 4.16, page 6.
Spherieal triangle ABC is on the surface of in Fig. 5-18. Sides a, b, c [which are arcs of measured by their angles subtended at tenter 0 of are the angles opposite sides a, b, c respectively. results hold. 5.96
Law of Sines sin a -z-x_ sin A
5.97
sin b sin B
a sphere as shown great circles] are the sphere. A, B, C Then the following
sin c sin C
Law of Cosines sinbsinccosA
cosa
=
cosbcosc
COSA
=
- COSB COSC +
+
Fig. 5-1’7
sinB sinccosa
with similar results involving other sides and angles.
Similar relations involving angles
2
T
0 L
5
o.
w
T
a
s
5
f 9
ri
s = &
S = +
f
e
E g
i
c
S(
8 n & + B
a
t(
t
&
a=
t(
4l
op
f r x F s ii
1e
o mn s
e
I
g
U
$ ) + n b )
sh a t ai v i
r)
G
e
aA
(
N
)
n
a A
n
O
n
C
T
t
u
n
u n h nl o d
N
s
i
l d e a g l e
t
r
O
(
rl v s
s
e i
u i hr
f e. o
.s
r)
A i rh
ra
0
FGR
RtGHT
o C it c na i b -va b i
m + ose
o sa t a i l i
r u n h n l dd
ld e g a e
t
r
lr s
0
( Se
RlJlES
a wn
- B
0
h+ B + C
NAPIER’S
a
t
1
w a
et
F
9
1+
h
.
S
w
9
w
5
a
i i
.
R
4o
f e. o
m g
.
meos
4
eu
ANGLED
rf , gh p e gor s i c gB e le, , u , . .
o sa t a i l i ,
l
SPHERICAL
ha e p t lef 9n A l
pv
Atr
r un h n l dd
ld e
ga
e
t
r
lr
a
TRIANGLES
t rwe , d
he i
Z aet
ih
ei f t r3s o a i
n r
h rC r nc
a
C
F S [
c
A a
t i
o p
5
uq ot
i h hn
o t n p n da t a t
-
g
a pu a
fi hce ri a m s ia ea d a e c to a om c r
of th y ac e rj n h w r p
T
s.
o
a
h m i1
fp
5.102
T
s
o
a
h m i
fp n ee i n
S
T
x
C
c
= 9i
o
ch
a-
n ee i n0 t
O C
s
a
s
( ba
=
t
n0= m 9A t =
o f
.
oe
F
9
ri p aar c s a en io nr Fi a n l p A a npi B
e m oc ayd
5
E
1
5
5e s n w s t a ir ndoc g . c
p rt ti tsl a ps r p ea Te
an cs laN
u d f oeh t oa a a l
a p y q d eo ht r rc
u d fo eht t ooo
O w c° h 0p,
B A
-
e e a °l n s
,
(
a
na
C o
C i
C a C
(
nO
Oo O C ~
uts
rl i rt 5 rhe o es p
a er fb
g
e 2p t
hd
ead
wl xi hr
pv le r l aao s
f p eh
B
dsp
et Cg.l h
eoc tn eu t
a -ei e p sr
.
0r t ri O ee e ni n s rl
da sl i n
ae ug j s
l e
ae
ve
r
uip s
r
frg di ie ce a n co
e:
i
AS =-r SC OaOs
2
ee
et
f eph dn d l e
a l
n = rOt
b it
-
w - a ehi ngc t dta l m p t
a p y q d eo1 ht r rt
--
i
O
-a B A
oa 1. s e n os a
n O -
mi 99 e
Bn
S i
) ug
a
)b
SB n
n .7
l e
e
A complex number is generally written as a + bi where a and b are real numbers and i, called the imaginaru unit, has the property that is = -1. The real numbers a and b are called the real and ima&am parts of a + bi respectively. The complex numbers a + bi and a - bi are called complex
6.1
a+bi
=
c+di
if and only if
conjugates
a=c
and b=cZ
6.2
(a + bi) + (c + o!i) =
(a + c) + (b + d)i
6.3
(a + bi) - (c + di) =
(a - c) + (b - d)i
6.4
(a+ bi)(c+
di) =
(ac- bd) + (ad+
of each other.
bc)i
Note that the above operations are obtained by using the ordinary rules of algebra and replacing 9 by -1 wherever it occurs.
21
22
COMPLEX
GRAPH
NUMBERS
OF A COMPLEX
NtJtWtER
A complex number a + bi cari be plotted as a point (a, b) on an xy plane called an Argand diagram or Gaussian plane. For example in Fig. 6-1 P represents the complex number -3 + 4i. A
eomplex
number
cari
also
be
interpreted
as
a
wector
p,----.
y
from
0 to P. -
0
X
* Fig. 6-1
POLAR
FORM
OF A COMPt.EX
NUMRER
In Fig. 6-2 point P with coordinates (x, y) represents the complex number x + iy. Point P cari also be represented by polar coordinates (r, e). Since x = r COS6, y = r sine we have
6.6
x + iy = ~(COS 0+
called
the poZar form
the mocklus
of the complex
and t the amplitude
i sin 0)
number.
L
We often
-
X
cal1 r = dm
of x + iy. Fig. 6-2
tWJLltFltCATt43N
[rl(cos
6.7
AND
DtVlStON
OF CWAPMX
el + i sin ei)] [re(cos ez + i sin es)] V-~(COSe1 + i sin el)
6.8
ZZZ 2
rs(cos ee + i sin ez)
If p is any real
number,
De Moivre’s [r(cos
rrrs[cos
1bJ POLAR
ilj 0”
FtMM
tel + e2) + i sin tel + e2)]
[COS(el - e._J + i sin (el - .9&]
DE f#OtVRtt’S
6.9
=
NUMBRRS
THEORRM
theorem
states
e + i sin e)]p
=
that rp(cos pe + i sin pe)
.
RCWTS
If
p = l/n
where
k=O,l,2
integer,
[r(cos e + i sin e)]l’n
6.10 where
n is any positive
OF CfMMWtX
k is any ,...,
integer. n-l.
From
this
the
=
n nth
NUtMB#RS
6.9 cari be written rl’n roots
L
e + 2k,, ~OSn of
a complex
+
e + 2kH
i sin ~
number
n cari
1 be
obtained
by
putting
”
In the following p, q are real numbers, CL,t are positive numbers and WL,~are positive integers.
7.1
cp*aq z aP+q
7.2
aP/aqE @-Q
7.3
(&y E rp4
7.4
u”=l,
7.5
a-p = l/ap
7.6
(ab)p = &‘bp
7.7
&
7.8
G
7.9
Gb
a#0
z aIIn
= pin
=%Iî/%
In ap, p is called the exponent, a is the base and ao is called the pth power of a. The function is called an exponentd function.
If a~ = N where a # 0 or 1, then p = loga N is called the loga&hm N = ap is called t,he antdogatithm of p to the base a, written arkilogap. Example:
Since
The fumAion
3s = 9 we have
y = ax
of N to the base a. The number
log3 9 = 2, antilog3 2 = 9.
v = loga x is called a logarithmic
jwzction.
7.10
logaMN
=
loga M + loga N
7.11
log,z ;
=
logG M -
7.12
loga Mp
=
p lO& M
loga N
Common logarithms and antilogarithms [also called Z?rigg.sian] are those in which the base a = 10. The common logarit,hm of N is denoted by logl,, N or briefly log N. For tables of common logarithms and antilogarithms, see pages 202-205. For illuskations using these tables see pages 194-196. 23
EXPONENTIAL
24
AND LOGARITHMIC
NATURAL LOGARITHMS
FUNCTIONS
AND ANTILOGARITHMS
Natural logarithms and antilogarithms [also called Napierian] are those in which the base a = e = 2.71828 18. . . [sec page 11. The natural logarithm of N is denoted by loge N or In N. For tables of natural logarithms see pages 224-225. For tables of natural antilogarithms [i.e. tables giving ex for values of z] see pages 226-227. For illustrations using these tables see pages 196 and 200.
CHANGE OF BASE OF lO@ARlTHMS
The relationship between logarithms of a number N to different bases a and b is given by
7.13
loga N
=
hb
iv
hb
a
-
In particular, = ln N
7.14
loge N
7.15
logIO N = logN
RElATlONSHlP
= 2.30258 50929 94.. . logio N =
0.43429
44819 03.. . h& N
BETWEEN EXPONBNTIAL ANO TRl@ONOMETRlC eie =
7.16 These are called Euler’s
COS 0 + i sin 8,
dent&es.
e-iO
=
COS 13 -
sin 6
Here i is the imaginary unit [see page 211.
7.17
sine
7.18
case =
=
eie- e-ie 2i
eie+ e-ie 2
7.19
7.20 2
7.21
sec 0
=
&O + e-ie
7.22
csc 6
=
eie
7.23
i
2i
eiCO+2k~l
From this it is seen that @ has period 2G.
-
e-if3
=
eie
k =
integer
FUNCT#ONS
;;
E
POiAR
T
p
XA
FORfvl OF COMPLEX
f
7
o h a co
o n
.
2
6
t
o
6
NUMBERS
.o hp r 2
(reiO)l/n E
LOGARITHM
7.29
COMPLEX
a.
l
OD
ym i a tm e
(
ffUMBERS
e7n ra m 2 t 1r t
(q-eio)Pzz q-P&mJ [
7.2B
OF
GU
EXPRESSE$3 AS AN
oxl + i r c u b w m a
WITH
7.27
PN
or rpe
N
AN 25
E
RC
N
EXPONENTNAL
n re b
[if lx 6
pi r e 2 st a ep .
a mr 2 et s x o6
g
4 6 + i sin 0) = 9-ei0 x + iy = ~(COS
OPERATIONS
F
fe
L
[~&O+Zk~~]l/n
q f og
M
t =
n
u
D
FORM
o 0eh uo ue
o
h
l
e
e
i
il
g
a
e
vl
v
h
s
o
NUMBER
k e=e i k
@n z
) t -
ao
r
rl/neiCO+Zkr)/n
OF A COMPLEX
= l r n + iT + 2
IN POLAR
e i
DEIWWOPI
OF HYPRRWLK
8.1
Hyperbolic
sine of x
=
sinh x
=
8.2
Hyperbolic
cosine
=
coshx
=
8.3
Hyperbolic
tangent
= tanhx
=
8.4
Hyperbolic
cotangent
8.5
Hyperbolic
secant
8.6
Hyperbolic
cosecant
RELATWNSHIPS
of x
of x
coth x
of x =
of x
AMONG
ez + e-=
2 ~~~~~~
2
ez + eëz
HYPERROLIC FUWTIONS
=
sinh x a
coth z
=
1 tanh x
sech x
=
1 cash x
8.10
cschx
=
1 sinh x
8.11
coshsx - sinhzx
=
1
8.12
sechzx + tanhzx
=
1
8.13
cothzx - cschzx
=
1
FUNCTIONS
2
= csch x = &
tanhx
8.7
# - e-z
ex + eCz = es _ e_~
= sech x =
of x
.:‘.C,
FUNCTIONS
cash x sinh x
=
OF NRGA’fWE
ARGUMENTS
8.14
sinh (-x)
=
- sinh x
8.15
cash (-x)
= cash x
8.16
tanh (-x)
= - tanhx
8.17
csch (-x)
=
-cschx
8.18
sech(-x)
=
8.19
coth (-x)
=
26
sechx
-~OUIS
HYPERBOLIC
AWMWM
FUNCTIONS
27
FORMWAS
0.2Q
sinh (x * y)
=
sinh x coshg
8.21
cash (x 2 g)
=
cash z cash y * sinh x sinh y
8.22
tanh(x*v)
=
tanhx f tanhg 12 tanhx tanhg
8.23
coth (x * y)
=
coth z coth y 2 1 coth y * coth x
8.24
sinh 2x
=
2 ainh x cash x
8.25
cash 2x
=
coshz x + sinht x
8.26
tanh2x
=
2 tanh x 1 + tanh2 x
=
* cash x sinh y
2 cosh2 x -
1
=
1 + 2 sinh2 z
HAkF ABJGLR FORMULAS
8.27
sinht
=
8.28
CoshE 2
=
8.29
tanh;
=
k
Z
sinh x cash x + 1
.4
[+ if x > 0, - if x < O] cash x + 1 -~ 2 cash x - 1 cash x + 1
’ MUlTWlE
[+ if x > 0, - if x < 0]
ZZ cash x - 1 sinh x
A!Wlfi WRMULAS
8.30
sinh 3x
=
3 sinh x + 4 sinh3 x
8.31
cosh3x
=
4 cosh3 x -
8.32
tanh3x
=
3 tanh x + tanh3 x 1 + 3 tanhzx
8.33
sinh 4x
=
8 sinh3 x cash x + 4 sinh x cash x
8.34
cash 4x
=
8 coshd x -
8.35
tanh4x
=
4 tanh x + 4 tanh3 x 1 + 6 tanh2 x + tanh4 x
3 cash x
8 cosh2 x -t- 1
2
H
8
YF
P
O
PU
HO
E N
FY& W
P
J
R C
E
E
B T
f
R
R
8
.
3
s
6=
&i c
2
-
4 na
8
.
3
c
7=
4 oc
2
+
$ sa
8
.
3
s
x
8=
&i s
3
-
8
.
3
c
x
9=
&o c
+
8
.
4
s
0=
8i -
4 c
2
n+
4 ca
4x
h
as
% 4
sh
x
h
8
.
4
c
1=
#o +
+ c
2
s+
& ca
4x
h
as
x 4
sh
x
h
S
D
8
U
.
AI
F
A
hs
zh
x
x
hs
zh
x
2 sn i
xx
ihn
nsh
2 cs o
x
ahs
ssh
K
NFO
x
W R
&
DFF F O P
Sl
h h3
E
x
UR D
R
s
4+
s
i
=
2 si2
& n
+ y
cn i
$ hx - y)
anh
(x
)
s hy
x
h
x
h
kR U
8
.
4s
-
s
3i
=
2 ci
n&
+ y
s an
$ hx - Y)
i sh
(x
)
n hy
8
.
4c
+
c
4o
=
2 co
is
+ y
c as
#(h
- Y)
a sh
xxx
)
s hy
8
.
4c
-
c
5o
=
2 so
$s
+ y
s is
$ (h - Y)
i nh
( xx
)
n hy
8
.
4s
x s
y 6i=
* i
n
{- n c
h
c ho
o
s
s
h
h
(
8
.
4c
x c
y 7 a=
+ a
s
{+ s c
h
c ho
o
s
s
h
h
(
s
x 4c
y
i=
+ a
n+ y
{- s s
x @ h- ) Y sl h i
) -i
n
} n
h
h
8
.
E
I
t
t
OX H
f
n
hw
.o
8 s
FP FY
x e>e 0 ls I
oa 1
x = u
i c
8(
= u
!R UPT
x < 0 u. l s t f
a
9
.
n o t
t s
x
i
n
h
c
x
a
s
h
t
x
a
n
h
c
x
o
t
h
s
x
e
c
h
c
x
s
c
h
= uh s a c
s ou h
s p
O
N ‘ E NEE
a e i wme
x = 1h n o s
i p b s fn i e
x =1 xu h t e c
h x
h
F OSC RR
g r 8y
o dn
x = xwh c s
T SB
n o .
rig
h c
HYPERBOLIC
GRAPHS
8.49
y = sinh x
OF HYPERBOkfC
8.50
29
FUNCltONS
8.51
y = coshx
Fig. S-l 8.52
FUNCTIONS
Fig. 8-2
y = coth x
8.53
/i
y
y = tanh x
Fig. 8-3
8.54
y = sech x
y = csch x Y \
X
1
7
10
X
0
-1
iNVERSE
HYPERROLIC
L
X
Fig. 8-6
Fig. 8-5
Fig. 8-4
0
FUNCTIONS
If x = sinh g, then y = sinh-1 x is called the inverse hyperbolic sine of x. Similarly we define the The inverse hyperbolic functions are multiple-valued and. as in the other inverse hyperbolic functions. case of inverse trigonometric functions [sec page 171 we restrict ourselves to principal values for which they ean be considered as single-valued. The following list shows the principal values [unless otherwise indicated] of the inverse hyperbolic functions expressed in terms of logarithmic functions which are taken as real valued.
8.55
sinh-1 x
=
ln (x + m
8.56
cash-lx
=
ln(x+&Z-ï)
8.57
tanh-ix
=
8.58
coth-ix
=
8.59
sech-1 x
8.60
csch-1 x
)
-m<x<m XZl -l<x
+ln
X+l ( x-l
)
x>l
O<xZl
=
[cash-r x > 0 is principal value]
ln(i+$$G.)
x+O
or xc-1
[sech-1 x > 0 is principal value]
HYPERBOLIC
30
FUNCTIONS
8.61
eseh-]
x
=
sinh-1
(l/x)
8.62
seeh-
x
=
coshkl
(l/x)
8.63
coth-lx
=
tanh-l(l/x)
8.64
sinhk1
(-x)
=
- sinh-l
x
8.65
tanhk1
(-x)
=
- tanh-1
x
8.66
coth-1
(-x)
=
- coth-1
x
8.67
eseh-
(-x)
=
- eseh-
x
GffAPHS
8.68
y =
OF fNVt!iffSft HYPfkfftfUfX
8.69
sinh-lx
FfJNCTfGNS
X
7 -ll
8.72
coth-lx Y
0
\
\
\
\
‘-. Fig. 8-9
L 11
x
y =
8.73
sech-lx
y =
Y
Il
0
I I’ Fig. 8-11
csch-lx Y
I
Fig. 8-10
\
Fig. 8-8
Fig. 8-7
l l l
x
-1 \
y =
tanhkl
l
Y
Y
8.71
y =
8.70
y = cash-lx
,
,
/
X
3
L 0
Fig. 8-12
-x
HYPERBOLIC
tan (ix) == i tanhx
sec (ix) = sechz
8.79
cet (ix)
8.81
cash (ix) = COSz
8.82
tanh (iz)
= i tan x
8.84
sech (ix) = sec%
8.85
coth (ix)
=
sin (ix) = i sinh x
8.75
COS(iz)
8.77
csc(ix)
8.78
8.80
sinh (ix) = i sin x
8.83
csch(ti)
-i
=
cschx
-icscx
In the following
31
8.76
8.74
=
FUNCTIONS
= cash x
== -
k is any integer.
8.86
sinh (x + 2kd)
=
sinh x
8.87
cash (x + 2kd)
=
cash x
8.88
tanh(x+
8.89
csch (x +2ks-i)
=
cschx
8.90
sech (x + 2kri)
=
sech x
8.91
coth (S + kri)
= i sinh-1 x
8.93
sinh-1 (ix)
2 i cash-1 x
8.95
cash-lx
tan-1 (ix) = i tanh-1 x
8.97
tanh-1 (ix) = i tan-1 x
8.98
cet-1 (ix)
8.99
coth-1 (ix)
8.100
sec-l x
8.101
sech-* x
8.102
C~C-1 (iz)
8.103
eseh-
8.92
sin-1 (ix)
8.94
Cos-ix
8.96
=
=
-icotz
= *i =
- i coth-1 x sech-lx - i csch-1 z
= =
(ix)
i sin-1 x
k i COS-~x
= - i cet-1 x =
*i =
sec-l x - i C~C-1 x
kri)
= tanhx =
coth z
9
S
QUAURATIC
9.1
S
o
I a b, c a
fa , i
(
r
a
u i
(
r
a
e i
(
c
I
9.2
r
a
t
L
9.3
Dr = eb2 - n 4
ei
I
u
t
2a
f i te
discriminant, a d a s ht
l t
i
cr
aeh
e
l
q
u
d
u)
l
a
l
n i
p
j )
o i
fr r
D
mf
t r
h x + ox , = -bla h e
Q
t
3a2 -
a; =
9
Xl
=
S + T -
x
=
-
=
-
7
,
w
x
x
x ah t
rt
s
-
-
( -
S
--
+3
-
+ &
a
T/ S
1
)Z +
( T
S )
-
discriminant, a d s ht
wo
l
o i a D >do 0 ot
eh
e i af y
mf l
dt e
n
) l
s mfp r
e
= 2
C
(
a
O @
x2 =
2
C
(
+ 1m
O +w
2C
=
2
C
(
+ 2G
O +
4
+ Ca + x r ,h ,
fa
i )
u s im o t
2 ar r e h
a
-
.
)f T
D 1n =C o 0. n
x
ar
,s
1
1a o u
xI + x2 + xs = -
a
a
Ti +
a i r
9.5
sx
x
a
r
3
g
+ S
a
x
e
l
+
(
if D < 0:
t
$
1a n o et e e )i awD = rd0t q a f l o
Solutions
en
&
1a o a l r i et
b
o
o er e
32
e p
n
j
l
e s u s
a t
s q
u
pu i
u
e l a
l tg
i ao
S )
= Qr r r
s
1
a i r
so
e
n
a
d
-
4
r
r i ei
nl
a
x
nce
-
re
u
tr s
- 2a
5
+
2
cs )e
- 2
-
n
a
i
to
=e c
o
f
an
oex x
9
R=
’
ra 2 i s Dr = eQ3 +, nR2, f i te i ri
are
l
o
o
d
(
c
h
af
Xl
9.4
2 ~/@-=%c-
(
D < 0,
r
-b
=
f a
iL rf
uz2 + bx -t c = 0
Q G U
Dn > 0n )
Solutions:
a a
EQUATION:
L
i e D =n 0 q i
e
I a a
Ef
lx
c i
x
o O A
o
Th 0 O e =S -RI&@
x s x e
S e
0
= - ,s ,
r z s
e
t
’
e
S
)
r
’ ax e
x r s
) ss
2 .
SOLUTIONS
QUARTK
Let y1 be a real root
9.7
Solutions:
ALGEBRAIC
EQUATION:
of the cubic
The 4 roots
OF
x* -f- ucx3 + ctg9 +
of ~2 +
xl, x2, x3, x4 are the four
u
3
+
a
3 =
0
4
3
$
equation
+{a1
2
a; -4uz+4yl}z
If a11 roots of 9.6 are real, computation is simplified a11 real coefficients in the quadratic equation 9.7.
where
EQUATIONS
by using
+ that
$&
* d-1
particular
= real
root
0
which
produces
roots.
-
FURMULAS Pt.ANE ANALYTIC
10
fram
GEOMETRY
DISTANCE d BETWEEN TWO POINTS F’&Q,~~) AND &(Q,~~) 10.1
d=
-
Fig. 10-1
10.2
mzz-z
EQUATION
10.3
OF tlNE JOlNlN@
Y -
Y1
x -
ccl
m
Y2 -
Y1
F2 -
Xl
TWO POINTS ~+%,y~)
Y2 - Y1 xz -
10.4
cjr
Xl
y = where
b = y1 - mxl =
XZYl xz -
EQUATION
XlYZ 51
tan 6
Y -
Y1 =
mb
ANiI
l%(cc2,1#2)
- Sl)
mx+b
is the intercept
on the y axis, i.e. the y intercept.
OF LINE IN ‘TEMAS OF x INTERCEPT a # 0 AN0 3 INTERCEPT b + 0 Y
b a Fig. 10-2
34
2
FORMULAS
FROM
ffQRMAL
10.6
ANALYTIC
FORA4 FOR EQUATION
+ Y sin a
x cosa
PLANE
=
where
p
=
perpendicular
and
a
1
angle of inclination positive z axis.
GEOMETRY
OF 1lNE
y
p
distance
35
from
origin
0 to line
of perpendicular
P/ ,
with
,
L LX
0 I Fig. 10-3
GENERAL
10.7
Ax+BY+C
KIlSTANCE
where
FROM
the sign is chosen
ANGLE
10.9
s/i BETWEEN
tan $ Lines are parallel Lines
POINT
SO that
=
(%~JI)
the distance
TWO
OF LINE
EQUATION
0
TO LINE
AZ -l- 23~ -l- c = Q
is nonnegative.
l.lNES
HAVlNG
SlOPES
wsx AN0
%a2
m2 - ml 1 + mima
=
or coincident
are perpendicular
if and only if mi = ms.
if and only
if ma = -Ilmr.
Fig. 10-4
AREA
10.10
Area
=
z=
where
*T
OF TRIANGLE
1
*;
the sign
If the area
Xl
Y1
1
~2
ya
1
x3
Y3
1
(Xl!~/2
+
?4lX3
is chosen
WiTH
VERTIGES
AT @I,z&
@%,y~), (%%)
(.% Yd +
Y3X2
SO that
is zero the points
-
!!2X3
-
the area
YlX2
-
%!43)
is nonnegative.
a11 lie on a line. Fig. 10-5
FORMULAS
36
TRANSFORMATION
1
10.11
FROM
PLANE
ANALYTIC
OF COORDINATES
x
=
x’ + xo
Y
=
Y’ + Y0
1 x’
or
y’
x
x
GEOMETRY
INVGisVlNG
x -
xo
Y -
Y0
PURE
TRANSlAliON
Y
l Y’ l
l
where (x, y) are old coordinates [i.e. coordinates relative to xy system], (~‘,y’) are new coordinates [relative to x’y’ system] and (xo, yo) are the coordinates of the new origin 0’ relative to the old xy coordinate system. Fig. 10-6
TRANSFORMATION
10.12
1
=
x’ cas L -
OF COORDIHATES
y’ sin L
or
-i y = x’ sin L + y’ cas L
x’ z
INVOLVING
PURE
x COSL + y sin a
ROTATION
\Y! \ \ \ \
yf z.z y COSa - x sin a
where the origins of the old [~y] and new [~‘y’] coordinate systems are the same but the z’ axis makes an angle a with the positive x axis. ,
,
,
,
Y
,
\o/ , ’ \
,
/
/
/
,x’
L
CL!
\ Fig. 10-7
TRANSFORMATION
OF COORDINATES
1 1
02 =
10.13
lNVGl.VlNG
TRANSLATION
x’ cas a - y’ sin L + x.
y = 3~’sin a + y’ COSL + y0
or
ANR
x’
ZZZ
(X - XO) cas L + (y - yo) sin L
y!
rz
(y - yo) cas a - (x - xo) sin a
1 \
ROTATION
/
,‘%02 \
where the new origin 0’ of x’y’ coordinate system has coordinates (xo,yo) relative to the old xy eoordinate system and the x’ axis makes an angle CYwith the positive x axis.
Fig. 10-8
POLAR
COORDINATES
(Y, 9)
A point P cari be located by rectangular coordinates (~,y) or polar eoordinates (y, e). The transformation between these coordinates is
10.14
x
=
1 COS 0
y = r sin e
or
T=$FTiF
6 = tan-l
(y/x)
Fig. 10-9
FORMULAS
RQUATIQN
10.15
FROM PLANE
OF’CIRCLE
(a-~~)~ + (g-vo)2
ANALYTIC
OF RADIUS
GEOMETRY
37
R, CENTER AT &O,YO)
= Re
Fig. 10-10
RQUATION
10.16
OF ClRClE
OF RADIUS
R PASSING
T = 2R COS(~-a)
THROUGH
ORIGIN
Y
where (r, 8) are polar coordinates of any point on the circle and (R, a) are polar coordinates of the center of the circle.
Fig. 10-11
CONICS
[ELLIPSE,
PARABOLA
OR HYPEREOLA]
If a point P moves SO that its distance from a fixed point [called the foc24 divided by its distance from a fixed line [called the &rectrkc] is a constant e [called the eccen&&ty], then the curve described by P is called a con& [so-called because such curves cari be obtained by intersecting a plane and a cane at different angles]. If the focus is chosen at origin 0 the equation of a conic in polar coordinates (r, e) is, if OQ = p and LM = D, [sec Fig. 10-121 10.17
T =
P 1-ecose
=
CD 1-ecose
The conic is (i)
an ellipse if e < 1
(ii)
a parabola if e = 1
(iii) a hyperbola if c > 1.
Fig. 10-12
38
FORMULAS
FROM PLANE
10.18
Length of major axis A’A
=
2u
10.19
Length of minor axis B’B
=
2b
10.20
Distance from tenter C to focus F or F’ is
ANALYTIC
GEOMETRY
C=d--
= c =
E__
10.21
Eccentricity
10.22
Equation in rectangular
a
-
~
0
a
coordinates:
(r - %J)Z + E b2 a2
Fig. 10-13
=
3
re zz
a2b2
10.23
Equation in polar coordinates if C is at 0:
10.24
Equation in polar coordinates if C is on x axis and F’ is at 0:
10.25
If P is any point on the ellipse, PF + PF’
=
a2 sine a + b2 COS~6
r =
a(1 - c2) l-~cose
2a
If the major axis is parallel to the g axis, interchange x and y in the above or replace e by &r - 8 [or 9o” - e].
PARAR0kA
WlTJ4 AX$S PARALLEL
TU 1 AXIS
If vertex is at A&,, y,,) and the distance from A to focus F is a > 0, the equation of the parabola is 10.26
(Y - Yc?
10.27
(Y - Yo)2 =
=
4u(x - xo)
if parabola opens to right [Fig. 10-141
-4a(x - xo)
if parabola opens to left [Fig. 10-151
If focus is at the origin [Fig. 10-161 the equation in polar coordinates is 10.28
T
=
2a 1 - COSe Y
Y
-x
0
Fig. 10-14
Fig. 10-15
x Fig. 10-16
In case the axis is parallel to the y axis, interchange x and y or replace t by 4~ - e [or 90” - e].
FORMULAS
FROM PLANE
ANALYTIC
GEOMETRY
39
Fig. 10-17 10.29
Length of major axis A’A
= 2u
10.30
Length of minor axis B’B
=
10.31
Distance from tenter C to focus F or F’
10.32
Eccentricity
10.33
Equation in rectangular
10.34
Slopes of asymptotes G’H and GH’
10.35
Equation in polar coordinates if C is at 0:
10.36
Equation in polar coordinates if C is on X axis and F’ is at 0:
10.37
If P is any point on the hyperbola,
e = ;
= -
2b =
c = dm
a coordinates:
=
(z - 2# os
(y - VlJ2 -7=
1
* a
”
PF - PF!
=
=
If the major axis is parallel to the y axis, interchange [or 90° - e].
a2b2 b2 COS~e - a2 sin2 0
22a
r =
Ia~~~~~O
[depending on branch]
5 and y in the above or replace 6 by &r - 8
11.1
E
i
p
qc
n r
E
1
1 i
1
A
b1
1
A
o 1o
r
l
+ y
An
o
r=
uo
= a c
2
. cn
u
q (
o
e
l 2
2 a
0
c
a
2 o
= C S - y* A e.
a
&f . n
B xga r
o
e
ao
a
o
’lx
o
B w
a
E
i
p q
fn [
C =
CE
L-
1y = a 1
A
1
A T
a r
o 1o l
a
1
2
o r ae
i a c dh a x ao
s l
.=
o
r 8f
(
F o o
E
1 i
r
q %
E
1
1
i
p q
A
11.11
A
u
b l
T i a c a i r o /t
brc o
e r
e
u
y
2 Z
a
=
a
s
li
dh s bu a p ei P o si t o o a n c4h n o rl f
d
A
o
’
s
\ l ’
eB,
/
xg
n
n
i 1
1
g-
l
m
i
2
,
tY
n-
nn
j
m i
O
:
e o
t n
n S
a
#
h
)
2
c g
h t
v i
ic f
a g
is
h er
nr n. F
1
d
c
ti g i
1
i
l
b g
-
u
ViflTH FOUR CUSf’S
/ Z
c 2
a
f
9o
o
3 Z
r
O
0
f= n6 c
a
1
o ss o r n n
8 o Z
l
u
a
r
a
a
t
m
3
ar ta
n gr
ya r c o ss o r n v i ai e s f al r.
i
d
m i
:
n
o
i
g
n
e o
t n
9
n
a
40
t 3
r
S
nu
t
3
i
o = & yeu ec
r
e
i
F
)
a
a
ya r c
. fn a u x = a C y
11.10
. cn +
c
7
HYPOCYCLOID
1
, e
o i
C
O
= 6e
nc rn
ei
p
a i
e
bu a p ei x l
r
a &
Y
- C
r = 3f . n
o
a u (s + +
r
i d\ ,
,
\
)!
5d
Y
r
t (
C
11.5
t
s)
t’=3 4 n
\
s
G
a 4e
tA r
z
2
v i
ic f a i sd
d
e
tv er c
r nr F d
1
e
d
he d i
e c l
ti i e
i 1 u
l e
b g
-
u s
.
SPECIAL
PLANE
CURVES
41
CARDIOID
11 .12
Equation:
11 .13
Area bounded by curve
11 .14
Arc length of curve
r = a(1 + COS0) = $XL~
= 8a
This is the curve described by a point P of a circle of radius a as it rolls on the outside of a fixed circle of radius a. The curve is also a special case of the limacon of Pascal [sec 11.321. Fig. 11-4
CATEIVARY
11.15
Equation:
Y z : (&/a + e-x/a)
= a coshs
This is the eurve in which a heavy uniform cham would hang if suspended vertically from fixed points A anda. B.
Fig. 11-5
THREEdEAVED
11.16
Equation:
ROSE \
r = a COS39
The equation T = a sin 3e is a similar curve obtained by rotating the curve of Fig. 11-6 counterclockwise through 30’ or ~-16 radians. In general n is odd.
v = a cas ne
or
r = a sinne
‘Y
\ \ \ \ \ , /
has n leaves if
/ +
,/
, Fig. 11-6
FOUR-LEAVED
11.17
Equation:
ROSE
r = a COS20
The equation r = a sin 26 is a similar curve obtained by rotating the curve of Fig. 11-7 counterclockwise through 45O or 7714radians. In general n is even.
y = a COSne
or
r = a sin ne has 2n leaves if
Fig. 11-7
a
X
42
SPECIAL
11.18
PLANE
CURVES
Parametric equations: X
=
(a + b) COSe -
b COS
Y
=
(a + b) sine -
b sin
This is the curve described by a point P on a circle of radius b as it rolls on the outside of a circle of radius a. The cardioid
[Fig. 11-41 is a special case of an epicycloid.
Fig. 11-8
GENERA&
11.19
HYPOCYCLOID
Parametric equations: z
=
(a - b) COS@ + b COS
Il
=
(a-
b) sin + -
b sin
This is the curve described by a point P on a circle of radius b as it rolls on the inside of a circle of radius a. If
b = a/4,
the curve is that of Fig. 11-3. Fig. 11-9
TROCHU#D
11.20
Parametric equations:
x =
a@ - 1 sin 4
v = a-bcos+
This is the curve described by a point P at distance b from the tenter of a circle of radius a as the circle rolls on the z axis. If
1 < a, the curve is as shown in Fig. 11-10 and is called a cz&ate c~cZOS.
If b > a, the curve is as shown in Fig. ll-ll If
and is called a proZate c&oti.
1 = a, the curve is the cycloid of Fig. 11-2.
Fig. 11-10
Fig. ll-ll
SPECIAL
PLANE
CURVES
43
TRACTRIX
11.21
PQ
x
Parametric equations:
u(ln cet +$ - COS#)
=
y = asin+
This is the curve described by endpoint P of a taut string of length a as the other end Q is moved along the x
axis.
Fig. 11-12
WITCH
11.22
Equation in rectangular
11.23
Parametric equations:
coordinates:
OF AGNES1
u =
8~x3
x2 + 4a2
x = 2a cet e y = a(1 - cos2e)
Andy
-q-+Jqx
In Fig. 11-13 the variable line OA intersects and the circle of radius a with center (0,~) at A respectively. Any point P on the “witch” is located oy constructing lines parallel to the x and y axes through B and A respectively and determining the point P of intersection.
y = 2a
FOLIUM 11.24
OF DESCARTRS Y
3axy
\
Parametric equations:
1
x=m
y =
11.26
Area of loop = $a2
11.27
Equation of asymptote:
3at
1
3at2 l+@ \
x+y+u
Z
Fig. 11-14
0
INVOLUTE il.28
Fig. 11-13
Equation in rectangular coordinates: x3 + y3 =
11.25
l
OF A CIRCLE
Parametric equations: x = ~(COS+ + @ sin $J)
I y = a(sin + - + cas +) This is the curve described by the endpoint P of a string as it unwinds from a circle of radius a while held taut. jY!/--+$$x . Fig. Il-15
I
44
S
11.29
E
i
r
q
(axy’3
+
P
11.30
e
x
(bvp3
1b = i t he u = 1e s z d
e
o
u tu3
-
1
P
of
6a
so h t i n r /i h lF 1a
+ qa4 .
a
s
z i
2
G 2
T I
i t 2 a
c
c d
hd s h i a c a p
i a ih F u 1 s t
b = u
cf
-
u b a p ie ib s o
t
u
A
C
a
m
r
R
N
I
V
E
A
a
i
d
t
e
n
o
i
g
2 u 3=
n
o he 2 s wg
lre
yr t sos t a 2s n
s[
a r
e1e
e lm l n. 1e F 1
F
o
i
rS
W
i oa 6d 1
p pi g
L t
i
m 1
v
i
r.
-k
o
1 7 eo
e
g
-
S
p u v h o i cih d r c e a f nrt e t i o hf trp t is ws d i .r t s t a t ] n a c
i g va1 - b
s- a1 rc
r
t
s
e -h
A
aO e a 4
ba
Pe s
i
)
OF CASSINI V
so nFe i r1 1
i , a Zh
U
t
)
tf the s eov v a nb o i 1s
l~i
2
o
_--\ ++Y !---
T [
e 1
a
u C
O 1
E
by3
COS3 z 8
- b ys
L
ELLIPSE
c
r
- b
(
C
OF Aff
q = (
P
EVOWTE
=
a c
T c + y
cn
P
8
1c
oo b
n
sr
.
1a
P
X
a
F
1
i
1
g
LIMACON 11.32 t
P
L c T
c
O b i t c i a c
e
o r = qb
l
-
.
1
i
1
g
-
u+
a
aa
r
tc
y i gai p f a n s
s os nFe i 1r 1 a r i g a1v -b >c a og .b s< -e a1 r c r F r 1 v id e ig 4 o
ii h r. .
t
io
os
r in a0 t t
aTn h c nt s. s
2 I9 e o1 = a 1 i
t
0 f sr , . d
-
F
1
i
1
.
OF PASCAL
a l ej Q eo i 0to t a rp n Q ioo an c io eo dnn h l u o a s ph oe P rs f 1t oe Pc = vub 1h i Q u . ec i a ih F u 1 u [ s a1
17
F
g
-
.
1
F
1
9
i 1
g-
m hg r h
SPECIAL
PLANE
C
11.33
Equation
in rectangular y
11.34
Parametric
OF
CURVES
Ll
IS
x
2a -
2
3
x
equations:
i
=
2a sinz t
?4 =-
2a sin3 e COSe
This is the curve described by a point P such that the distance OP = distance RS. It is used in the problem of duplicution of a cube, i.e. finding the side of a cube which has twice the volume of a given cube.
SPfRAL
Polar
BS
coordinates: ZZZ
x
11.35
45
equation:
Y =
a6
Fig. 11-21
OF ARCHIMEDES
Y
Fig. 11-22
OO
C
FORMULAS APJALYTK
12
SCXJD GEOMETRY from
Fig. 12-1
RlRECTlON
12.2
COSINES OF LINE ,lOfNlNG
1 =
COS L
=
% - Xl
~
d
’
m
=
where a, ,8, y are the angles which line PlP2 d is given by 12.1 [sec Fig. 12-lj.
FO!NTS &(zI,~z,zI)
COS~
=
Y2 d,
Y1
n
=
AND &(ccz,gz,rzz)
c!o?, y
=
22 -
-
21
d
makes with the positive x, y, z axes respectively
and
RELATIONSHIP EETWEEN DIRECTION COSINES
12.3
or
cosza+ COS2 p + COS2 y = 1
lz + mz +
nz
=
1
DIRECTION NUMBERS
Numbers L,iVl, N which are proportional The relationship between them is given by
12.4
1 =
L dL2+Mz+
to the direction cosines 1,m, n are called direction
M
m= N2’
dL2+M2+Nz’
46
n=
N j/L2 + Ar2 i N2
numbws.
FORMULAS
OF LINE JOINING
EQUATIONS
12.5 These
FROM
x-
x,
% -
Xl
are also valid
Y-
~~~~ Y2 -
Y1
z -
Y1
752 -
ANGLE
are
also valid
+ BETWEEN
if 1, m, n are replaced
TWO
LINES WITH
12.7
12.8
x -
OF PLANE
AND
y
=
Y-
12.9
xz -
Xl
x3 -
Xl
2 -
Y1
m
FORM
Zl
n
IN PARAMETRIC
1 =
FORM
.zl + nt
by L, M, N respectively.
DIRECTION
mlm2
THROUGH
X
x -
Y =p=p
I’&z,y~,zz)
y1 + mt,
EQUATION
PASSING
Xl 1
COSINES
L,~I,YZI
AND
h
,
+ nln2
OF A PLANE
.4x + By + Cz + D
EQUATION
IN STANDARD
~&z,yz,zz)
or
47
by L, M, N respeetively.
COS $ = 1112 +
GENERAL
GEOMETRY
21
I’I(xI,~,,zI)
x = xI + lt, These
AND
.z,
if Z, m, n are replaced
12.6
ANALYTIC
~I(CXI,~I,ZI)
OF LINE JOINING
EQUATIONS
SOLID
=
[A, B, C, D are constants]
0
POINTS
Y1l
2 -
.zl
Y2 -
Y1
22 -
21
Y3 -
Y1
23 -
Zl
(XI, 31, ZI), (a,yz,zz),
=
(zs,ys, 2s)
cl
or
12.10
Y2 -
Y1
c! -
21
Y3 -
Y1
z3 -
21
~x _ glu
+
EQUATION
z+;+;
12.11 where a, b,c respectively.
are
the
z2 -
Zl
% -
Xl
23 -
21
x3 -
Xl
OF PLANE
z intercepts
~Y _
yl~
+
IN INTERCEPT
xz -
Xl
Y2 -
Y1
x3
Xl
Y3 -
Y1
-
(z-q)
FORM
1 on
the
x, y, z
axes
Fig. 12-2
=
0
48
FOkMULAS
FROM
E A
z
t
N
YB
X”
A
N t A B C
OQ
P
x -
-
Yn
P z
-
F I
2
P
T
T
”
R( S y
O
2 w
.
t
s
i hc
x
=
N
I
(
x,, + At,
r oe
T
O
+ B
F
A N
x
R
T E
+ C q+ D 3 d
EO
= yo + Bf, z =
y
R
y
O
I
N
.z(j
,
oees s
N
M
R
T
,
r e ir n
,NC ~
,
, B
n e t
FUM L
U
E
+ t
,
ct
+
A
OQ R P
O
nA b o +rhB c +l C + eDx =e p 0ey at a z
z
nas
o
E
I AZO + eM By L A ,+ Cz N + L ~ =A N0,
s teh d e Ogh i nhro i
F
H
ti mt et e pr
O xP T ,
1 k
h S it
GEOMETRY
R Ax O+ By L + C.z P + L =A 0
a al u ep
A 1
FU
PD
or
C
i ft
ANALYTIC
L
E
d o h n h , , .
D
SOLID
n na
A
A A
e
L
T N
1
1
2 x cas L + y COS,8 . i- z COSy w P a a
p = p C/ y a
h an x
de Xb3
=1
ef On d a e
r
4
p
0 i tr p r r a ,e,p e P xg y nz s
s op l to
eo t ,l , d .
t e a ws
m e
a n n ei
n d e et
s
Fig. 12-3
T
R
22
1
=
2y = z
w ( t t r t t x o t n s
=
x’ +
O
A
F
x’
x()
y’ + yo d
C
c
x -
y’ ZZZ 1Y -r
. o +
O IN
x
ON PS
(
T
RV UF
R
DO RO
A
l
J
5
Y0
z
(
a h o c% r e [l oc , e rird oy i (o y z y a v n a c z’ ’ s r e e[ ? o , ) t e s i oa ’ ( y vz a n yt q c s0 ee r d ’ h , o t, o f 0h r e r t t ’ e o e wq i c o h l l z go y s t
s
‘
J
e.e ro, rw , o ) e ze e da e
e o e io
el e
m
X
Fig. 12-4
d r~ r ’ t
m r m nr
.a l
d i) d i
.
] d ] d
FORMULAS
FROM
TRANSFORMATION
x
=
1
2
=
n
+ &
+
ANALYTIC
OF COORDINATES
+ 1
n
l+
y
1
y = WQX’+ wtzyf+
12.16
SOLID
3
r
INVOLVING
!
x
n
n
2
x
y
'
z
PURE ROTATION
*
% 1
p
3
49
GEOMETRY
? '
\
’
%
\
'
\
O i
=
Z
+' m
+I
y'
l=
1
+
x
=
z
+' m
m ?
T
1
X
z
y
l
+2
n
2
x
p
y
.
+z
?
a
x
%
y
g
\
Z
\ \
z
where the origins of the Xyz and x’y’z’ systems are the same and li, ' n 1 mm nl 1 2 m 2 l n 2 ; are 3, 3 the , , sdirection ; , , cosines of the x’, ,y’, z’ axes relative to the x, y, .z axes respectively.
3
1
, ,
,
\ X
’
\
, Y
, ?/‘ ’ ’ ~
’
Y
,,/ X Fig. 12-5
TRANSFORMATION
z
12.17
OF COORDINATES
Z
=
+ &
+ l&
+ I x.
INVOLVING
y
X
TRANSLATION
’
’
y = miX’ + mzy’ + ma%’ + yo 2
or i
=
n
+
n
l+
2+
zX
3
y
.'
y!
zz &z(X- Xo) + mz(y - yo) + n&
- 4
x’
=
- zO)
d-
y
x I+
F’
\
\
=
+t m
z
ROTATION
'
X
4
-' X
n
AND
n
-d z t
&(X - X0) + ms(y - Y& + 42
z
'
l
d y
COORDINATES
/
/
‘X’
(r, 0,~)
A point P cari be located by cylindrical coordinates (r, 6, z.) [sec Fig. 12-71 as well as rectangular coordinates (x, y, z). The
transformation x
12.18
=
between
these
coordinates
is
r COS0
y = r sin t
or
0 =
tan-i
r
(y/X)
z=z
Fig. 12-7
-
Y
1 '
$
l
Fig. 12-6
CYLINDRICAL
/
o
/
where the origin 0’ of the x’y’z’ system has coordinates (xo, y,,, zo) relative to the Xyz system and Zi,mi,rri; cosines of the la, mz, ‘nz; &,ms, ne are the direction X’, y’, z’ axes relative to the x, y, 4 axes respectively.
y
,
\ b
,
,
l
'
"
FORMULAS
50
FROM
SPHERICAL
[sec
SOLID
ANALYTIC
COORDINATES
GEOMETRY
(T, @,,#I)
A point P cari be located by spherical coordinates (y, e, #) Fig. 12-81 as well as rectangular coordinates (x,y,z). The
transformation
12.19
between
those
=
x sin .9 cas .$J
=
r sin 6 sin i$
=
r COSe
coordinates
is
x2 + y2 + 22 or
$I =
tan-l
(y/x)
e =
cosl(ddx2+y~+~~) Fig. 12-8
EQUATION
12.20
OF
SPHERE
(x - x~)~ + (y - y# where
the sphere
has tenter
IN
RECTANGULAR
+ (,z - zo)2 =
COORDINATES
R2
(x,,, yO, zO) and radius
R.
Fig. 12-9
EQUATION
12.21
OF
SPHERE
CYLINDRICAL
COORDINATES
rT - 2x0r COS(e - 8”) + x; + (z - zO)e where
the sphere
If the tenter
has tenter
(yo, tio, z,,) in cylindrical
is at the origin
the equation
12.22
7.2+ 9
EQUATION
12.23
OF
SPHERE
rz + rt where
the sphere
If the tenter
12.24
IN
has tenter
IN
and radius
= Re
SPHERICAL
COORDINATES
2ror sin 6 sin o,, COS(# - #,,)
the equation r=R
R’2
is
(r,,, 8,,, +0) in spherical
is at the origin
coordinates
=
is
coordinates
=
Rz
and radius
R.
R.
FORMULAS
E
FROM
OQ E
SOLID
ANALYTIC
C
tA (L
FW U L
51
GEOMETRY
E
A TTx I S
N
N HI ~P a Eb
T
D O, ,S M,
E
N y O dI
Fig. 12-10
E
1
C
2 w I
L
W Y
.
a I a
sh
b = a
i b
A I xL A X I
2
, f
A L
o re ee ac
c
ST
I X PI
H N I S T
D S I
6 fs e l
t e c
mr
r
e l
io r c y u
rf
ie
o
c i
a o l .
c
-
s
t p
d mi
u
a i
s
i t
en
l
x u
sd
Fig. 12-11
E
1
2
C
.
L
W
AO
2
L A I z A XN
J ST
X IE
P
H
I S
T
S
7
Fig. 12-12
H
1
2
$
.
Y O
z+
1
2
$
O
S
P F
8
_
N
H
E
E
$
Fig. 12-13
E
R
E
B
I
5
2
FORMULAS
FROM
SOLID
H
Note
orientation
of axes
ANALYTIC
YO
in Fig.
T
GEOMETRY
S
IF
W
H
’
O
E
E
E
12-14.
Fig. 12-14
E
1
2
P
.
L
3
A
L
R
I
A
P
0
Fig. 12-15
H
1
2 Note
xz --a2
orientation
y2 b2 of
axes
= .
PY
_z
3
AP
RE
AR
1
C
in Fig.
12-16.
/
-
Fig. 12-16
X
D
If y = f(z),
OE
A D
FF
E
t
R
N
t
~
lim f(X+ ‘) - f(X)
=
d
+h
hX
=
G
R
a
O
where h = AZ. The derivative is also denoted by y’, dfldx called di#e~eAiatiotz.
E
O
D
f
+ A
or f(x).
l
- fi
(
~
(r
~
)
~
F E
A
Ax
Ax-.O
The process of taking a derivative
N
F
t
t
E
F k
R
is
In the following, U, v, w are functions of x; a, b, c, n are constants [restricted if indicated]; e = 2.71828. . . is the natural base of logarithms; In IL is the natural logarithm of u [i.e. the logarithm to the base e] where it is assumed that u > 0 and a11 angles are in radians. 1
g(e) =3
1
&x)
0
=
3
c
.
2
.
3
1
3
.
4
1
3
.
5
c u
1
&
3
1
&
3
1
$-(uvw) 3 =
1 1 1 1
= =
du dx -H
v
3
du
_
ijii
-
du -=-
2
dv
3
du
du dx
1
dyidu
3
-
dxfdu
z n
c
6
.
u
7
dv + dx
vw-
u(dv/dx)
V
&
.
uw.
+
v(duldx)
dxfdu
=
v
-
3
dx
dy z
uv-
_
3z &
-
V
the derivative of y or f(x) with respect to z is defined as
13.1
1
l
$
-(Chai?
.
.
Z
du dx
gu gv g
8 9 1
0
. rule)
1
1
.
1
2
.
1
3
j
5
3
) )
+
E S
54
DERIVATIVES
AL”>. 1 _. .i
”
.,
13.14
-sinu
d dx
=
du cos YG
13.17
&cotu
=
-csck&
13.15
$cosu
=
-sinu$
13.18
&swu
=
secu tanus
13.16
&tanu
=
sec2u$
13.19
-&cscu
=
-cscucotug
13.20
-& sin-1
13.21
&OS-~,
13.22
u
-%<
=$=$ =
&tan-lu
-1du qciz
=
13.23
&cot-‘u
=
13.24
&sec-‘u
=
&
csc-124
[O < cos-lu
dx
< i
C
+&
[O < cot-1 u < Tr] 1
du zi
-I
1
< z-1
LJ!!+ 1 + u2 dx
ju/&zi 13.25
sin-‘u
< tan-lu
1
< t
if 0 < set-lu
d -log,u dx
-
=
13.27
&lnu
13.28
$a~
=
13.29
feu
=
~l’Xae u
=
=
-du
if 0 < csc-l
=
I
u < 42
< csc-1 u < 0
1
ig
aulna;<
TG
d"
fPlnu-&[v
13.31
gsinhu
=
eoshu::
13.32
&oshu
=
13.33
$
=
tanh u
< r
a#O,l
dx
-&log,u
< 7712
if 7712 < see-lu
=
+ if --r/2
13.26
du
lnu]
=
vuv-l~
du
+ uv lnu-
dv dx
13.34
2
cothu
=
- cschzu ;j
sinh u dx
13.35
f
sech u
=
- sech u tanh u 5
sech2 u 2
13.36
=
- csch u coth u 5
du
A!- cschu
dx
dx
dx
DERIVATIVES
13.37
d - sinh-1 dx
13.38
-dx cash-lu
13.39
-tanh-1
13.40
-coth-lu d
u
d
d dx
u
dx
13.41
=
~
=
~
=
--
=
-- 1
+ if cash-1 u > 0, u > 1 if cash-1 u < 0, u > 1
-
du
1
[-1
1 - u2 dx
1 _
-&sech-lu
55
du dx
u2
71
=
- d csch-‘u
if sech-1 u > 0, 0 < u < 1 + if sech-lu
[ -
du
-1
=
du
[-
dx
HIGHER The second, third
and higher
derivatives
13.43
Second derivative
=
d dy ZTz 0
13.44
Third
=
&
13.45
nth derivative
derivative
13.46
DERtVATlVES
are defined =a
if u > 0, + if u < 0]
d’y
=
as follows. f”(x)
=
y”
f’“‘(x)
LEIBNIPI’S Let Dp stand
< u < 11
[u > 1 or u < -11
u-z 13.42
RULE FOR H26HER
for the operator
&
uD%
+
D+.w)
=
so that
II
y(n)
DERIVATIVES
OF PRODUCTS
= :$!& = the pth derivative
D*u
;
(D%)(D”-2~)
of u. +
...
0
where
0n
1 ’
As special
0n
2
‘...
coefficients
are the binomial
[page
31.
cases we have
13.47 13.48
DlFFERENT1ALS Let
y = f(x)
and
Ay = f(x i- Ax) - f(x).
13.49
AY x2=
where
e -+ 0 as Ax + 0. Thus
13.50 If we call
13.51
the differential
Then
f(x + Ax) - f(x) Ax
AY Ax = dx
1
=
=
f/(x)
+ e
f’(x) Ax -t rzAx
of x, then we define the differential
dv
=
=
j’(x) dx
of y to be
Then + wDnu
1
DERIVATIVES
56
RULES
The rules
for differentials
are exactly
FOR DlFFERENf4ALS
analogous
13.52
d(u 2 v * w -c . ..)
13.53
d(uv)
13.54
d2 0
=
udv
=
-
d(sinu)
=
cos u du
13.57
d(cosu)
=
- sinu
=
du
PARTIAL
x and y. Then
af
lim
az=
derivative
2
13.59
with
derivatives
of higher
order
respect
of f(x,y)
is defined
dx =Ax
Extension
and
x constant,
a2f
of more
than
to be
a df ax 0 ay 9
a2f -=ayiG ayax
and its partial
a af 7~ 0 ay a
af 0
derivatives
as =
$dx
+ $dy
dy = Ay.
to functions
is defined
as follows.
a1/2=
df
where
to y, keeping
a af TGFG' 0
The results in 13.61 will be equal if the function case the order of differentiation makes no difference. The differential
of f(z, y)
AY
can be defined
a2f -=--axay
13.61
derivative
Ax
AY'O
@f -= a22
13.60
the partial
lim fb, Y + AY) - fb, Y)
-
dY
Partial
we define
I
fb + Ax, Y) - f&y)
Ax-.0
of f(x,y)
_^.1 :“” _
i”
DERf,VATIVES
Let f(x, y) be a function of the two variables respect to x, keeping y constant, to be
the partial
du?dvkdwe...
nun- 1 du
13.56
Similarly
that
udv
d(e)
13.58
we observe
212
V
13.55
with
As examples
+ vdu
vdu
=
to those for derivatives.
two variables
are exactly
analogous.
are continuous,
i.e. in such
If
2
= f(z),
or the indefinite
then
y is the function of f(z),
integral
Since the derivative
derivative
is f(z)
and is called
of a definite
integral,
see page
94.
The
process
of finding
In the following, u, v, w are functions of x; a, b, p, q, n any constants, e = 2.71828. . . is the natural base of logarithms; In u denotes the natural logarithm that u > 0 [in general, to extend formulas to cases where u < 0 as well, replace are in radians; all constants of integration are omitted but implied. 14.1
14.2
14.3
14.4
S
S
S
S
adz
=
uf(x)
dx
ax
=
a
(ukz)kwk udv
14.6
14.7
14.8
Sf(m) S
=
WV -
dx
F{fWl
dx
undu
=
14.10
=
_(‘udx
vdu
S
integration
”
svdx
*
[Integration by parts,
.(‘wdx
*
by parts]
see 14.48.
aSf(u) du
-
=
F(u)2
S
du
=
F(u) f’(z)
S
du
where
u =
.&a+1
S
du -= S
S
s
n-t
n#-1
1’
In u
U
...
1
=
= 14.9
f(x) dx
. ..)dx
For generalized 14.5
S
if
[For n = -1, see 14.81
u > 0 or In (-u)
if
u < 0
In ]u]
eu du
=
eu
audu
=
S
@Ina&
the anti-derivative
denoted by if y = f (4 dx. Similarly f (4 du, then s S is zero, all indefinite integrals differ by an arbitrary constant.
of a constant
For the definition integration.
whose
=
eUl”Ll
-=-
In a
au
In a ’ 57
a>O,
a#1
f(z)
an integral
of f(s) $
=
f(u).
is called
restricted if indicated; of u where it is assumed In u by In ]u]]; all angles
INDEFINITE
58
du
=
- cos u
cosu du
=
sin u
tanu
du
=
In secu
14.14
cot u du
=
In sinu
14.15
see u du
=
In (set u + tan u)
=
In tan
csc u du
=
ln(cscu-
=
In tan;
=
#u
-
=
j&u + sin u cos u)
14.11
14.13
14.16 14.17 14.18
14.19 14.20 14.21 14.22 14.23
14.24 14.25 14.26 14.27 14.28 14.29 14.30
sinu
INTEGRALS
I‘
I‘
I‘ .I'
tanu
=
-cotu
tanzudu
=
tanu
cot2udu
=
-cotu
sin2udu
=
- 2
=
;+T
du
=
secu
=
-cscu
S S S s
' co532u du
S
secutanu
s
cscucotudu
S I‘ I‘ J
U
-
sin 2u
du
=
coshu
coshu
du
=
sinh u
tanhu
du
=
In coshu
coth u du
=
In sinh u
sechu du
=
sin-1
csch u du
=
In tanh;
(tanh u)
J
sechzudu
=
tanhu
14.32
I‘
csch2 u du
=
- coth u
tanh2u
=
u -
s
du
u
sin 2u 4
14.31
14.33
cosu
u
sinhu
S S
-In
cotu)
=
sec2 u du * csc2udu
I
=
tanhu
or
or
sin u cos u)
2 tan-l
- coth-1
eU
eU
INDEFINITE
14.34 14.35 14.36 14.37 14.38
S S S S s
sinheudu
=
sinh 2u --4
coshs u du
=
sinh 2u ___ i- t 4
59
cothu u 2
- sech u
csch u coth u du
=
- csch u
=
+(sinh
=
Q(sinh u cash u + U)
u cash u - U)
du ___ = u’ + CL2
14.42
s
14.43
u -
=
14.41
14.40
=
sech u tanh u du
S S S
14.39
cothe u du
INTEGRALS
u2 =
-
du ___ @T7
s
>a2
u2 < a2
=
ln(u+&Zi?)
01‘
sinh-1
t
14.44 14.45
14.46 14.47 14.48
S
f(n)g dx
This
=
is called
f(n-l,g
-
generalized
f(n-2)gJ
+
integration
f(n--3)gfr
-
. . .
(-1)”
s
by parts.
Often in practice an integral can be simplified by using an appropriate and formula 14.6, page 57. The following list gives some transformations 14.49 14.50 14.51 14.52 14.53
S S S S S
F(ax+
b)dx
F(ds)dx
F(qs)
1 a
= = dx
=
F(d=)dx
=
F(dm)dx
=
S S S S S
fgcn) dx
F(u) du
transformation and their effects.
where
u = ax + b
i
u F(u) du
where
u = da
f
u-1
where
u = qs
F(u) du
a
F(a cos u) cos u du
where
x = a sin u
a
F(a set u) sec2 u du
where
x = atanu
or substitution
INDEFINITE
14.54 14.55 14.56 14.57
F(d=)
I‘
F(eax)
dx
F(lnx)
s
=
dz
a
F(u)
s =
apply
x, cosx)
tan u) set u tan u du
F(a
where
x = a set u
where
u = In 5
where
u = sin-i:
s
dx
results
F(sin
s
$
=
F (sin-l:)
Similar 14.58
=
s
I‘
s
dx
INTEGRALS
dx
e” du
oJ
F(u)
for other =
cosu
inverse
du
trigonometric
2
functions. -
du
1 + u?
where
u = tan:
Pages 60 through 93 provide a table of integrals classified under special types. The remarks page 5’7 apply here as well. It is assumed in all cases that division by zero is excluded.
14.59
dx s
‘, In (ax + a)
as= xdx ax + b
14.60
X
=
-
a
dx
x3
14.62
S i&T-%$-
14.63
S z(az
14.64
S x2(ax
14.65
I‘
14.66
S ~(ax
14.67
S ~(ax
14.68
Sm
b
- ;E- In (ax + 5)
(ax + b)2 --ix---
2b(az3+
(ax + b)s ---m----
3b(ax + b)2 + 3b2(ax + b) _ b3 2 In (ax + b) 2a4 a4
b, + $ In (ax + b)
dx
= dx + b)
=
b)
=
dx x3(ax+
dx
-1
+ b)2
=
a(ux + b)
x dx + b)2
=
a2(af+
=
ax + b --- a3
=
(ax + b)2 _ 2a4
x2 dx x3
dx
14.69
~ S (ax
+ b)2
14.70
S x(ax
dx + b)2
14.71
S
xqax
dX
+ by
b)+ $ In(ax+ b) a3(ax
b2 + b)
3b(ax + b) + a4
$
In (ax + b) bs + z aJ(ax + b)
In (ax + b)
given
on
INDEFINITE
14.72 14.73 14.74 14.75
14.76
dx
s
x3(az+
s
dx ~(ax + b)3
s
14.79 14.80 14.81
S S S S
14.82
2b a3(az+
=
b)2
+ +3 In (as + b)
b3 2u4(ax+
6x2 =
2b2(u;a+
=
2b5(ux + b)2 -
b)ndx
n = -1, -2,
b)2 -
b3(ux + b)
a4x2
4u3x b5(ux + b) -
(ax + b)n+l
=
(n+l)a =
If
*
(ax + b)n+2 ~-
+ b)n dx
(n
(ax + b)n+3 + 3)a3
=
_
'
2b(ux +. b)n+2 (n+ 2)u3
nZ-1*--2
t b)” dx
+
b2(ux + b)n+’ (nfl)u3
see 14.61, 14.68, 14.75. + b)n
=
xm(ux
(m +
nb
+
m+n+l x”‘(ux
see 14.59.
b(ux + b)n+l (n+l)u2
-
(n + 2)u2
n = -1,
see 14.60, 14.67.
n = -l,-2,-3,
S
In (ax + b)
b3(ax + b)
xm+l(ax
14.83
- 2
by
2ux
2b3(ux + b)2 -
=
x(ux + b)ndx
If
b2 2a3(ax+
b) -
3b2 u4(ux + 6) +
5-
dx
x3(ux + bJ3
SX~(UX
b)
2u
+ bJ3
(ax+
b4(:3c+
b
dx
x2@
If
b4x
a2(as + b) + 2a2(ax + b)2
=
dx + bJ3
x(ax
3
3a(az + b) _
-1
=
x3 dx ~(ax + b)3
61
-1 2(as+ b)2
=
x dx (ax + b)3 dx ~ (ax + b)a
(ax + b)2 + -2b4X2
=
~
S x2 S
14.77 14.78
b)2
INTEGRALS
S
xm(ux
mfnfl
n
+ b)n+’
+ 1)~
-xm+l(ux+b)n+l
(n + 1)b
_
mb
(m + n + 1)~ .f +
xm--l(ux
(n S
m+n+2
+ 1)b
+ b)n-1
xm(ux
dx
+ b)“dx + b)“+’
dx
62
INDEFINITE
14.89 14.90
14.93
=
xd-6
s
14.91
14.92
dzbdx
s
dx
x%/G
s
“7 =
dx
2(3a;z;
14.96
14.98
dx
=
&dx
2d&3
X
&T”
s
X+GT3
=
dx
=
t2;$8,,
.(‘
Xm
x(ax
-
=
-(ax + b)3/2 (m - l)bxm-’
+ b)““z
(as +xbP”2
(ax + b)m’z
x(ux
-
x-l:= X~-QL-TTdX
c2;“+b3,a
s
=
dcv
dx
X2
dx + b)m/2
1) s
x--l:LTT >T gm--1
_ (2m - 5)a (2m - 2)b s
2b(ux -
2(ax + b)(m+s)lz u3(m -I- 6) ~(CLX + b)““z m
=
=
2(mf
2(ax + b)(“‘+Q/z a2(mf4)
=
=
dx
+
dx
-
4b(ux
+ b)(m+4)/2 4)
+
2b2(ax
a3(m+
(ax + b)(m-2)/2
+ b s
(ax + b)(m+2)‘2 bx
(m - Z)b(ax
_
+ b)(m+z)/z aym + 2) + b)(“‘+2)‘2 2)
u3(m+
dx
X
(ax + b)m’2
+z
2 + b)(m-2)/2
’
INVOLVfNC
S S 1 5
X
x(ax
dx
dx + b)(“‘--2)/z
c&z + b AND
p;z! + q
>:“:
dx
14.105
14’109
b)3’2
dx
2(ax + b)(“‘+z)lz a(m + 2)
INTEGRALS
14.108
+
(m-l)xm-’
=
z2(ax + b)m’2 dx
S S
_ (2m - 3)a (2m - 2)b s (as
=
c (ax + b)m’2 dx
s
dXGb
&&x5 dx
Xm
l/zT-ii -----dx
s
[See 14.871
X&iZT
2mb (2m + 1)a s
-
\/azfb
-
s
14.107
s
(m - l)bxm-1
xmd=
s
(2m + 1)~
= dx
14.102
14.106
dx
+;
2LlFqz s
[See 14.871
x&zz
&zTT
=
s
14.104
dx
+ b
x2
14.100
14.103
+ 8b2) ,,m3
s
s
14.99
14.101
;$a;bx
‘&zT J
&iTx
14.97
2b’ l&a@
2(15a’x2
=
14.94
14.95
INTEGRALS
(ax + b)(w x dx . (‘ (ax + b)(px
S S j-
+ d + d
dx (ax + b)2bx
+ d
xdx (ax + b)2(px
+ 4
x2 ds (ax + b)z(px
+ q)
=
=
&
g In (ax+
(bp - aq;&ux+
b) -
b) +
(b-
% In (px+
’ ad2
q)
b(bp ,Z 2uq) In (uz + b)
INDEFINITE
14.110
dx (ax + bpqpx
I’
+ qp
-1 (Yz - l)(bp
=
INTEGRALS
63
1 - aq)
(ax + b)+l(pz
1
+ q)“-’
+ a(m+n-2) ax + b -dds s PX + Q
14.111
=
7
dx (ax + bpqpx
s
-1 (N
-
l)(bp
-
(ax (px
uq)
+ +
bp+’ q)“-l
+
(x-m
-
va
s
1 (ax
14.112
+
bp
(px+
s
q)n dx
-1
=
(m
I
14.113
S
14.114
s
-E!C&.Y d&zT
dx
=
+ q)n-1
+ yh(px+q)
-
m
(ax + bp -
(n--:)p
l)p
i
{
(px
+
q)n-l
(ax + ap (pxtqy-1
+
-
m@p
-
aq)
s
,E++q;!Tl
dx >
(ax + b)m- 1 (px+ 4”
dx
>
(ax + by-1 \ (px + qy- l dx1
S
mu
2(apx+3aq-2bp)Gb 3u2
dx
(Px + 9) &ii-G
14.115
14.116
14.117
Jgdx
=
(px + q)” dn~
s
=
S S
=
=
(n - l)(aq
=
S,
+ q)n-l
+ 2n(aq - W (2n + 1)a
(2n + 1)u -&m (n -
+
(Px + q)” dn
2(n ‘“^I),;)”
+ qy-
l + 2(n ” 1)p s
INVOLVING
ds
bp) s
* (px + q)“s
l dx
&ii%
1
l)p(pz
INTEBRAES
14.120
- bp)(px
2(px + q)n &iTT
da
14.119
b - aq (2n + 3)P s
daxi-b
-bx + dn dx
&zTiT Smdx
I
(2n + 3)P
dx
(px + 9)” &z-i
14.118
2(px + q)n+ l d&T?
dx
dx (px + qp-’
AND
~GzT
J/K
&ln(dGFG+~)
dx
ZI
(ax + b)(w + q') i
14.121
xdx (ax + b)(px
= + q)
dbx
+ b)(px UP
+ 4
b + w --x&T-
dx (ax + b)(w
+ q)
dx
dx (px + q)n-1
&-TT
INDEFINITE
64
INTEGRALS dx
14.122
(ax + b)(px + q) dx
.
14.123
.('
j/sdx
=
=
(ax + b)(px + 4
‘@‘+
y(px+q)
+ vj-
(ax+;(px+q)
2&izi 14.124 (aq - W d%=i
lNTEGRALS
14.125
s--$$
=
$I-'~
14.126
J-$$$
=
+ In (x2 + a2)
14.127
J$$
=
x -
14.128
s&
=
$
‘4-l
J
x2(x?+
14.131
J
x3(x?+a2)
14.132
J
(x2d;Ga2)2
14.137 14.138
($2)
-
$ln(x2+az)
+3 tan-l:
2a2(xf+
S
14.140
S
(~2
14.141
S
dx x(x2 + a2)”
.
(x2+ a2)"
-~
x -- 2:5 tan-l: 2a4(x2 + a2)
1 2a4x2
1 2a4(x2 + u2) 2n - 3 + (2n- 2)a2
X
=
2(n - l)a2(x2 + a2)%-*
xdx
S
dx
(x2 + a2)n-1
-1 2(n - 1)(x2 + a2)n-1
+ a2)n=
dx
1 2(12 - l)a2(x2 + uy--1
=
xm dx dx S x9z2+a2)n
+ &3 tan-':
a2)
--- 1 a4x
dx S x2(x2 + c&2)2 = dx + a2)2 = S x3(x2 (x2d+za2)n
14.143
1 2a2x2
-=
14.139
14.142
six
=
=
x’ + a2
a tan-13c a
--30
INVOLVtNO
S
= =
xm--2 dx (x2
+
a2)n-l
-
a2
+ $
S
dx 1 2 S 33x2 + a2)n--1
S x(x2 + a2)n-1
x*--2 dx (x2 + a2)" --
1 a2 S
dx
xme2(x2 + a2)”
INDEFINITE
:INTEORAES I.
14.144
14.145
s
~ x2 - a2
xdx
14.147
s m--
14.150
14.151
14.156
14.157
14.158
s
x2(x2 - a2) =
s
x3(x2-a2)
s
(x2?a2)2
s
(x2 - a2)2
s
(x2--2)2
dx
dx
14.162
=
__ 1 2a2x2
=
2a2(sta2)
=
-1 2(x2-a2)
=
2(xFTa2)
=
2(x2 - a‘9
xdx
Lln
-
z
~~3
(
>
x2 dx
'
+
-a2
x3dx
(,Zya2)2
&ln
+ i In (x2 - a2)
dx
s
x(x2 - a2)2
s
x2(x2-a2)2
=
dx
S S
=
dx
-
---
1
xdx
=
dx
u2)n
-
--x
$5'"
2n - 3 s (2~2 - 2)a2
dx (x2
-
a2p-
1
-1 2(n - 1)(x2 - a2)n--1
S - = =S S --a?)" S S x(x2
+
2(n - 1)u2(x2 - a2)n-1
a2)n
(X2-a2)n
2a4(xi-a2)
2~~4x2
=
dx
(x2 -
--
=
x3(x2-a2)2
s
14.161
+ $ In (x2 - a2)
dx
x(x2 - a2) =
14.159
14.160
$
s
14.154
14.155
;
Jj In (x2 - ~22)
x3 dx
14.152
14.153
z2 > a2
x2 dx s n--
14.149
=
65
ix2 - a’,
1 - a coth-1
or
m=
14.146
14.148
INVOLVlNO
dx
*
INTEGRALS
-1 2(n - l)dyx2 - dy-1
x77-2 dx
xm dx
-
1 az
S
x(x2-
S - u2p-S a?
a2)n--1
dx
xm--2 dx
(x2
(x2-a2)n-1
dx Xm(X2qp=
1
dx
,z
xm-2(x2
+
a2
(x2-a2)n
1
xm(x2-
dx u2)n-l
INDEFINITE
66
tNVOLVlNO
IWTEGRALS
14.163
S
~ dx a2 - x2
14.164
S
__ a2 - x2
14.165
S
g-z-p-
14.166
Sm
14.167
S
x(a2 - 22)
14.168
S
22(d
=
= ---2
dx
S
-
S
$ In (a2 - x2)
=
22)
22 x3(,Ex2)
=
-&+
&lln (
dx
22>
-
5
2a2(a2 - x2)
=
2(a2--x2)
=
2(Lx2)
-
a2 2(&-x2)
+ i In (a2 - x2)
1
x dx
(a2 -
__ a2
=
x2)2
S
22 dx (&-x2)2
14.173
S
(CL2- x2)2 =
14.174
S
14.175
S
14.176
S
14.177
S
(a2 -dx x2)n
S
(a2 - x2)n =
14.179
S
x(a2
14.180
S
(,2-x2p
14.181
i tanh-I$
dx
14.172
14.178
xz
- f In (a2 - x2)
x2
x3 dx
(a2-x2)2
14.171
or
u~--~,
x2 dx
14.169J 14.170
=
x dx
INTEGRALS
x3 dx
=
-
5”’ dx
S
(a2
-
dx
=
dx
1
-
x‘p
(2n2n - 2)a2 3
+
l)a2(;2-x2)n-l
1 2(n - l)(a2 - x2)n-1
xdx
dx
qn-
2(n - l)a2(a2
a2
xm -2dx
S
j- xmc,~xp)n =
+2s
- x2)n-1
(a2 - x2p
S
+f
x(u2 - xy--1
x*-2dx
-
xm(a2?z2)n-~
s
(a2-
x2)n-l
+$f
x--$-x2)n
x2)n-l
INDEFINITE
14.182
In (x + &&?)
INTEGRALS
or
sinh-1s a
S
x dx ___ ~~
S
lfzT-2
S
~I2xz
14.183
14.184
II
x2 dx
-- a2 2 In(x+@Tz)
2 dx
x3
14.185
x 7 2 +a
=
(x2 + a2)3/2 3
=
-
a2&GZ
14.186 S
14.187 S
=
.2&F&i
s 14.189 14.190 14.191 14.192
14.193
14.194
=
dx
14.188
-
J/Xa22 ~~
+ k3
-2a2x2
x3~~5
S S s
a+&3T2
In
X
+ $l(x+~W)
xdmdx
(x2 + a2)3/2
=
x%jmdx
3
=
ad-g-q
dx
=
S
x(x2 + a2)312 4 (x2 + a2)5/2 _ 5
a2x&T2
&T S-dx S
a2(x2 + a2)3/2 3
= --&G-G
+ ln(z+drn)
&s-T-z
X
dx
14.196
S
(x2
14.197
s
(%2
14.198
.f
(x2
s
(x2
s
x(x2 +
S
x2(x2
dx +
x3(x2
dx + a2)3/2
14.199
14.200
14.201
14.202
a2)3/2
x dx
+
a2)3/2
x2 dx
+
x3
a2)3/2
dx
+ a2)3/2
= = &is = d& =
+ ln(x
@TTP
1 a2)3/2
a2)3/2
+ d&i7)
im+a2
dx
S
a-l-&372
- $a In
x3
+
a4
sln(x+j/~)
8
=&qgwalIn
s
X2
14.195
>
=
=
-
a+JZ2
In
2
(
~~ - ~ a4x
-1 =
f
a2&SiZ
2a2x2>
x -
a4&FS
-
3
2a4&FiZ
3
+ s5ln
a+&-TS 2
INDEFINITE
68
14.203 14.204 14.205 14.206 14.207
14.208 14.209
14.210
14.211 14.212 14.213
S S S S S S S
(x2
+
x(x2
a~)312
+
dx
dx
u2)3/2
x(x2
=
+
3&q/~
u2)3/2
4 (x2
=
+
+
~2)3/2
ds
=
x3(x2
+
u2)3/2
dx
=
u2)5/2
x(x2
+
u2)5/2
_
+
u2)3/2
dx
(22
=
+
(x2+
ds
=
u2)3’2
~2(~2
+
U2)3’2 x3
dx
=
(x2 +
-
-
--
u4x@TF2
~~ln(~+~2xq
16
~2)5/2
CL+@-TT?
+ u2@T2
-
x a2)3/2
a3 In
x
>
+ 3a2 ln (x + q-&-T&) 2 U-kdlXS
2x2
x
S
In (x + j/277),
S
u2)3/2
5
2
s ~ x2 dx &G=z
+
-
_ (x2 + u2)3’2 + 3x-
x2
(x2
+
24
~247’2
3
X
(x2 + UT’2
u2x(x2
6 7
(x2
+~a4ln(x+~2TTq
8
5
x2(x2
’
+
INTEGRALS
5 P--x-a
=
2
x3dx
s G= 1 5
asec-l
x2- u2
X
I
U
I
14.214
14.215
14.216 14.217 14.218 14.219
14.220
@=2 S
=
x3(&
s
dndx
S Sx2@73 S
=
xda~dx
,“d~
s-dx
+ k3 see-l xU I I
2u2x2 x =
dx
=
dx
=
7
x2-a
(x2
_
-$ln(x+dm) u2)3/2
3
x(x2
cAq/m~ - a2)3/2 +
4
cx2
-
~2)5/2
+
8 ~2(~2
-
5 =
dm-
~2)3/2
3 a see-l
I;1-
-- “8” ln(x
+ +2TS)
>
INDEFINITE
INTEGRALS
69
14.224
14.225
14.226
14.227
14.228 14.229
S 22 dx S S S S (~2 -
a2)3/2
=
x3 dx (22 - a2)3/2
=
-~
-1 a2@qp
=
dx
z2(s2
-
lJZ2
a2)3/2
=
-_
x3(x2
-
(~2 -
x a+iGZ
&)3/z
&
x(x2 - a2)3/2 4 -
z
x(52 - a2)3/2 dx
(x2
=
-
14.236
x2(99 -
a2)3/2
x3(52 -
a2)3/2
S
dx
2(x2 -
=
a2)5/2
14.238
dx
(22 -
=
a2)7/2
a2x(x2
+
-
14.240
az(x2
-
In (5 + &372)
a4x&FS 16
-
a2)5/2
@2 _ a2)3/2
S S S
X
(x2 _ a2)3/2
dx
=
tx2
-
a2)3'2
-
a2da
+
a3 set-'
c
3 dx
=
-
(x2
,jx
=
_
(x2;$33'2
I
-xa2)3'2
+
3xy
+
"y
_
ia
I
ln (1 + da)
X2
@2 -
a2)3/2
_
ga
sec-'
x3
Sda& =
lNVC)LVlNG
<%=??
sin-l:
xdx ____ = -dGi ___x2 dx ).lm x3 dx ____
S
sjlzz
[El a
x 7 a-x
=
-
=
(a2 - x2j312 _ 3
2
a2dpz3i
a+&KG X
14.243
:
Ia I
5
7
14.241
14.242
a2)3/2
24
@G?
14.239
see-l
+ :a4
8
+
6
1NtEORAtS
14.237
3a2x&iF2
2a5
a!2)5/2
S
14.235
3
--
5
14.233 14.234
3
=
* I
2 IaI
-
a4x
1
a2)3/2
S
14.232
1 -- a3 set-1
dx
14.230 14.231
- dx2aLa2
GTZ-
dx
4x2 - a2P2
+ ln(x+&272) &z
dx
Sx743x5 -~ 2a2x2
-
&3 In
a + I/-X; 5
+ $
In (z + $X2 - a2 )
INDEFINITE
70
14.244
14.245
+
xqTF-2
s
dx
x+s-?5
s
14.249
&AT s ~ x2
14.250
S~
14.252
14.253
14.254
14.255
14.256
14.257
14.258 14.259 14.260 14.261
dx
-
x(a2-
=
_
a2(a2-
=
x2 dx (a2 ex2)3/2
=
*
x3 dx (a2-x2)3/2
=
daz_,Z+d&
g
a
a+@=-2 (
1
_sin-1: x a +
2x2
xdx (,2mx2)3/2
sin-l
8
x2)3/2
>2
-~
x3
+ g
3
~~-CLln
=
a+@=2
&In
(
X
>
X
>
X
.3Lz2 &A?
-
x2)3/2
-
a2&z
=
2
sin-l-
dx
a
a+&GS
i31n
(
diFT1
dx x2(a2-x2)3/2
S
a2xF
8
dx= _~
x(a2-
+
5
dx @2ex2)3/2
s
x2)3/2
4
(a2 - x2)5/2
=
Wdx=
S S S S S
-x2)3/2
3
=
@=z -dx
S
-ta2
=
x3dmdx
s
14.248
14.251
sin-l:
s
14.246
14.247
$f
INTEGRALS
=
x
614x
dx
a4&iGz
-1
x3(a2-x2)3/2
=
3
+
x(a2
-
&51n
2a4&FG
2a2x2@T2
S($2 - x2)3/2 dx= Sx(&-43/2& = Sx2(& - &)3/2 ,&= S x2)3/2 dx=
-
+
3a2x&Ci3 8
x2)3/2
4
a+@? (
X
>
ia4 sin-l:
+
(a2-x2)5/2
s
(a2 -xx2)3'2
14.263
S
14.264
s
(a2-
x2)3/2
-
x2)5/2
+
a2x(a2--2)3/2
6 x2)7/2
=
(a2 -3x2)3'2
dx
=
-(a2-x2)3/2
+
_
a2(a2-
=
+
x2)5/2
a2dm
3x&z%
-
2
a3 ln
_
(a
+ y)
;a2sin-1~ a
_
“7
+ gain
a+&PZ X
.
+ igsin-l;
5
_
_ ta2 ;x;2)3’2
a6
16
X
dx
a4xjliGlF
24
7
dx
x2
la2 -x;2)3’2
x(a2
(a2 -
x3(&2 -
14.262
5
>
x
INDEFINITE
INTEOiRALS
INTEGRALS
71
ax2 f bz + c
LNVULWNG
2 14.265
s
&LFiP
dx bx + c
ax2+
=
2ax + b - \/b2--4ac $-z
If results
14.268
14.269
14.270
14.271
14.272
14.273
14.274
14.275
s
xdx ax2 + bx + c
=
&
s
x2 dx ax2 + bx + c
=
--X a
s
ax2-t
x”’ dx bx+c
S s
dx + bx + c)
xz(ax2
S S S S S
xn(ax2
ax2 + bx + c
(
dx + bx + c)
14.277
14.278
14.279
X2
1 =
-(n
- l)cxn-l
-- b c
b 2c
--
( ax2 + bx + c )
&ln
2ac
~“-1 dx ax2 + bx + c
I b2 - 2ac 23
x”-l(ax2
(4ac -
x dx (ax2 + bx + ~$2
=
- (4ac -
=
2c (b2 - 2ac)x + bc f4ac - b2 a(4ac - b2)(ax2 + bx + c)
=
- (2n - m - l)a(ax2
2ax + 6 2a +b2)(ax2 + bx + c) 4ac - b2, f
x”’ dx
+ bx + c)n--l
(n - m)b (2n - m - 1)a s
dx ax2 + bx + c
S
xnp2(ax2
dx ax2 + bx + c
S S
dx ax2 + bx + c
(m - 1)~ (2n-m1)a s
’
~“‘-2 dx (ax2 + bx + c)n
xm-1 dx (ax2 + bx + c)fl
+bx+c)n= $S(a392f~~3~~)“-I - $S(ax:";;:!+ -iS S S S S S S .I S Sx~-~(ccx~ s
x2n--1 dx (m2
dx x(ax2 + bx f
x2n-2
dx
(ax2 + bx -t- c)n
c)~
dx x2(ax2 f bx + c)~
xn(ax2
dx + bx + c)
dx ax2 + bx + c
b
-4ac
xWL-l
(ax2 + bx f CP
S
dx -- a + bx + c) c
S
=
$2 dx
use
S
dx (ax2 + bx + c)2
(ax2 + bx + c)2
b = 0
dx ax2 + bx + c
J
_ 1 cx >
bx + 2c b2)(ax2 + bx + c)
If
dx ax2 + bx + c
s
x”-2 dx -- b ax2 + bx + c a
s
60-61 can he used.
dx ax2 + bx + c b2 -
X2
$1,
=
a
:i
on pages
+ T
C
--
(m-l)a
=
s
&ln(ax2+bx+c) x?T-l
=
dx + bx + c)
x(ax2
In (ax2 + bx + c) - $
-
14.276
i( 2ax + b + dn
b2 = 4ac, ax2 + bx + c = a(z + b/2a)2 and the results on page 64. If a or c = 0 use results on pages 60-61.
14.266
14.267
In
dx f bx $
1 -2c(ax2 + bx + c)
=
1
=
- cx(ax2
+ bx + C)
b 2c
-- 3a c
dx +$ (ax2 + bx + c)2
dx -- 2b (ax2 + bx + c)2 c
1
c)~
=
-(m
- l)cxm-l(ax2
_ (m+n-2)b (m - 1)~
+ bx + c)n--l
-
(m+2n-3)a (m - 1)c
dx + bx + c)n
dx x(ax2 + bx + c)
x(6x2
dx + bx + c)2
x-~(ux~
dx + bx + c)”
72
INDEFINITE
INTEGRALS
In the following results if b2 = 4ac, \/ ax2 + bx + c = fi(z + b/2a) and the results be used. lf b = 0 use the results on pages 67-70. If a = 0 or c = d use the results $
ax
14.280
=
ax2+bx+c
a
In (2&dax2
-&sin-l
on uaaes 60-61 can on pages 61-62.
+ bx + e + 2ax + b) (J;rT4ic)
or
&
sinh-l(~~~c~~2)
14.281 14.282
x2 dx s,
ax2+bx+c
14.283
dx
14.284
=
-
ax2 + bx + c
14.285
ax2+bx+cdx
(2ax+
=
14.286
b)
ax2+ 4a
bx+c
+4ac-b2
16a2
14.288
14.289 14.290 14.291 14.292 14.293
=
6az4a25b
bx+c
(ax2 + bx + c)~/~ +
“““,,,“”
J
d ax2f
bx+c
dx
ax2+bx+c
S“
X
ax2+bx+c X2
S S ax2 Scax2 x2 S+x2+%+c)3’2 = cdax2 : bx+e+: SJ s S, S dx (ax2 + bx + c)~‘~
2(2ax + b)
(4ac - b2)
x dx
(ax2 + bx + dx + bx +
x2(aX2
ax2 + bx + c
2(bx + 2c)
~)3’~
(b2 - 4ac) \/
43’2
a(4ac - b2)
+ bx + c
(2b2 - 4ac)x
+ 2bc
dx + bx +
c)~‘~
=
ax2 + 2bx + c - &?xdax2 + bx + c +
2c2
(ax2 + bx + c)n+1/2dx
=
dx
1~x2 + bx + c
-- 3b
14.295
ax2 + bx + c
axz+bx+c
x
14.294
.
(ax2 + bx + c)3/2 b(2ax + b) dp ~ ax2+ 3a 8a2 dx - b(4ac - b2)
=
14.287
dx 8a
ax2+bx+c
S +ifif+
dx
(QX~
axz+bx+c
b2 -
26
2ac
Scax2
dx + bx +
43’3
dx
x
ax2+bx+c
(2ax + b)(ax2 + bx + c)n+ 1~2 4a(nf 1) + (2% + 1)(4acb2) (a&+ 8a(n+ 1)
S
bx + c)n-1’2dx
4312
.
INDEFINITE
14.296 14.297
S s’(ax2-t
x(uxz + bx + C)n+l/z dx
=
(ax2 + bx + C)n+3'2 cq2n+ 3) 2(2ax
dX
bx + ~)n+l’~
=
dx + bx + ++I’2
x(ux2
s
73
.
_ $
(ax2 + bx + ~)~+l’zdx
s
+ b)
(2~2 - 1)(41x - b2)(ax2 + bx + +--1/z 8a(n1) dx (2~2 - 1)(4ac - b2). (‘ (61.x2 + bx + c)n--1E
+
14.298
INTEGRALS
1 =
(2~2 - l)c(ux’J
+ bx + c)n--1’2 dx
JPJTEORALS Note 14.299
14.300
14.301 14.303
14.304
14.305
that
14.308 14.309 14.310
dx
~
s
involving
=
+
X3
x2 - ax + c-9 + 1 (x + c-42
=
__ = x3 + CL3
$ In (x3 +
ClX
s
.(
s
x2(x3
u3)
=
'(z3yu3)2
'
1
-+
u3)2
%(X3
+ a3)2
s
x2(x3
dx +
+ &In
14.312
=
1 -
--
=
CL62
xdx + u4
=
x4
S
x3
dx
~ x4 + a4
3u5fi
(x + a)2
x2 xm-3
-
m-2
a3
~
x3
-1 1)x+-’
c&3@-
-
1
In
4u3fi
&
-L 4ufi 14.314
2x-u a \r 3
x2 - ax + a2
2x +
=
tan-l
In
-4-.---
3u6 s
3a6(x3 + u3)
xm-2 -
=
a3)
=
~
tan-l
3utfi3 tan-’
3
&,3(x3 + as)
=
u3)2
x4 + a4
S
2
-
1
u3
dx
+
F
- 3(x3 + US)
dX
I'
+3tanP1
x2 - ax + a2
+ &n
a3)
x2 + axfi x2 - uxfi
x3
x dx + u3
[See 14.3001
dx + a3 dx
-2
JNTEORALS
14.311
-
(x + a)2
=
x(x3+u3)
(xfcp
x2 3a3(x3 +
=
x-’ dx
s
G-4 In
3u3(s3 +a3)
dx
x9x3+
dx
s
1
s
s
43
x2 - ax + u2
1
-
x2 dx (x3+
+
2x-u 7
tan-l
X
=
s
x3
a by --a.
14.302
~3)
a32
xdx (x3 + c&3)2 =
~
(ax2 + bx + c),+ l/i
3ea+ a3
a\/3
x2 dx
s
x3 - u3 replace
2”~ s
x2 - ax + cl2
u3
~ x dx x3 + a3
s
14.306 14.307
for formulas
JNVOLVING
dx
--
x(ux2 + bx + c)“-~‘~
s
u3
s
+
xn-3(x3
INVOLYJNG + a2
c?+* a* 1
--
u3)
tan-1
2aqi
+ c&2
-!!tC-LT
22 - CL2
$
x2 - axfi
+ a2
x2 + ax&
+ u2
$ In (x4 + a4)
--
1 2ckJr2
tan-1
-!!G!- 6
x2 - a2
a
74
14.315 14.316
INDEFINITE
INTEGRALS
dx x(x4 + d)
s s
dx x2(x4
14.317
+ u4)
+-
=
1 2a5&
dx x3(x4
.
+ a4)
=
14.322
14.323
14.324
14.325
14.326
14.327
14.328
14.332
dx
.I’ x(xn+an) fs
=
S
xm dx (x”+ c&y
I’
dx xm(xn+ an)’
xn + an
‘, In (29 + an)
=
s
xm--n dx (xn + (yy-l 1
=
2
x”’ dx s-- (xn - an)’
14.333
14.334
&nlnz
=
=
an S
s
-
an s
x”’ --n dx (xn + an)T
dx xm(xn + IP)~--~
xm--n dx (~“-a~)~
1 an -s
xm--n dx
+ s
(xn-an)r-l
=
S
dx = m..?wcos-~
!qfzGG
m/z
dx xmpn(xn + an)r
tan-l
CiXfi ___ x2 -
a2
INDEFINITE
14.335
xp-1 dx
INTEGRALS
75 x + a cos [(2k - l)d2m]
1 ma2m-P
I‘----=xzm + azm
a sin [(2k - l)r/2m]
x2 + 2ax cosv where 14.336
xv- 1 dx X2m
s
-
m-1
1
a2m
=
2ma2m-P PI2
cos kp7T In km sin m
x
(’
2*
ka
x -
tan-l
+ a2 a cos (krlm)
a sin (krlm)
k=l
+
.
2ax ~0s;
m-1
1
where
x2 -
m
k=l
-&pFz
14.337
+ a$!
0 < p 5 2m.
{In (x - 4
+ (-lJp
> ln (x + 4)
0 < p 5 2m.
x2m+l
xP-ldX + a2m+l 2(-l)P--1 (2m + l)a2m-P+1k?l
=
m
sin&l
x + a cos [2kJ(2m
a sin [2krl(2m
+ l)] + l)]
m
(-1p-1 (2m + l)az”-“+‘k?l
-
tan-l
cossl
In
x2 + 2ax cos -$$$+a2
+ (-l)p-l In (x + a) (2m + l)a2m-P+ l where
14.338
O
s
x2m+l
-
dx a2m+l 77,
1
(zrn+
x -
2kpr
kzlSin
l)a22m-P+l
2m + 1 Iian-’
a cos [2krl(2m
a sin [2k7;/(2m
+ l)] + l)]
>
m
+
s
14.340
sinaxdx
=
=
%sinax+
14.342
=
(T-
siyxdx
=
14.345 14.346
14.347
=
-$)sinax
sin ax
s
dx
+ a S Ydx
= =
sin2 ax dx
+ (f-f&--$)
cosax
5*5!
X
S sin ax xdx S sin ax
s
cos ax
3*3!
dx
sin ax
ax-(aX)3+(a2)5-...
s sinx;x
lNVOLVlNC3
x cos ax ___ a
y-
14.341
14.344
2ax cos
-- cos ax a
=
‘ssinaxdx
14.343
x2 -
O
INTEGRALS 14.339
In
In (x-a) (2m + l)a2m-n+1
+ where
cos&
(2m + 1)ta2m-p+ ‘,li,
=
: _ sin 2ax 2
4a
[see 14.3731
a2
76
INDEFINITE
14.348 14.349
x sin2 ax dx sin3 ax dx
s
X2
=
-
x sin 2az 4a
-
4
=
_ cos ax -+-
=
3x 8
14.351
~
=
- 1. cot ax a
__ dx sin3 ax
=
-
14.352
s
14.353 14.354
1 -
dx sin ax
sin 4ax 32a
cos ax 2a sin2 ax
sin px sin qx dx
s
3a
sin 2ax -+-t 4a
sin4 ax dx
cos 2ax 8a2
--
cos3 ax
a
14.350
INTEGRALS
sin (p - q)x 2(P - 4)
=
=
_ sin (p + q)x
[If
2(P + (I)
p = *q,
see 14.368.1
‘, tan
14.355 14.356
14.357 14.358
p tan *ax
14.360
14.361
14.362
dx p + q sin ax
I‘
ad&2
tan-’
a&2
In
+ q
@q
= ptan+ax+q--
If
p = *q
see 14.354
s
dx (p + q sin ax)2
If
p = *q
( p tan +ax
-
+ q + dm
and 14.356. q cos ax a(p2 - q2)(p + q sin ax)
=
see 14.358
t--J---p2
and 14.359.
dx p” + q” sin2 ax
p2 -
s
14.364
14.367 14.368
xmsinaxdx
1
.I’ s
=
sijlnuxdx = sinn ax dx
s
=
-’
In
m cos ax mxm--l + a
sin ax (n - 1)xn-l _ sinn--l
tanax
tan
ap&2
dx q2 sin2 ax
_ 2wdF7z
14.366
dx p + q sin ax
s 1
14.365
1
q2
-1 dm
14.363
)
+a
- dx sinn ax
=
- cos ax a(n - 1) sin”-’
ax
~ xdx sinn ax
=
-x cos ax a(72 - 1) sinn--l
ax -
dn
tan ax +
( dm
tan ax -
sin ax
a2 =$
n-1
ax cos ax an
P
m(m - 1) -7 dx
s
xmp-2 sin ax dx
[see 14.3951
s +-
72-l n
s
sinnp-2 ax dx
dx sin”-” ax az(n - l)(n
1 - 2) sinnez
ax
+-
n-2
n-1
xdx sinnP2 ax
.
INDEFINITE
14.369
14.370
' cosax
*
xcosaxdx
s
- xzcosaxdx
14.372
'
cosax
x3
dx
Fdx
a
=
$,,,a.
=
(T---$)cosax
";,'
dx
+
(axY
--GE-=
14.376
=
- cos ax _ a
2*2!
14.377
x dx cos ax
-
14.378
14.379
14.380
14.381
s
=
=
cos4
ax dx
=
dx
=
14.383
cos ax cos px dx
14.384 14.385
dx 1 - cosax
s
x dx 1 - cos ax
=
14.387
xdx 1 + cos ax
=
14’389
S
dx cos ax)2
dx (1 + cosax)2
=
sin3 ax
3a
sin (a - p)x
2(a - P)
-- x cot E a 2
=
dx 1 + cosax
_
...
cos 2ax 8cG
+
sin (a + p)x
2(a + P)
=
14.386
JtI
+
=
s
14.388
+2
tan ax a
-
ax
x sin 2ax -+4a -sin ax a
14.382
cos3
=
sin 2ax 4a
f+-
=
ax dx
COG ax
[See 14.3433
En(ax)2n
cos3
___dx
dx
...
(2n-k2)(2n)!
x co9 ax dx
s
'y S'
=
co532ax dx
s
-+(axF 6*6!
+ 4*4!
$ In (see ax + tan ax)
cos ax
S
--kd4
Ins--
X
14.375
sin ax
+ ($-$)sinax
=
s
14.374
77
-cos ax ~x sin ax a2 + a
=
14.371
14.373
=
dx
INTEGRALS
=
=
2 + - In sin ax a2 2
[If
a = *p,
see 14.377.1
78
INDEFINITE
I
INTEGRALS
ad-2tan-’
14.390
dx p+qcosax
s
=
dt/(p - Mp tan *ax
&j&2
14.391
dx (p + q cos ax)2
s
=
a($
dx
14.392
=
! tan &ax -
tan-l
s
tan-l p2-
ptanax-dm ( ptanax+dv
14.395
14.396
14.397 14.398
14.399
14.400
14.401 14.402
14.403
14.404
14.405 14.406 14.407
ydx
=
-
s
s
co@ ax dx
S
S
S
s
S
S
co@ ax xdx COP ax
-
sinax
a cos ax -n-1 (n - 1)x*- 1
cos ax -
sin ax cosn--I ax +?Z-1 n an
=
=
x sin ux a(n - 1) COP--I ax
=
-
sin px cos qx dx
=
_ cos (P - q)x VP - 4
sinn ax cos ax dx
=
COP ax sin ax dx
=
dx S sin ax cos ax
s
xm-2
S
sinn + 1 ax (n + 1)~ -cosnflax
(n + 1)a
=
co@-2ax
dx
dx
COP-2 ax 1 - l)(n - 2) cosnP2 ax
a2(n
+n-2
cos (P + q)x VP + 9)
_
[If
n = -1,
[If
n = -1,
sin 4ax 32a
X
- 8
dx sin2 ax cos ax
=
-
S
dx sin ax ~052 ux
=1 ;lntan
1 a sin ax
A In tan a
-2cot2ax a
y
+ &
cos ax dx
[See 14.3651
=1 a In tan ax
=
a2
2a
S
dx S sin2 ax cos2 ax
mtm - 1)
sin2 ax
cosax dx
sin2 ax cos2 ax dx
>
sdx
S’
sin ax +n-2 a(n - 1) co@--I ax n-l -s
.-AL= s
xm sin ax mxm--l +a a2
=
[If p = *q see 14.388 and 14389.1
E
In
1
xm cos ax dx
s
dx p + q cos ux
P 42 - P2 s
q2
I WdFT2 14.394
- P)
dn7
ap
=
-PI
d(q + dl(q --
[If p = *q see 14.384 and 14.386.1
P tan ax
w/FS
+
dx p2 - q2 cos2 ax
+ d(q + dl(q
q sin ax - $)($I + q cos ax) 1
s p2 + q2 cos2 ax
14.393
In
+ 4 tan ?px
see 14.440.1 see 14.429.1
n-1 -s
xdx
cosn-2 ax
INDEFINITE
14.408
INTEGRALS
79
s
14.409 s cos ax(1
dx C sin ax)
=
.
sinax(1
dx 2 cosax)
-
S
dx sin ax rfr cos ax
14.410 14.411
14.412
14.413
14.414
s
L a&
sin ax dx sin ax * cos ax
=
I
cos ax dx sin ax f cos ax
=
2:
sin ax dx p+qcosax
=
14.416
cos ax dx p+qsinax
=
14.417
14.4 18
1 f sin ax)
2a(l
1 * cos ax)
k
=
14.415
2a(l
i
- $
$
In tan
T $a In (sin ax * cos ax)
+ +a In (sin ax C cos ax)
In (p + q cos ax)
In (p + q sin ax)
S
sin ax dx (p + q cos axy
=
aq(n - l)(p
1 + q cos axy-1
s
cos ax dx (p + q sin UX)~
=
aq(n - l)(p
-1 + q sin UX)~--~
dx p sin ax + q cos ax
14.4 19
=
adi+
q2 In tan
ax + tan-l 2
(q/p)
2 dx p sin ax + q cos ax + T
14.420
p + (r - q) tan (ax/z)
a&2-p2-q2tan-1 =
T2
1
ln
aVp2 + q2 - ~-2 If
14.421
I‘
r = q see 14.421.
If
q + p tan 5
=
ax + tan-’ 2
=
1 In 2apq - sinmP1
14.425
I‘
(q/p)
dx p2 sin2 ax + q2 cos2 ux dx p2 sin2 ax - q2 COG ax
14.424
q2
dp2 + q2 - r2 + (r - q) tan (ax/2)p + dp2 + q2 - r2 + (T - q) tan (ax/2)
dx
S
-
p -
psinax+qcosax*~~
14.423
p2
~~ = p2 i- q2 see 14.422.
dx p sin ax + q(1 + cos ax)
14.422
-
sinm uz COP ax dx
p tanax - q p tan ax + q ax co@+ l ax m-l +a(m + n) mfn
sinm-2
ux cosn ax dx
= sin”
+ l ax cosnwl a(m + n)
ax +
n-l m+n
s
sinm ax COS”-~
ux
dx
80
14.426
INDEFINITE
_r’s
dx
=
INTEGRALS
sinm-l ax a(n - 1) co??--1 ax -
m-l -n-l
sinm + 1 ax a(n - 1) cosn--1 ax
m--n+2 n-l
- sinme ax I a(m - n) cosnel ax - cosn-l
ax
a(n - 1) sinn--l 14.427
S
Ed,
ax
a(n - 1) sinn--l COP-~
14.428
S
ax
14.430
14.431
14.432
14.433
14.434
14.435
14.436
14.437
14.438
14.439
S S S S
tan ax dx
1
=
-ilncosax
tan3 ax dx
=
tan2 ax 2a + $ In co9 ax
=
edx
dz= tan ax
ydx
s;;;”
2”zx dx
S
‘;?&l,az
dx
m+n-2 n-1
S S
dx
sinm ax cosnw2 ax sinm-2
dx
ax COP ux
tamuzc
ax
x
tarP + 1 ax (n + 1)a
=
1 (ax)3 ;Ei 1 3
=
I 2(ax)7 105
-2 tan ax + $ In cos uz - f a
=
p2 + 42
PX
tan”-’
ax
(n _ l)a
+
Q
ah2 + q2) -
S
I . . . + 22922n-
l)B,(ax)*~+' (2n + 1) !
2*n(22n - 1)B,(ax)2n-1 (2n- 1)(2n)!
~(cLx)~
=
dx
+ q tanax
I (ax)5 15
(ad3 a~+~+~+-+
=
tann ax dx
s
dx
ilntanax
=
xtanzaxdx
Sp
z;;:;;z
i In sin ax
xtanaxdx
s
‘-, lnsec
tan ax a
S S S s
1
=
=
tann ax sec2 ax dx
s
INVOLVING
dx
tanzax
m-l m-n
dx
-1 mtn-2 + 1 a(m - 1) sinm--l ax ~0.9~~~ ux m-l
INTtkRAlS 14.429
+-
dx
sic”;;z;x
n-l
ax
c;;:;;x
S S
1 + ~(72 - 1) sinmP1 ax cosn--l ax
=
sinm ux co@ a5
-- m-l 72-l
ax
I a(m - n) sinn--l
dx
m-l m-n
.s
_ m-n+2
-coSm+lax
=
f-
sinme ax cos”-!2ax dx
S
In (q sin ux + p cos ax)
tann--2 ax dx
+ *”
+
...
INDEFINITE
14.440 14.441
14.442
14.443 14.444 14.445
14.446
14.447
14.44% 14.449
14.450
14.451 14.452
14.453 14.454
14.455
14.456
14.457
14.458
cot ax dx
s s
s
=
=
-- cot ax a
cot? ax dx
=
- -
=
dx
cot ax
=
+%dx
qcotax
set ax dx
S
S
ax
(n-1)a
-
cotn--2 ax dx
i In (set ax + tan ax)
=
tan ax
sec3 ax dx
=
set ux tan ax + & In (set a2 + tan ax) 2a
se@ ax tan ax dx
x secax
=
S
x sec2 ax dx
a
=
se@ ux na
-sin ax a
dx
ydx
...
cot--l S
-
dx
-
Q In (p sin ax + q cos ax) a(p2 + 92)
p2’Tq2
-
.**
sin ax - g
=
S set ax
S
+ -$ln
--
=
(2n-1)(2n)!
sec2 ax dx
S S
a
x cot ax
=
=
135
- -
-
22nBn(ax)2n--1
ax
=
+1
(2n+l)!
-~-!$%-i!?%..,-
dx
cotn ax dx
2w3n(ux)~~
ax
a2
dx
p+
ax ax
1
=
x cot2ax
S
-iIncot
=
1 In sin ax a
-cotnflax (n + 1)~
=
--a Incas
zcotaxdx
x
cots ax 2a
cotn ax csc2 ax dx
sdx
81
i In sin ax
cotzaxdx
S S S S S S S S
INTEGRALS
=
(ax)2 + (ax)4 + 5(ax)6 8 - 144 W2 lnx+T+-gg-f-
=
=
5(ax)4
E tan ax + 5 In cos ax
E,(ax)2n +2
+
Gl(ax)s 4320
**.
+ (2n+2)(2n)!
+
. . . + E,(ax)2” 2n(2n)!
+
.”
+
**’
INDEFINITE
82
14.459 14.460
dx
S s
dz P Q s p + q cos ax
=x ---
q + p set ax
Q
se@ ax dx
INTEGRALS
secne2 ax tan ax n-2 +n-1 a(n - 1)
=
se@--2 ax dx
s
; 1NTEQRALS
14.461
14.462
14.463 14.464
14.465
s
s
csc ax dx
csc2ax
S
csc3
s s
14.466
=
k In (csc ax -
--
ax dx
=
- csc CL5 2a
csc
cot
+ z
1
UX
In tan T
_ cscn ax na
=
a
,jx
=
$
ax
.l
14.467
c&x
-- cos ax
=
ar
$ In tan 7
a
CSC” ax cot ax dx
- x
=
ax
=
dx csc r&x
cot ax)
cm az
cot
dx
-
INVOLVING
+
k$
+
!k$
+
. . .
+
2(22n-’
-
S S csc2 S ?%!!?
dx
=
1)B,(ax)2n+’
+
. . .
(2n + 1) !
f
_ &
+ $? + !&I?$
+
... +
2’22’;;n-m1$$;‘2’-
’ +
...
5
14.460
14.469
14.470
x
ax dx
=
- ~x
=
E-I?
dx
q + p csc ax
s
Q
CSC” ax dx
=
S
sin-1
14.473
14.474
Ed%
=
s
sin-l
(x/a)
z & a dx
= =
=
[See 14.3601
n-2 n-1
S
csc”-2
ax dx
IRZVRREiZ TR100NQMETRfC
a
x3
j- sin-l z+-
z + a
X&Z? 4
(x2 + 2a2) &K2
z +
9
(x/aj3
- sin-1
+
(x/u) X
1 * 3(x/a)5 2.4.5.5
1 3 5(x/a)7 + 2*4*6*7*‘7 l
l
a-kdG2
-
$l
-
2x + 2dm
X
2
14.476
fl&CtlONS
ZZ + dm
2*3*3 dx
+-
sin-l
5
14.475
dx p + q sin ax
=
39 sin-1
In sin ax
S
P
5 sin-l
U
S
+ $
lNVotVlN@
‘xsin-lzdx
14.472
ax
a
CSC~-~ ax cot ax a(n - 1)
-
INTEORALS
14.471
cot
sin-l
z
+
...
“’
INDEFINITE
14.477
cos-1 :dx a
.(‘
=
zc,,-l~&
x cos-1%
-
INTEGRALS
@?2
=
cos-ls
_
xr a
a
39 cos-l
14.479
: a
,&
cos-1 (x/a)
14.480
x
cos-;;xln)
14.481
=
cos-1
fj
;lnx
-
dx
=
_ cos-1 (x/a)
tan-1Edx
14.484
x tan-1
14.486
14.487
=
Edx
=
x2 tan-1
z dx
=
tan-~(xiu)
dx
=
-
a
&(x2+
(x/a) x
&i72
-
dx
[See 14.4741
a+~~~
+ iln
z ( cos-1 xa)2 xtan-1E
2a2)
+
9
x
=
-5
4
sin-1
=
s
(x2
-
dx
ds
X
2x -
cot-‘?dx
14.489
x cot-’
=
a zdx
2dz&os-'~
zIn(xzfa2)
a2) tan-1
(x/u)3 ; _ 32
cot-* (x/u)
14.491
X
(x/a)
cot-1
14.492
x2
s
see-*z
a
x cot-l
=
52 cot-’ ; dz
x a
7
+ ~(xla)5
_ -(x/a)772
+ *.*
z + % In (x2 + a2)
4(x”
a2) cot-1 E + 7
+
=
;
dx
=
g In x -
dx
=
_
tan-’
(x/a)
dx
[See 14.4861
X
(x/a)
cot-' X
dx
!
=
2 set-l
z -
a In (x + &?C3)
x set-*
z + a In (x + dm)
o<sec-*:<; 5 < set-*
2 see-lE - a 7x-a 14.494
>
.
14.488
14.493
i?3
83
S
x set-1
z dx
x2
see-* f +
x3 ,secelz s
x2 see-1:
a
ds
0 < set-1 z < i
=
z
14.495
2 < i7
-
= X3 i
ysec-1
z +
t < set-*
2 ax&F2
6 ax&2G3 6
-
$In(x
-t $ln(x+da)
t
< T
+ dZ72)
0 < see-1 i
< set-11
i
< g < T
INDEFINITE
84
14.496
set-l
(x/a) X
.I’
dx
=
;1nx
dx
=
+ ; + w3
_ see-l
14.497
set-l s
(da) X2
14.499
s
s
* csc-1
2 dx a
x csc-1:
a
x2 csc-1
5 < set-1
dx
dx
(x/a)
xcsc-1:
-
-5
E +
a 7 x-a
22 y csc-l
% -
2
csc-l
X ;
csc-1
a -
E ,
(dx)3 2-3-3
-5
14.506
s
=
_
dx
(x/u)
- csc-1
5 a
z < 0
< ;
dx
-5
I 1 ’ 3(a/x)5 204-5.5
+
1 ’ 3 5(a/x)7 2*4*6*7-T l
(x/u)
=
< csc-1;
...
0 < w-1 z < ;
+
; < csc-1:< 0 ___
mt1
s
Stan-l:
-
xm cot-1 f dx
=
-$+eot-l~
+ -&.I'=""
set-l mfl
&Jsdz
(x/u)
0 < s,1:
< 5
i
< set-l%
< T
0 < csc-1
E < ;
= xm+l
see-1 (x/a)
+ -
a m+l
mS1
xm+l csc-1(x/u) m+l dx
+
-
=
xm see-1 z dx
< 0
X
x dx a
' x"'tzsc-1:
< ;
+
=
xm+l
14.508
< ,se-1;
X
(x/a)
xm tan-1
< T
< csc-1
0 < csc-1;
Xlnfl
I'
t
=
X
14.505
< i
0 < csc-1;
uln(x+~~)
X2 2 csc-1
X
sin--l
...
=
f dx
X2
xm
+
ax
_ csc-1 .s
1*3*5(a/2)7 2-4.6-7-7
0 < set-lz
&ikS
+ aIn(x+@=2)
X
14.502
+ &GFG
x csc -1:
3
CSC-~
+
1~3(cLlX)5
2-4-5-5
=
X3
s * w-1
+
.
X
x3 3
14.501
(x/u)
.
_ sec-lx(xiu) 1
14.498
INTEGRALS
xm+l
csc-1 mfl
a m+1
=
i
I
(x/a)
~ xm dx
s d=
S
xm dx @qr
-;
< 0
INDEFINITE
14.509
14.510
14.511
14.512
eaz dx
=
a
xeaz dx
=
e””
s dx
Z2eaz
=
"" a
xneaz dx
$dx
=-
eaz xn---+ a (
Inx
~
P + waz
S
dx (p + qeaz)2
14.516
14.517
dx
S
n
a
I taxJ2 .
X
- P
S
14.519 S
14.520
. . . ~(-l)%! an
S
14.522
&
1 + WY
a&
-
$2
t ...
w 1
In (p + qeaZ)
2?em Q >
tan-l
eaz - jLjFp
In
eaz + &G&
e” sin bx ds
=
eaz(a sin bx - b cos bx) a2 + b2
eaz cos bx dx
=
eQz(a cos bx + b sin bx) a2 + b2
xem sin bx &
=
xeaz(a si~2b~~2b
xeax cos bx dx
=
xeax(a cos bx + b sin bx) a2 + b2
eaz In x dx
=
14.523 S
14.524 S
integer
In (p + qeaz)
=
e”lnx --a
S
n = positive
3*3!
‘OS bx)
_ ea((a2
S
14.521
if
a
___ 1 2&G
14.518
-
--ssdx n-l
1 adiG
peaz + qe-a.%
I taxj3
Z-2!
+
;+
=
dx n(n - 1)xn-2 a2
nxnel
(n - 1)x”-’
=
>
xn--leaz
a S
+ la;, -
dx
S
a2
-eaz
z
S
14.515
a
Pea2 --a
S
14.514
%2-&+Z
(
=
=
Fdx
1 a>
X--
a (
s
14.513
85
e""
s
s
INTEGRALS
1 5 a S
- b2) sin bx - 2ab cos bx} (a2 + b2)2
_ eaz((a2 - b2) cos bx + 2ab sin bx} (a2 + b2)2
dx
eu sinn bx dx
=
e”,2s~~2’,~
eaz co@ bx dx
=
em COP--~ bx (a cos bx + nb sin bx) a2 + n2b2
in sin bx - nb cos bx)
+
+
n(n
- l)b2
a2 + n2b2 S n(n - l)b2 a2 + n2b2
S
eu sin”-2
em
bx dx
cosn--2 bx dx
86
INDEFINITE
INTEGRALS
HWEOiRA1S 1NVOLVfNO 14.525 14.526 14.527 14.528 14.529 14.530 14.531 14.532 14.533 14.534 14.535 14.536
s
14.538 14.539
=
S S S$Qx
xlnx
xlnxdx
=
xm lnx
dx
-
$1
=
2
nx-4)
--$ti
1 m+1
-
lnx (
=
14.541 14.542
see 14.528.1
;lnzx
P 1+x dx J
=
x ln2x
~Inn x dx
=
-lP+lx
s
dx
xln
=
x
-
lnnx
[If
In (lnx)
=
In x
+ 2x
n = -1
+ lnx
ln(lnx)
dx
xlnnx
=
+ $$
+ s
* .
-
n
m = -1
see 14.531.
S S
S
Inn-1
=
x ln(x2+&)
In (x2 - ~2) dx
=
x In (x2 - u2) xm+l
=
sinh ax dx
x sinh ux dx
x2 sinh ax dx
+ (m+3!)~~x
x dx
In (x2 + ~2) dx
xm In (x2 f a9 dx
.*a
.
+ (m+2t)Iyx
n xm+l Inn x -m+l m+1
=
+ l
+ (m+l)lnx
xmlnnxdx
S S S
see 14.532.1
In (lnx)
=
xm dx
2x lnx
nfl
X
S Sf& S S S
-
-
In (x2* m+l
INTEGRALS
14.540
[If m = -1
s
If
14.537
lnxdx
Inx
xm Inn-1
s
2x + 2a tan-1
&)
--
2
m+1
!NVOLVlNO
~
=
x cash ax -- sinh ax U
=
u2
coshax
z
2x + a In
cash ax a
=
x dx
-
$sinhax
S
Y$gz
sinh (cx
c-lx
+ a**
INDEFINITE
14.543
'14.544
14.545
14.546
14.547
14.548
14.549
14.550
sinLard
14.552
14.553
14.554
sinizax
*
i In tanh 7
xdx sinh ax
=
1 az
ax
sinhz ax dx
=
sinh ax cash ax 2a
-
s
s
x sinha ax dx
,I'
dx sinh2 ax
~
I‘
I
.(
cash 2ax 8a2
x2 4
a
px dx
sinh (a + p)x %a+p)
=
p)x aa - P)
sinh (a -
' sinh ax sin px dx
=
a cash ax sin px -
' sinh ax cos px dx
=
a cash ax cos px +
p sinh ax sin pz a2 + p2
ax+p--m qeaz + p + dm
1
s
(p +
=
ad~2
dx
S
p +
q sinh ax
=
dx -
’ sinh” ax dx
dx
S sinhn ax ~ x dx
.I’ sinhn ax
xrn
cash a
ux
--
m a
+ -n-l
=
- cash ax a(n - 1) sinhnP1
ax
=
- x cash ax a(n - 1) sinhn--l
ax -
dx
=
I’
p + dm
tanh ax
p - dm
tanh ax
xm--l
sinhn--l ax coshax _ -n-1 n an - sinh ax (n _ l)xn-’
sinh ax Xn
2apGP =
=
In
1
=
q2 sinh2 ax
xm sinh ax dx
~
a(p2 +
dx
q2 sinh2 ax
p2 +
-
=
>
- q cash ax +” q2)(p + q sinh ax) P2 + 92
dx
q sinh ax)2
p sinh ax cos px
c&2+ p2
dx
p + q sinhax
S
2
a = *p see 14.547.
s
14.558
-~--
X
--
-- coth ax
=
sinh ax sinh
.I'
x sinh 2ax 4a
=
[See 14.5651
=Fdx
s
=
87
,. . . .
I a
dx sinh ax
I‘ p” S
14.561
=
-
S
14.556
14.560
dx
x
S
14.559
I jJ$: / 05 * . 5*5!
s
14.555
14.557
ax
s
For
14.551
=
INTEGRALS
cash ax dx
S
sinhnP2
cash ax
a
S QFr -- n-2 92-l
[See 14.5851
ax dx
[See 14.5871
dx dx
S sinh*--2
as(n - l)(n
ax
1 - 2) sinhnP2
ax
-- n-2 n-l
~- x dx
S sinhnP2 ax
88
INDEFINITE
INTEGRALS
INTEGRALS
14.562
cash ax dx
14.563 14.564
cash -& ax
14.565
s
a
x sinh ax -- cash ax a a2
=
- 22 cash ax a2
=
z
*
dx
=
- dx cash ax
14.570 14.571 14.572 14.573 14.574
14.575 14.576 14.577
14.580
14.581
=
xcosh2axdz
s
dx
cosh2 ax
s
=
s
4+
P)
%a + P)
=
a sinh ax sin px - p cash ux cos px a2 + p2
cash ax cos px dx
=
a sinh ax cos px + p cash ax sin px a2 + p2
dx
s
dx cash ax - 1
s
cash ax + 1
=
$tanhy
=
-+cothy
=
!? tanh a
xdx
x dx
cash ax - 1
--$coth
=
dx
(cash ax + 1)2 dx
(cash ax - 1)2
7
7
-$lncosh + -$lnsinh -
&tanh3y
=
& coth 7
-
&
coths y
tan-’ ln
s war + p - fi2
( qP
s
7
&tanhy
p + q cash ax
dx (p + q cash ax)2
f
=
S dx =
14.582
+
sinh (a - p)z + sinh (a + p)x
=
cash ax sin px dx
cash ax + 1
s
. . . + (-UnE,@42n+2 (2%+2)(272)!
~tanh ax a
2(a -
s
S
5(ax)6 + 144
x sinh 2ax cash 2ax 4a -8a2
X2
=
+
(ad4 8
sinh ax cash ux 2a
;+
cash ax cash px dx
s
[See 14.5431
s
-
S
14.570 14.579
cosh2 ax dx
s
. . .
= -
14.569
+
; a
X
s
(axP 6*6!
4*4!
.
cash ax
s
14.567
+
lnz+$!!@+@+-
X
cos&ax
14.566
-
x2 cash ax dz
.
cash ax
sinh ax
=
x cash ax dx
.
INVOLVING
=
+ p + @GF
q sinh ax -a(q2 - p2)(p + q cash as)
)
P 42 -
P2
dx p + q coshas
S
***
INDEFINITE
In
1
14.583
p2 -
s
dx q2 cosh2 ax
INTEGRALS
p tanh ax + dKz p tanh ax -
2apllF3
=
89
I
14.584
dx
!
p tanh ax + dn
In
p tanh ax - dni
2wdFW
=
s p2 + q2 cosh2 ax
1
1
tan
--1 p tanhax
dF2
14.585 14.586
xm cash ax dx
.
coshn ax dx
s
coshnax
14.587
dx
coshn--l
= =
ax sinh ax
14.591 14.592 14.593
s s s
14.594 s 14.595
I
ax +
(n-
=
sinh2 ax ~ 2a
sinh px cash qx dx
=
cash (p + q)x 2(P + 9)
sinhn ax cash ax dx
=
sinhn + 1 ax (n + 1)a
coshn ax sinh ax dx
=
coshn+ l ax (n + 1)a sinh 4ax ~ 32a
dx sinh2 ax cash ax
=
14.597
S
______ dx sinh ax cosh2 ax
zz -sech a2 + klntanhy
S
14.600
S
14.601
S
z
dx
=
sinh
;s,hh2;;
dx
=
cash ax + ilntanhy a
dx cash ax (1 + sinh ax)
[See 14.5591
i tan-1
ax
- 2,‘a2 coshn--2
ax ’
cash (p - q)x
[If
n = -1,
see 14.615.1
[If
n = -1,
see 14.604.1
ax _
- 2 coth 2ax a
-
,jx
n-2 -n-l
sinh ax AND c&t USG
a
a
ax dx
-- x 8
S
=
coshn--2
ax
_ t tan - 1 sinh
[See 14.5571
2(P - 9)
14.596
14.599
1 In tanh a
+
dx sinh ax cash ax
dx sinh2 ax cosh2 ax
l)(n
INVOLVCNG
S
14.598
S
dx coshnPz
sinh ax cash ax dx
=
n-1 n ?$!?
x sinh ax a(n - 1) coshn--l
=
sinh ax dx
s
ax
sinh2 ax cosh2 ax dx
f-
xn--l
a n-1
sinh ax a(n - 1) coshn--l
INTEGRALS
,('
_ m a s
an
-cash ax (n - l)xn-1
s
14.590
l.h=7
xm sinh ax a
=
>
sinh ax
csch ax a
J
~- xdx coshn--l:
ax
.:,".'
INDEFINITE
90 14.602 14.603
S S
dX
sinh ux (cash ax + 1) dX
sinh ax (cash
14.604
14.605
14.606
14.607
14.608
14.609
14.610
14.611
14.612
14.613
14.614
14.615
14.616
14.617
14.618
14.619
14.620
S S S S S S S S S S S
tanhax
dx
x
=
tanhs ax dx
=
=
=
ax
tanhn + 1 (72 + 1)a
1 2
1
X2
=
=
(ax)3
3
- 2
-
bxJ5 +
-
-2k47 105
15
ax _ k!$
dx
=
+ ?k$
tanhn ax dx
cothax
dx
=
- PX
P2 -
_
dP2 - q2)
- tanhn--l ax + a(?2 - 1)
=
x -
coths ax dx
=
i In sinh ax -
cothn ax csch2 ax dx
- dx coth ax
dx
=
S
=
-
=
-
-coth2 ax 2a
cothn + 1 ax (n + 1)a
- i In coth ax
$ In cash ax
...
...
(-l)n--122n(22n - l)B,(ax)2n+ (2n + 1) !
(-l)n--122n(22n - l)B,(ax)2n-’ (2% - 1)(2?2) !
In (q sinh ax + p cash as)
tanhnw2 ax dx
coth ax a
coth2 ax dx
s
Q
-
42
i In sinh ax
=
-
x tanh ax + -$ In cash ax a
X
S S
1 2a(cosh ux - 1)
tanh2 ax 7
k In cash ax -
=
p+qtanhax
S S S
-
‘, In sinh ax
xtanhzaxdx
s
-&lntanhy
ilntanhax
xtanhaxdx
tanh ax dx ___
=
1 2a(cosh ux + 1)
+
tanhax a
tanhn ax sech2 ax dx
=
7
i In cash ax
tanhe ax dx
~ dx tanh ax
klntanh
- 1)
ux
=
edx
=
INTEGRALS
-t . . .
1
+
... >
INDEFINITE
14.621
14.622
14.623
s
s
x coth ax dx
1 i-2
=
x coth2 ax dx cothaxdx
1
ax
x2 -
=
-
2
x coth ax + +2 In sinh ax a b-d3 135
-$+7-v
X
14.624 14.625
14.626
14.627
14.628 14.629
14.630
14.631
14.632 14.633
14.634
14.635
14.636
14.637
14.638 14.639
S S
dx
p+
qcothax
cothn ax dx
S S S S S S S S Sq + p S
- PX
=
sech ax dx
cothn--l ax + a(n - 1)
-
=
+
i tan-l
. . . (-l)n22nBn(ux)2n--1 (2n- 1)(2n)!
9 In a(P2 - q2)
-
P2 - !I2
=
cothn-2
tanh ax ___ a
sech3 ax dx
=
sech ax tanh ux + &tan-lsinhax 2a
xsechaxdx
na
+ 5(ax)s + 144
=
x sech2 ux da
x tanh ax a
=
=
sechn ax dx
=
=
“-2 9
9
S
Gus 4320
dx
i In tanh y coth ux a
csch2 ax dx
=
- -
csch3 ax dx
=
- csch ax coth ax 2a =
cschn ax na
- -
+
. . (-lP~,kP 2n(2?2)!
[See 14.5811
p+qcoshax
sechnP2 ax tanh ax + n-2 a(n - 1) m-1
cschn ax coth ax dx
. . . (-1)n~&X)2”+2 (2n + 2)(2n)!
+
...
$ In cash ux 5(ax)4
lnx--m++-- (ad2
=
dx sechas
csch ax dx
- ~sechn ax
=
sinh ax a
.A!-= sech ax
S S S S
ax dx
eaz
=
sechn ax tanh ax dx
+ ---
(p sinh ax + q cash ax)
sech2 ax dx
“e”h”“,-jx
91
INTEGRALS
$lntanhy
ssechnm2
ax dx
+ ** *
INDEFINITE
92
14.640 14.641
14.642
14.643
S S
ds= csch ax
i cash ax
x csch ax dx
S Sq + p S
csch*xdx
1 2
=
x csch2 ax dx
s
= =
X
14.644 14.645
14.646 14.647 14.648
dx csch ax
cschnax
S S S
sinh-1
=
S
a
-
$+f
=
sinh-1
S
(x/a)
dx
I
14.650
14.651
14.652
14.653
14.654
sinh;~W*)
dx
S S
E dx
S
; dx
S S
14.656
14.657 14.658
cash;:
S S S r
(u/x)2 2.2.2
--
- ln2 (-2x/a) 2 -
1 3 5(a/xY 2*4*6*6*6 l
+
1x1 < a
+
l
...
l-3 * 5(alx)6 2*4*6*6*6
_
x>a
...
*Jr&F2
:In
X
(
)
(x/a)
-
d=,
cash-1
(x/a)
> 0
i x cash-1
(x/a)
+ d=,
cash-1
(x/a)
< 0
&(2x2 - a2) cash-1
(x/a)
-
i a(222 - a2) cash-1
(x/a)
+ $xdm,
=
f
(x/a) > 0, dx
E dx a
= =
x tanh-19
dx
x2 tanh-1
z dx Il.
ix@??,
4x3
cash-1 (x/a)
-
$x3
cash-1
+ Q(x2 + 2a2) dm,
-
C f
ln2(2x/a)
if cash-1
_ cash-1
(x/a) X
tanh-1
...
cash-1
(x/a)
> 0
cash-1
(x/a)
< 0
3(x2 + 2~2) dm,
cash-1 (x/a)
> 0
cash-1
< 0
=
dx
(da)
_
1*3(a/x)4 2*4*4*4
-
+
l
+ 1. 3(a/x)4 2.4.4.4
+ __ (a/~)~ 2.2.2
(x/a)
1.3 5(x/a)’ 2*4*6*7*7
x cash-1
i
cosh-;W*)
_
l
=
x2 cash-1 E dx
+ if cash-1 14.655
+ 1 3(x/a)5 2.4~505
(xlaJ3
2.3.3
_ sinh-1
=
&FT2
9
=
a
x cash-’
cschn--2 ax dx
x m x 4 +a
-
X
cash-1
S
(2a2 - x2)
z +
ln2 (2x/a) 2
=
X
...
[See 14.5531
)
g sinh-1
+
dm~ sinh-1;
(
f dx
dX
p + q sinhax
xsinh-1: =
a
Q
cschnm2 ax coth ax -- n-2 a(n - 1) n-l
-
z dx
x2 sinh-1
E-P Q
=
g dx a
x sinh-1
x coth ax + -$ In sinh ax a - 1)B,(ax)2n-1 v*x)3 + . . . (-l)n2(22n-1 e&-y+1080 (272 - 1)(2n) ! -
=
dx
ax
X --a
14.649
INTEGRALS
x tanh-1 = =
7 F
(x/a)
+(a/5)2 +
1. 3(a/x)4 2-4-4-4
292.2
+ 1.3 * 5(a/x)6 2*4*6*6*6
+
...
1
(x/a) < 0 r
1 ln a + v a X (
z + % In (a2 - x2)
+ # x2 - ~2) tanh-1: + $tanh-1:
(x/a)
a
+ $ln(a2-x2)
[- if cash-1 (x/a) > 0, + if coshk1 (x/a) < 0]
x < -a
INDEFINITE
tanh-1
14.659
14.660 14.661 14.662 14.663
14.664
14.665
S S S S S S
tanhi:
14.669 14.670
=
“+@$+&f$+... a
(z/u)
dx
=
_ tanh-1
!! dx a
x coth-’
'Oth-i
(x/u)
=
7
a
(xia)
' sech-'2
a
x sech-1
+ +(x2 - ~2) coth-’
dx
=
F
+ fcoth-1:
dx
=
_
;
dx
=
_ coth-1
dx
J? dx
(x/a)
(x/u)
+ a sin-l
(x/u),
sech-1
(x/u)
> 0
r x sech-1
(z/u)
-
(x/u),
sech-1
(x/u)
< 0
=
dx
(x/u)
-
+a~~,
sech-1
(x/u)
> 0
+x2 sech-1
(x/u)
+ +ada,
sech-1
(x/u)
< 0
-4
=
14.674 14.675
4 In (a/x)
S S
csch-1
” dz
=
x csch-1
U
x ds a
x csch-’
S
csch-;
(x/u) dx
S
xm sinh-15
s
xm cash-’
S S
a
x”’ coth-’
dx
E
U
U
T
=
=
14.677
xm sech-1
S
1 * 3Wu)4
_
...
2.4.4.4
’
sech--1
(s/u)
z+--
=
xm csch-’
: dx a
U
5
> 0
if x > 0, -
if x < 0]
[+
if z > 0, -
if
1. 3(d44
+
-
sech-1
(x/u)
x < 0] ...
O<x
2-4.4-4
+
...
1x1 > a
-
cash-’
E -
cash-’
i
--&s$=+
xmfl
coth-’
+
? U
-
dx
~
a
mt1
E -
-J?m+l
+ am
xm+1
m+lswh-‘s
U
cash-1
(x/a)
> 0
cash-1
(x/u)
< 10
S x2 SCL2- x2 S Zm+l
dx
u2 -
Zm+l
+ 1
xm dx ~~
dx
seckl
(da)
> 0
sech-1
(s/a)
< 0
xm+l
m+l
csch-1:
c a
< 0
- $T$$ + ' '3(x/u)4 -.... -u<x
-
a
tanh-1
mS-l
=
=
f. .,
@+l
i : dx
+
[+
1. 3W45
U
U
3(x’u)4
2.4.5.5
Xmfl
m-tl
5 dx
”
2.4.4.4
2.404.4
(a/xl3
nz+lSinh-lE
+
+ +@$.$ . .
ln (-x/4u)
2.3.3
=
=
2
In (4alx)
+ In (-x/a)
U
dx
a&FTS
k
U
s s dx
xm tanh-15
S
-
.
+ -$$$ . .
z k a sinh-1
x2 csch-‘z
=
In (4ulx)
~sp&l% 14.676
- a.
In (4ulx)
In (u/x)
X
i
14.673
a sin-1
&x2 sech-1
--
14.672
+ $In(xZ--2)
(x/a)
4 In (x/u)
14.671
x
x sech-1
=
U
sech-1
+ tIn(xz-u2)
a
coth;~(xlu)
S
xcoth-lx
U
x2 coth-1:
S
=
” dx
.(
14.668
dx
93
X
coth-’
14.666
14.667
(z/a) x
s
INTEGRALS
[+
if
x > 0, -
if x < 0]
15
DEFINITE
DEFINITION
OF A DEFINITE
INTEGRAL
a)/n.
Let f(x) (b -
INTEGRALS
the interval into n equal parts be defined in an interval a 5 x 5 b. Divide Then the definite integral of f(x) between z = a and x = b is defined as
of length
Ax
=
b
15.1
f(x)dx
s a
The limit If
will f(x)
=
certainly
S
if f(x)
f(x)dx
S
dx
=
lim
dx
S S S S f(x)
b-tm
dx
=
b-m
continuous. theorem
=
g(x)
calculus
the above
definite
integral
a
=
c/(b)
-
in the interval, the definite limiting procedures. For
integral example,
s(a)
dx
f(x)
dx
dx
=
lim t-0
f(x)
dx
=
lim
f(x)
a
dx
if b is a singular
point
if a is a singular
point
b
f(x)
c-0
dx
a+E
F6RMULAS
INVOLVING
b
DEFINITE
INTEGRALS
b
{f(x)“g(s)*h(s)*...}dx
S
=
a
f(x)
dx *
a
b g(x) dx * s a
Sb h(x) dx a
2
* **
b
b
cf(x)dx
=
c
S
where
f (4 dx
c is any constant
cl
a
15.9
of the integral
a
GENERAL
15.8
f(a + (n - 1) Ax) Ax}
b--c
f(x)
a
S S Sa S Sb Sb
. . . +
a
iim n-r--m
b
15.7
+
b
Cc f(x) -m
a
15.6
Ax
or if f(x) has a singularity at some point and can be defined by using appropriate
b
15.5
+ 2Ax)
b
m f(x)
a
15.4
f(a
b
b d -g(x) (I dx
=
If the interval is infinite is called an improper integral
S S S S
is piecewise
f
the result
a
15.3
Ax + f(a + Ax) Ax
then by the fundamental
by using
b
15.2
{f(u)
exist
= &g(s),
can be evaluated
lim
n-m
f(x)
dz
=
0
=
-
a
b
15.10
f(x)dx
a
15.11
a f(x)dx
b
f(x)dx
=
a
15.12
S
f(z)dx
=
SC f(x) a
dx + jb
(b - 4 f(c)
f(x)
dx
c where
c is between
a and b
a
This aSxSb.
is called
the mearL vulzce theorem
for
94
definite
integrals
and is valid
if f(x)
is continuous
in
DEFINITE
b s
15.13
f(x) 0)
dx
=
$
This is a generalization g(x) 2 0.
of 15.12 and is valid
LEIBNITZ’S
RULE FOR DIFFERENTIATION
S
a
a and b
* a
dlz(a)
15.14
95
where c is between
f(c) fb g(x) dx
a
and
INTEGRALS
if j(x)
and g(x)
are continuous
in
a 5 x Z b
OF lNTEGRAlS
m,(a) aF
F(x,a)
dx
S
=
xdx
f
F($2,~)
2
-
F(+,,aY)
2
m,(a)
6,(a)
APPROXIMATE
FORMULAS
FOR DEFINITE
INTEGRALS
In the following the interval from x = a to x = b is subdivided into n equal parts by the points a = ~0, . . ., yn = j(x,), h = (b - a)/%. Xl, 22, . . ., X,-l, x, = b and we let y. = f(xo), y1 = f(z,), yz = j(@, Rectangular
formula b
S(I f (xl dx
15.15 Trapezoidal
=
h(Y, + Yl +
i=
$(Y,
Yz
+ ..*+
Yn-1)
formula b
S
15.16
j(x)
dx
+ 2yi
+
ZY,
+
...
+
%,-l-t
Y?J
a
Simpson’s
15.17
formula
(or
b
I‘ a
f(z)
dz
DEFINITE
15.18
15.19
15.21
INTEGRALS
xp-ldx
1+x
=
-
o
for
--?i
sin p7r ’
=
RATiONAl
+ 4Yn-l f
Yn)
OR IRRATIONAL
O
,an+l-n
n sin [(m + 1),/n]
cos p + x2
=
sin
’
o<m+1
mi7
77
xm dx
1 + 2x
n even
INVOLVING
xmdx
~ + an xn
o
formula)
; (y. + 4y, + 2Y, + 4Y, + . . . + 2Y,-2
=
dx x2 + a2 Som---z-g y; S~ = S S 0
15.20
parabolic
sin m/3 sin /3
15.22
15.23
15.24
a ,,mdX s
=
?$
0
S
a xm(an - xn)p dx
0
am+*+n~l?[(m+l)ln]~(p+l) =
nl’[(m
+ 1)/n
+ p + l]
(-l)r--17ram+1-nrr[(m 15.25
n sin \(m + l)nln](r-
l)! l’[(m
+ 1)/n] + 1)/n - T + l] ’
o<m+1
EXPRISS!ONS
DEFINITE
96
DEFINITE All letters 15.26
15.27
s
IM’fEGRdiLS~JNVdLVSNO
are considered
positive
0
S
mxcos nx
S TTsin
and m f n
i r/2
m, n integers
and m = n
0
m, n integers
and m # n
sin2 x dx 0
m, n integers
and m + n odd
m, n integers
and m + n even
1.3.5...2rn-l1 2-4-6..*
2m
??I2
a/2
sin2mx
dx
=
cos2”‘x
S
0
dx
n/2
n/2
si$m+l
s
15.32
jr12
x dx
co+“+12
=
0
s
sin2P-1
"-dx
=
2 r(P
0
p
m sin p:;in
qx dx
<
0
0
15.37
15.38
0
S 0
15.39
S
0 < p < q
i iTI4
p = q > 0
i uql2 m sin2px -
dx
=
X2
x2 m cos px -
cos qx dx
dx
5 q
p 2 q > 0 15.41
15.42
2 2 =
=
m sinmx dx s o X(x2+ a2)
15.43
49 2- P)
15.44
S
cosmx
0 277
S
:e-ma
ike-ma
15.45
S
0
=
s(l-e-ma)l
=
cos-1
dx
dx
a + b cos x ii/2
=
=
a + b sin x
0
* ___ o x2 + u2 dx
S
m -dx x sin mx
211
ln 9 P
0
15.40
S 0
2
m~o~p~-/sq~
0
9 2 =
"l--osPxdx s
p>q>o
d2
apl2
=
S S
9)
X
0
15.36
+
p=o
=
m sin px cos qx dx
0
15.35
m=l,2,...
X
0
S
2*4*6..*2m ... 2m+l’ 1.3.5
)...
p > 0
-%-I2
15.34
=
m=1,2
2’
UP) r(4)
=
x cos29--1z dx xl2
0
dx
0
0
s
=
0
15.31
15.33
= ;
cot325 dx
0
15.30 s
n
0
a/2
S
=
and m =
2mf (m2 - 4)
II
mx cos nx dx
FUNCTIONS
indicated.
i 7~12 m, n integers
T/2 s
otherwise
m, n integers
0
15.29
TR10ONOMETRIC
0
=
dx
0
15.28
unless
=
ii sin mx sin nx dx
D cos
INTEGRALS
dx
a + b cosx
$2-3
(ala)
DEFINITE
15.46
2r; S S S S o
(a + b sin x)2
257
15.47
0
15.48
S
=
o
dX
o
dx l-2acosx+az
=
cos mx dx l-2acosx+a2
Ial
ram l-a2
> 1
a2 < 1,
m = 0, 1,2, . .
r
sin ax2 dx
S naYn
cos ax2 dx
=
=
i
S S S S S S
w sinaxn
=
-
m cos axn dx
=
---&
0
15.52
1
dx
2 II-
0
15.51
r(lln)
sin & ,
rfl/n)
cos
jc sin
0
15.54
S
m cos x dx 6
dx=
-@/dx
=
0
15.55
-!?$i?dx
=
0
15.56
=
0
6
2Iyp)
Sk (pn/2)
’
2l3p)
c,“, (pa/2)
’
m sin ax2 cos 2bx dx
=
k
=
i
O
O
0
15.57
S
m cos ax2 cos 2bx dx
0
15.58
S
* sin3 x
-
15.60
S S S S
=
* -tanxdx
0
$f
=
T z
x
VT/2
15.61
&y
x3
0
0
dx 1 + tarP
=T 4
2
?r/z
15.62
0
1
15.63
tan-'
1
S
sin-'x
dx
ll-cosxdx
S
15.66
s:
15.67
S
X
(h 5, tan-l
0
'+$A+... _ 32
$
=
;ln2
_
S
X
0
15.65
=
X
0
15.64
x dx
-
cosx)'$
px - tan-lqx X
m cos x -dx
=
X
= dx
y =
p
n>l n>l
2,
0
15.53
b’)312
laj < 1
77 In (1 + l/a)
=
(az-
O
(57/a) In (1 + a) =i
97
227-a
(a Jr b cos x)2 27r 1--’
=
x sin x dx iT o 1 - 2a cos x + a2 Tr
15.49
27r
dX
INTEGRALS
y
e-axcosbx
15.69
15.70
s
=
a2 + b2
m e-az sin bx dx
=
b ~ a2 + b2
0
S S S S S
m e-az
sin
mC-az-
dx
=
tan-l
k
e-bz dx
=
In!! a
X
0
15.72
bx
X
0
15.71
a
dx
0
15.73
m ecaz2 cos bx dx
1
=
15.74
-
b2/4a
5
0
e-(az2tbz+c)
dz
erfc - b
=
2fi
0
where co
15.75
S S
,-&tbztc)
ds
=
--m
cc
15.76
0
xne-azdx
Iyn + 1)
=
an+1
cc
Xme-azz dx
15.77
r[(m
=
s 0
15.78
S S S
m e-k&+b/z2)
+ 1)/2]
2a(mfl)/Z
dx
=
;
15.79
"-g+
=
;e2'6
a
d-
0
A+$+$+$+
***
=
f
0
15.80
-
xn-l s
dx
=
L+&+$+ ln
l'(n)
. . . >
(
0
For even n this can be summed 15.81
15.82
15.83
= -12
S-
1
m xdx
ez + 1
0
m
S o
xn-l
dx
eZ+l
=
$+$-$+
r(n)
For some positive
integer
S
=
“cdl:
$ -&+
S
co e-z2-e-*dx
0
15.86
&-
values
+coth;
X
=
&
of Bernoulli ..*
(
0
15.85
in terms
***
=
numbers
9 12
>
of n the series can be summed -
&
[see pages 108-109 and 114-1151.
[see pages 108-109 and 114-1151.
DEFINITE
m e-az
15.87
15.88
_
m e-~x
s
@-bs
_
e-bz
dx
x csc px
0
m e-“‘(lx;
s
‘OS
‘)
,jx
1
xm(ln x)” dx
i -
cot-l
=
tan-l%
a
(--l)%! (m + l)n+l
=
s 0
If
n#0,1,2,...
replace
S o
l - lnx 1+x
dx
=
-$
&
=
-$
’ In (1 + x) dx
S S S S S
2
0
15.94
tan-1
=
-
;
’ ln(l-x) x
0
(a2
+
dx
=
$
=
-?
m > -1,
1)
n = 0, 1,2, . . .
n! by r(n. + 1).
6
1
15.95
In
0
15.90
15.93
99
x set px
15.89
15.91
INTEGRALS
In x In (1 + x) dx
=
2-2ln2-12
In x In (l-x)
=
2 -
572
0
1
15.96
dx
c
0
15.97
- 772WC pn cot pa
O
0
’ F
dx
=
In s
=
-y
m e-xlnxdx
=
-5(-y
dx
=
S S
In sin x dx
=
RI2
lncosx
=
-l
(In cos x)2 dx
=
dx
In2
a/2
(ln sin x)2 dx
=
0
15.104
S S 0
0
15.103
$
n/z
n/2
15.102
+ 2 ln2)
0
srxlnsin
x dx
=
-$ln2
0
S
7712
15.105
sin x In sin x dx
In 2 -
=
1
0 2a
15.106
S 0
2n
In (a + b sin x) dx
=
S 0
In(a+bcosz)dx
=
2rrIn(a+dn)
DEFINITE
100 7r 15.107
ln(a
s
+ b cosx)dx
U+@=G
T In
=
(
0
2
7i 15.108
INTEGRALS
2~ In a,
a 2 b > 0
2~ In b,
b 2 a > 0
=
In (a2 - 2ab cos x + b2) dx
.(‘ 0
)
T/4
15.109
S
In (1 + tan x) dx
=
i In2
0
dx
=
+{(cos-~u)~
sin 2a
y
+ T+
-
(cos-1
sin 3a 32
b)2}
+ ...
(’
See also 15.102.
“.
:
DEFiNlTi
ti!tThRAl.S
1NVOLVlNG
S
m - sinaz dx sinh bx
=
$ tanh $
15.113
p -cos ax dx s o cash bx
=
&
15.114
S
15.112
0
-6
=
$
=
Sr(n+
NYPERBQLIC
FUNCTtC?NS
a7 sech%
0
15.115
m xndx o sinh az
S
If n is an odd positive m ___ sinh ax dx ebz + 1 0
15.116
S
15.117
S
* sinh ux ebz
dx
m ftux)
i ftbx)
1) integer,
=
2
csc $
=
&
-
5
the series can be summed
-
[see page 1081.
1
2a
cot %
0
15.118
S
&
=
{f(O) - f(m)}
ln i
0
This 15.119
is called ’ dx
S
-
Ia
(u+x)m-l(a--x)-l&
0
15.120
Frulluni’s
--a
It holds
integral.
=
22
=
(2a)m+n-1;;'f;;
if f’(x)
is continuous
and
s
- f(x) - f(m) dx converges. 1
x
16
THE GAMMA
DEFINITION
OF THE GAMMA
16.1
r(n)
FUNCTION
FUNCTION
r(n)
cc S
tn-le-tdt
=
FOR n > 0
n>O
0
RECURSiON
FORMULA
16.2
lT(n + 1)
=
nr(n)
16.3
r(n+l)
=
n!
THE GAMMA For
n < 0 the gamma
function
r(n)
GRAPH
by using
=
n=0,1,2,...
where
O!=l
FOR n < 0
FUNCTION
can be defined
16.4
if
16.2, i.e.
lyn + 1) ___ n
OF THE GAMMA
FU
CTION
Fig. 16-1
SPECIAL
VALUES
FOR THE GAMMA
r(a)
16.5 16.6 16.7
r(m++) r(-m
FUNCTION
= 6
= 1’3’5’im * em - 1) 6 + 22
=
(-1p2mG 1. 3. 5 . . . (2m
101
m = 1,2,3,
... _
m = 1,2,3,
...
k&n\! MI
- 1)
Y-
ti 6
THE
102
GAMMA
RELAT4ONSHIPS
FUNCTION
AMONG
GAMMA
16.8
r(P)r(l--pP)
16.9
22x-1 IT(X) r(~ + +) This is called
the duplication
=
* =
Gr(2x)
formula.
r(x)r(x+J-)r(x+JJ-)...r(..+)
16.10 For
m = 2 this reduces
OTHER
=
r(s+
OF THE QAMMA .
1) =
JE
-=1 r(x)
16.12 This
is an infinite
mM--mz(2a)(m-l)‘2r(rnz)
to 16.9.
DEflNIflONS
16.11
FUNCTIONS
product
. ..k
(x + 1:(x”+ 2”, . . . (x + k) kZ
xeY+il
representation
DERWATIVES
.
FUNCTION
{(1+;)r.‘m) for the gamma
Of
THL
function
GAMMA
where
y is Euler’s
constant.
FUNCTION
m
16.13
r’(1)
=
e-xlnxdx .(’
m4 r(x)
16.14
_ -
-y
+ (p)
ASYMPTOTIC
r(x+l)
16.15
+ (;-A)
+
EXPANSIONS
=
&iixZe-Z
=
-y
0
.**
+ (;-
FOR THE OAMMA
..t,_,>
If we let [e.g. n > lo]
is called
Stirling’s
x = n a positive is given by Stirling’s
asymptotic
integer
._ t
16.17
that
where
>
series.
in 16.15,
n! - is used to indicate
n!
1+&+&-a+...
then
a useful
&n
nne-n
approximation
for
formula
16.16 where
-.*
FUNCTION
-i This
+
the ratio
-
of the terms
MISCELi.ANEOUS Ir(ix)p
=
on each side approaches
RESUltS i7
x sinh TX
1 as n + m.
n is large
17
THE BETA
FUNCTION 7
DEFINITION
OF THE BETA
FUNCTION
B(m,n) =s1
17.1
P-1
(1 - t)n--l
dt
B(m,n)
m>O,
n>O
0
RELATIONSHIP
OF BETA
17.2
FUNCTION
Extensions
of B(m,n)
to
m < 0, n < 0 is provided
SOME
by using
IMPORTANT
17.3
B(m,n)
=
B(n,m)
17.4
B(m,n)
=
2
17.5
B(m,n)
=
17.6
B(m,n)
=
FUNCTION
r(m) r(n) r(m + n)
=
B(m,n)
TO GAMMA
16.4, page 101.
RESULTS
n/2 s0
sinzmp-1 e COF?-1 e de
T~(T-+ l)m
103
.ltm-l(l-
.(
0
Ql-1
(T + tp+n
dt
,,
fiASlC
18 tWFERfNtfAL 18.1
EQUATION
Separation
Linear
SOfJJTfON
of variables
fl(x) BI(Y) dx + f&d C&(Y) dy
18.2
difF’ERENTIA1 EQUATIONS and -SOLUTIONS
first
order
=
s
g)dx
0
equation
Bernoulli’s
ye.!-J-‘dz = I‘
equation
P(x)Y
J-P&
where
Q(x)Y”
=
=
Exact
M(x,
QeefPdxdx
U-4
If
v = ylen. lny
18.4
=
c
-t- c
I 2)e(l--n)
2 +
Sz(Y) -dy g,(y)
s
I
2 + P(x)y = Q(x)
18.3
+
=
f
Qe (1-n)
jPdz&
n = 1, the solution
.
(Q-P)dx
+
c
is
+ c
equation
y) dx + N(x,
where
aivflay = m/ax.
18.5
Homogeneous
y) dy
=
~iV~x+j+‘-$L3x)dy
0
=
where ax indicates that the integration with respect to x keeping y constant.
equation
c
is to be performed
I
dy z = F:0
S
lnx= where
104
v = y/x.
-
F(v)
If
F(w)
dw
- w
fc
= V, the solution
is y =
CX.
BASIC
DIFFERENTIAL
DIFFERENTIAL
EQUATIONS
AND
105
SOLUTIONS
EQUATION
SOLUTION
18.6 y F(xy)
dx + x G(xy)
dy
=
0
lnz where
Linear, second
18.7
homogeneous order equation
$$+ag+by
w = xy.
If
=
Case 1.
and distinct:
real y =
0 m,,me
real
constants.
y
where
nonhomogeneous order equation
the solution
is
:cy = c.
m2 + am + 6 = 0.
Then
+ c2em2J
and equal: =
clemP
+ e2xemlz
m2=p-qi:
=
epz(cl cos qx + c2 sin qx)
p = -a&
There above.
of
clemP
m,=p+qi, y
Linear, second
= F(v),
roots
Case 3.
18.8
G(v)
m2
mi,m,
+ c
S wCG(4
Let m,, be the there are 3 cases.
Case 2. a, b are real
G(v) dv - F(v))
=
q = dm.
are 3 cases
corresponding
to those
of entry
18.7
Case 1. $$+a$+ a, b are
by
real
=
=
Y
R(x)
cleWx
+ c2em2z
emP +----ml - m2
constants.
S c-ml% em9
+-
m2 - 9
R(x)
dx
S e-%x
R(x)
dx
Case 2. =
Y
cleniz
+ c2xenG
+ xernlz -
s
e-ml= R(s)
emP
S
dx
xe-mlx
R(x)
dx
Case 3. Y
=
ePz(cl cos qs + c2 sin qx) +
18.9
Euler
or Cauchy
equation Putting
x2d2Y
dx”
+ ,,dy
dx
+ by
epx sin qx e-c; R(x) cos qx dx S P - epz cos qx c-pz R(x) sin qx dx S P
=
S(x)
x = et, the equation 3
and can then
+
(a-l)%
be solved
+ by as in entries
becomes =
S(et)
18.7 and 18.8 above.
BASIC
106
18.10
Bessel’s
DIFFERENTIAL
EQUATIONS
AND
SOLUTIONS
equation
x2=d2y + Z&dy + (A‘%-n2)y
=
0
Y
=
C,J,(XX)
+ czY,(x)
See pages 136-137. 18.11
22%
dx2
Transformed + (2 +1)x& ’
dx
Bessel’s
equation --
+ (a%Pf~2)y
0
Y = x-’ {CLJo (@ where
18.12
(l-zs’)$$
Legendre’s
-
2x2
+ c2ypls (;c)}
q = dm~.
equation
+ n(n$-1)y
=
0
Y See pages 146-148.
=
cup,
+ czQn(4
19
SERIES
of CONSTANTS
ARlTHMEtlC 19.1
a + (a+d) where Some special
+ (u+2d)
I = a + (n - 1)d
+
**.
SERIES
+ {a + (n-
l)d}
=
dn{2u
+ (n-
l)d}
=
+z(a+
I)
is the last term.
cases are
19.2
1+2+3+**.
19.3
1+3+5+*.*+(2n-1)
+ n
=
+z(n + 1) =
GEOMETRIC
n2
SERIES
19.4 where If
-1
1 = urn-1
is the last term
and
r # 1.
< r < 1, then
19.5
a + ur + ur2 -I- a13 +
...
=
ARITHMETIC-GEOMETRIC a + (a+@.
19.6
where If
+ (a+2d)r2
SERIES
**a + {a+(n-l)d}rrt-1
=
!G$+Tfl
+ rd{l-nr"-'+(n-lPnl (1 - r)2
r P 1.
-1 < r < 1, then
19.7
a + (a+
SUMS
19.8
+
lnr
1p + 2p + 3* +
...
d)r + (a+
2d)r2 +
OF POWERS
...
=
OF POSITIVE
*
+ (1 ?r),
INTEGERS
+ ?zp =
where the series terminates numbers [see page 1141.
at n2 or n according
107
as p is odd or even, and
B, are the Bernoulli
108
SERIES
Some special
OF
CONSTANTS
cases are
19.9
1+2+3+...+n
19.10
12
+
22
+ 32 +
... +
%2
=
n(n+1g2n+1)
19.11
13
+
23
+
33
+
... +
n3
=
n2(n4+ ‘I2
=
19.12
14 + 24 +
34
+
... +
%4
=
n(n+
+iA(3n2
If
19.13
Sk = lkf (“+
=
2k+
3k+
...
+ (“;‘)S2
dy
+ nk where +
*..
lNzn
(1 + 2 + 3 + * * * + 72)s + 3n-
k and n are positive
+ (“:‘)Sk
=
l)
integers, (n+l)k+‘-
then (n+l)
SERIES
OF
CONSTANTS
~'P-'~~PB
P
(2P)! 19.36
&
+ &
+ &
+ &
+
...
(22~ - 1)&B
=
P
2(2P)! - l)&‘B
(22~-'
P
(2P)! 19.38
&
-
-!-
+
32~+1
1
__ 1
-
52~+1
+...
79 + ‘E,
=
72p+1
22Pf2(2p)!
MlSCEI.LANEOUS
1 2
19.39
-+cosa+cos2a+~*~+cosna
19.40
sina
19.41
1 + ?-cos(u
19.42
r sina
19.43
1 + rcosa
19.44
rsincu
+
sin2a
+
+ r2cos2a
...
+ r3cos3a
+ r2cos2a +
+ ...
2 sin (a/2)
+ sinna +
+ r2 sin 2a + + sin 3a +
+ r2sin2n
sin (n + +)a
=
+ sin3a
***
a**
SERIES
sin [*(n
= ..*
sin &na
=
1-‘2,,‘,‘,“,“;r2,
ITI < 1
r sin (Y l-22rcosafr2’
=
+ r”cos?za.
+ msinm
+ l)]a sin (a/2)
b-1 < 1
m+2COSnLu-?-r”+1cos(n+l)a-~rosa+1 1 - 2r cos a + ?-2
=
-
rsincu-V+1sin(n+l)cu+rn+2sinncu 1 - 2r cosa + r2
=
THE EULER-MACLAURIN
SUMMATION
FORMULA
n-1
19.45
&
F(k)
=
j-&k)
dk
-
f P’(O)
+ F(n)1
0
+
& {F’(n) +
- F’(O)}
&{F(v)(n) , +
THE POISSON
19.46
,=iii,
F(k)
-
...
&
- F(v)(o)}
(--lF1
SUMMATION
=
,J--,
{F”‘(n)
3 {F (Zp) !
- F”‘(O)} -
&
t
?
(ZP-~)(~)
-
FORMULA
{S” --m eznimzF(x)
dx
>
{F(vii)(n) F(~P-l,(O)}
- F(vii)(O)) +
. . .
20
TAYLOR
TAYLOR
f(x)
20.1
=
SERIES
FOR
f@&) + f’(a)(x-
SERIES
FUNCTIONS
a) + f”(4(2z,-
OF
42
+
ONE
1
.
VARIABLE
. . . + P-“(4(x
-4n-’
+ R,
(n-l)! where R,, the remainder 20.2 20.3
Lagrange’s Cauchy’s
after
n terms,
form form
R,
=
R,
=
is given
by either
f’W(x
of the following
forms:
- 4n n!
f’“‘([)(X
-p-y2
- a)
(n-l)!
The value 5, which may be different in the two forms, continuous derivatives of order n at least.
lies between
a and x.
The result
holds
if f(z)
has
If lim R, = 0, the infinite series obtained is called the Taylor series for f(z) about x = a. If tl-c-3 a = 0 the series is often called a Maclaurin series. These series, often called power series, generally converge for all values of z in some interval called the interval of convergence and diverge for all x outside this interval.
BINOMIAL
20.4
(a+xp
=
&I
+
nan-lx
=
an
+
(3
Ek$a
an--15 .
Special
+
+
20.5
(c&+x)2
=
a2 + 2ax + x2
20.6
(a+%)3
=
a3 +
3a2x
+
3ax2
20.7
(a+x)4
=
a4 +
4a3x
+
6a2x2
20.8
(1 + x)-i
=
1 -
x + x2 -
x3 + 24 -
20.9
(1+x)-2
=
1 -
2x
-
20.10
(1+x)-3
=
1 -
3x + 6x3 -
20.11
(l$
20.12
(1 fx)i’3
20.13
(1 +x)-l'3
20.14
(l+z)'/3
=
x)-l'2
=
=
+
dn--
1,‘,‘”
an--2z2
+
(‘;)
@--3X3
23
+ +
4ax3
+
x4
...
4x3 + 5x4 -
-l<x
10x3 + 15x4 -
-1<2<1 * *a
1-;x+~z2-~x3+. 1 + 2” 1
=
3x2
(3
an-2x2
. I
I
cases are
+
SERIES
-
2x31
-l<sSl + 2.4.6 l-3
x3 -
..,
l-;x+~x2-~x3+.~
1 + 3x 1
-
&x2
-l<x
-l<xZl -l<xCzl
+ $&x3
-
110
***
-l<xSl
-
2)
an-3z3
+
. **
+
. . .
TAYLOR
SERIES FOR EXPONENTIAL
20.15
e= =
l+x+~+~+*.*
20.16
a~ =
@Ina
20.17
ln(l+x)
20.18
$
=
=
=
(
AND
LOGARITHMIC
23
+ k$.d!+
+ _“3” -
$
-“4” +
5 + g + f
k-!&d!
...
Ins
=
2{(~)+;(++;(~)5+
20.20
Inx
=
(s?+)
+ q + . ..
+ ~(~)”
x-2”+“-sc’+ 3!
--m<2<m
-l<x
+$(z$!)“+
SERIES =
+ **.
-l<xzzl
)
20.19
sin 2
20.22
cosx
20.23
tanx
=
20.24
cotx
= 1 _ : _ f 5
20.25
secx = l+g+g+!g+...+-
20.26
cscx = ;+~+~+A!??+
20.27
sin-l
20.28
cos-lx
= T2
20.29
tan-lx
=
5!
2>0
...
X2+
FOR TRIGONOMETRIC ...
20.21
FUNCTIONS
--m<x<m
1 + xlna
x -
‘2
ln
22
111
SERIES
FUNCTIONS
--m<x<m
7!
--m<x<m 2922n
- 1)&x+-1 (2n) !
+
- . . . _ 22n~~2inp1 - . . .
im
x+$+z+E+.*.+
_ g
...
0 < 1x1< P
E,x2" (2n)!
. .. +
15,120
1.3.5 ~-
x’ 7 +
2.4.6
=
1p,x2n--1 (2n) !
x
sin-lx
1x1
+ ."
2(2+‘-
x+2y+=5+ 1 x3
T2
x-$+$-$+
.**
x5
3x3
5x5
...
0 < 1x1 < ?r
I4 < 1 I4 < 1
. ..
*E-1+1-L+ 2 x
+
/xl < 1
*.*
1.3
I4 < ;
[+ if 5 2 1, -
...
if 5 zZ -11
1x1< 1 20.30
cot-lx
= 9 - tan-12
=
2
[p = 0 if x > 1, p = 1 if x < -11 0:
20.31
see-l x
=
cos-‘(l/x)
=
E2 -
20.32
csc-1 x
=
sin-1
=
k+‘-
(l/x)
I4 > 1 2-3x3
+
2
*l-3 4 * 5x5
+
...
14 > 1
/
TAYLOR
112
SERIES
SERIES FOR HYPERBOLIC 20.33
sinh x
=
x+g+g+g+
20.34
cash x
=
l+$+e+e+...
20.35
tanh x
=
x-if+z&rg+...
20.36
cothx
=
~+fA+E+
20.37
sechx
=
l-~+~x&+
20.38
cschx
=
1 -
FUNCTIONS
-m<x<m
***
--m<x<m (-l)n-l22n(22n
...
(-I)*-
122nBnx2n-
+
. . . (-l)nEnx2n (2n) !
; + g
X
-
E.
sinh-lx
-
G
+
2.4.5
1
+
, l
+
1x1 <x
l)B,Gn--1
...
+
20.40
cash-1x
20.41
tanh-1~
=
20.42
coth-1s
=
=
-r-
lnj2xl
+ A--
1x1 < 1.
‘*’
k{In(2x)-
l-3*5 + 2.4.6.6~6
1*3 204.4~~
(
esinz =
20.44
ecosz
[‘ii
L-
1
E~~~I:~~~:
:::I
I4 < 1
x2
SERtES
x5 + . . .
x4
--m<X<m
l-$+x!pz!+...
--m<x<m
(
)
20.45
etanz
20.46
ez sin x
20.47
e2
20.48
In lsin xl
=
In(x(
20.49
~nlcosxl
=
-$
20.50
In ltan x1
=
x2 7& In 1x1 + -py + g-
20.51
In - (1 +x) 1+x
=
x -
x
+ifxZl if x 5 -1
>
1x1 > 1
e
=
’ ‘.
x+$+g+$+...
1+x+;i--s-z
=
-
(&+&+,.::“,Y”,x6+.**))
MlSCELLAN(KMJS 20.43
2
0 < 1x1 < x
= 1
2
0 < /xl < a
(2n) !
1.3 ’ 5x7 + 2.4.6.7
3x5
...
...
(-l)n2(22”-l-
-0.
1x1
. . .
+
1
(2n) !
x3
cos
1px2n-1
(2n) !
’
20.39
-
1+.+;+g+y+...
=
~+x2++3+~+
=
1+x-+$+...+
1x1
. . .
2nf2 sin (m/4) ?Z!
+
2ni2 cos (m/4) n! 22n-
-
-
f
-
go
$
-
$
(1 +&)x2
-
&
-
17x* - 2520 62x6 + 2835
... -
**.
+
+ (1 + & + #a+
+
22922n--1n(S)
* **
Jr
2
--m<x<m --m-LX-Cm
...
22n- 1(22” - l)B,xz” n(k) !
** * +
...
...
1Bn52n
n(2n) !
-
-
xn +
xn +
<
l)B,xzn !
0 < 1x1 < ?r +
...
+
...
I4 < ; 0 < 1x1 < ;
I4 < l
If
20.52
y = qx + c‘@ +
+
c323
+
c424
c525
+ I+?9 + . * *
then
20.53
x = c,y +
+
C2Y2
+
c3y3
cqy4
+ C5y” + Csy6 + * * -
where
20.54
c,cl = I
20.55
c;C,
20.56
c;C3 =
20.57
c;C4 = Sc,c,c, -
20.50
c;C, =
20.59
c;'C, = 7cfc2c5+
20.60
= -c2 2~;
- clc3
6cfc,c,
5$
+
- c1c4 2
3cFc,2
- $c5 +
84qc~c,
14~24 - 21c,c~c3
+ 7cfc3c4-
fb, Y) = f@, b) + (z - dfzb, b) +
(?I
- W&
+ $ {(x - 4‘Vi,b, b) + 2(x where
fz(a, b), f,(a, b), . . . denote
partial
- ctc6 -
28cfc2ci
derivatives
-
a)(~
with
28cfo~c4
-
42~;
b) -
bYi&,
respect
b) +
(Y -
Wfyy(%
b))
to 5, y, . . . evaluated
+
. **
at z = a, y =
b.
21
BERNOlJtLI
and
&
DEFINITION The Bernoulli
numbers
21.1
- x ez -
1
21.2
1 -
: cot 5 2 2
=
1 -
OF BERNOULLI
B,, B,, B,, . . . are defined f
+ A?!$
_ B;r’
\ B;;”
B,x2 B2x4 ~+~+-y-+*-
=
numbers
E,x2
sechx
=
l--
21.4
set x
=
1+
2!
E,x6
E,x4
+-G---
6!
E1x2 E,x4 F+qr+F+*-
TABLE
OF FIRST
Bernoulli
by the series -
...
OF EULER
El, E,, E,, . . . are defined
21.3
NUMBERS
B,x6
DEFINJTION The Euler
EULER NUMBERS
+
NUMBERS
by the series 1x1 < 9 2
*.-
E,x6
1x1 -cE
FEW BERNOUttl
AND
numbers
Euler
2
EULER NUMBERS
numbers
Bl
=
l/6
El
=l
B2
=
l/30
E,
=
5
B3
=
l/42
~93
=
61
B4
=
l/30
E4
=
1385
B5
=
5/66
E5
=
50,521
B6
=
691/2’730
E6
=
2,702,‘765
B7
=
716
E?
=
199,360,981
63
=
3617/510
E3
=
19,391,512,145
B,
=
43,867/798
E,
=
2,404,879,675,441
ho
=
174,611/330
EIO
=
370,371,188,237,525
41
=
854,513/138
El1
=
69,348,874,393,137,901
B12
=
236,364,091/2730
E12
=
15,514,534,163,557,086,905
114
BERNOULLI
21.6
E,
=
('2")Enm1
21.7
B,
=
22.($m1,{(2n.+,
21.12
- (y)E,-,
AND
EULER
+ (;)E,-,
- (‘3Env2
-
NUMBERS
115
. ..(-l)n
+ (2n;1)Ene,
-
... (-l)n-1)
FORMULAS from VECTOR ‘ANALYSIS
22
VECTORS
AND
Various quantities in physics such as temperature, Such quantities are called scalars.
SCALARS volume
and speed can be specified
by a real number.
Other quantities such as force, velocity and momentum require for their specification a direction as by an arrow or directed well as magnitude. Such quantities are called vectors.~ A vector is represented The magnitude of the vector is determined by the length of the arrow, line segment indicating direction. using an appropriate unit.
NOTATION
A.
FOR VECTORS
A vector is denoted by a bold faced letter such as A [Fig. 22-l]. The magnitude is denoted by IAl or The tail end of the arrow is called the initial point while the head is called the terminal point.
FUNDAMENTAL 1. 2.
3.
Equality of vectors. Two vectors magnitude and direction. Thus Multiplication (scalar), then magnitude of to A according called the zero
DEFINITIONS
are equal if they have the same A = B in Fig. 22-l.
If m is any real number of a vector by a scalar. mA is a vector whose magnitude is ]m] times the A and whose direction is the same as or opposite as m > 0 or m < 0. If m = 0, then mA = 0 is or null vector.
A B
/ /
Fig. 22-l
Sums of vectors. The sum or resultant of A and B is a vector C = A+ B formed by placing the initial point of B on the terminal point of A and joining the initial point of A to the terminal point of B [Fig. 22-2(b)]. This definition is equivalent to the parallelogram law for vector addition as inThe vector A - B is defined as A + (-B). dicated in Fig. 22-2(c).
Fig. 22-2
116
FORMULAS
FROM
VECTOR
Extensions to sums of more than two vectors the sum E of the vectors A, B, C and D.
117
ANALYSIS
are immediate.
Thus
Fig.
22-3 shows how to obtain
B I D Y\
(b)
(4 Fig. 22-3 4.
Unit vectors. the direction
A unit vector is a vector with unit of A is a = AfA &here A > 0.
magnitude.
LAWS OF VECTOR If A, B, C are vectors
and m, n are scalars,
22.2
A+(B+C)
22.3
m(nA)
22.4
(m+n)A
=
mA+nA
Distributive
law
22.5
m(A+B)
= mA+mB
Distributive
law
= =
(mu)A
Commutative (A+B)+C = n(mA)
a unit
vector
in
then
A+B
B+A
then
ALGEBRA
22.1
=
If A is a vector,
law for addition
Associative
law for addition
Associative
law for scalar
COMPONENTS
multiplication
OF A VECTOR
A vector A can be represented with initial point at the origin of a rectangular coordinate system. If i, j, k are unit vectors in the directions of the positive x, y, z axes, then 22.6
A
where A,i, Aj, i, j, k directions
= A,i
Y
+ A2j + Ask
A,k are called component and Al, A,, A3 are called
vectors of A in the the components of A.
Fig. 22-4
DOT 22.7 where
A-B B is the angle
between
A and B.
OR SCALAR =
ABcose
PRODUCT 059Sn
Index of Special Symbols and Notations The following list shows special symbols and notations used in this book together with pages on which Cases where a symbol has more than one meaning will be clear from they are defined or first appear. the context.
Symbole Berri (x), Bein
(xj
B(m, n) 4l (34 Ci(x) e elp e2, e3
natural
Euler
7, T-l
Fourier
elliptic
Hermite
in curvilinear
unit vectors
In(x)
modified
Jr, (4
Bessel
in rectangular
Bessel
function
function
complete
kind, 138
coordinates,
117
integral
136
of first kind, 179
140 Bessel
or loge x
natural
logarithm
common
function
polynomials, and inverse
pn (4
Legendre
f%4
associated
Qn (4
Legendre
Qt’b)
associated
Legendre
cylindrical
coordinate,
polynomials,
functions
kind, 148
functions
of second
22, 36
sine integral,
50 184
183
polynomials
of first kind, 157
Chebyshev
polynomials
of second
function
kind,
49
Chebyshev Bessel
transform,
of first kind, 149
of second
coordinate,
sine integral,
155 Laplace
146
Legendre functions
coordinate,
Fresnel
153
Laguerre transform
spherical
kind, 139
.of x, 23
polynomials,
Laplace
polar
of second
of x, 24
logarithm
Laguerre associated
r
175, 176
124
of first kind, 138
of first kind,
elliptic
modified
L?(x)
transform,
151
of first and second
Wr)
<,-Cl
Fourier
eoordinates,
unit, 21
i, i, k
J%(r)
of first kind, 179
and inverse
functions
imaginary
160
integral
polynomials,
Hankel
kind, 1’79
114
transform
HA)
kind, 179
of second
183
function,
elliptic
scale factors
124
183
of second
integral
integral,
&Y h
or logl”x
function,
integral
numbers,
incomplete
Kern (x), Kein (x)
errer
hypergeometric
F(k, @)
eoordinates,
183
elliptic
exponential
1
in curvilinear
function,
Ei(x)
K = F(k, 742)
184
184
base of logarithms,
unit vectors
incomplete
H’;‘(x)
114
integral,
integral,
E(k, $)
i
logx
cosine
complete
En F(u, b; c; x)
lnx
Fresnel cosine
103
numbers,
complementary
erfc (x) E = E(k, J2)
H’;‘(x),
Bernoulli
errer
erf (x)
h
140 beta function,
of second
263
kind, 158
kind, 136
150
161
INDEX
264
OF SPECIAL
SYMBOLS
AND NOTATIONS
Greek Sym bols Euler’s constant, 1
6
spherical coordinate, 50
lW
gamma function, 1, 101
77
1
Hr)
Riemann zeta function, 184
ti
spherical coordinate, 50
Y
e
cylindrieal coordinate, 49
e(P)
the sum 1 + i + i + - *. +;,
e
polar coordinate, 22, 36
@(xl
probability distribution function, 189
-a(O)=O, 137
Notations A=B
A equals B or A is equal to B
A>B
A is greater than B [or B is less than A]
A
A is less than B [or B is greater than A]
AZB
A is greater than or equal to B
ASB
A is less than or equal to B
A-B
A is approximately
A-B
A is asymptotic to B or A/B approaches 1, 102
absolute value of
equal to B
A
=
AifA -A
if A 5 0
factorial n, 3 binomial coefficients, 3
Y
,, --
d2Y
D
derivatives of y or f(x) with respect to x, 53, 55 =
etc.
f’(x),
pth derivative with respect to x, 55 differential of y, 55 partial derivatives,
56
Jacobian, 125
s-
1(x) ch
lJ J a f(x) dx A * dr
line integral of A along C, 121 dot product of A and B, 11’7
AXB
cross product of A and B, 118
vs=v-v =
definite integral, 94
A-B V
v4
indefinite integral, 57
V(V2)
del operator, 119 Laplacian operator, 120 biharmonic operator, 120
I
N
Addition formulas, for Bessel functions, 145 for elliptic functions, 180 for Hermite polynomials, 152 for hyperbolic functions, 27 for trigonometric functions, 15 Agnesi, witch of, 43 Algebraic equations, solutions of, 32, 33 Amplitude, of complex number, 22 of elliptic integral, 179 Analytic geometry, plane [sec Plane analytic geometry] ; solid [see Solid analytic geometry] Angle between lines, in a plane, 35 in space, 47 Annuity, amount of, 201, 242 present value of, 243 Anti-derivative, 57 Antilogarithms, common, 23, 195, 204, 205 natural or Napierian, 24, 226, 227 Archimedes, spiral of, 45 Area integrals, 122 Argand diagram, 22 Arithmetic-geometric series, 10’7 Arithmetic mean, 185 Arithmetic series, 107 Associated Laguerre polynomials, 155, 156 [sec uZs0 Laguerre polynomials] generating funetion for, 155 orthogonal series for, 156 orthogonality of, 156 reeurrence formulas for, 156 special, 155 special results involving, 156 Associated Legendre functions, 149, 150 [sec also Legendre functions] generating function for, 149 of the first kind, 149 of the second kind, 150 orthogonal series for, 150 orthogonality of, 150 recurrence formulas for, 149 special, 149 Associative law, 117 Asymptotes of hyperbola, 39 Asymptotic expansions or formulas, numbers, 115 for Bessel functions, 143 for gamma function, 102 Base of logarithms, 23 change of, 24 Ber and Bei functions, 140,141 definition of, 140 differential equation for, 141 graphs of, 141 Bernoulli numbers, 98,107,114, asymptotic formula for, 115 definition of, 114 relationship to Euler numbers, series involving, 115 table of first few, 114
for
D
E
Bernoulli’s differential equation, 104 Bessel functions, 136-145 addition formulas for, 145 asymptotic expansions of, 143 definite integrals involving, 142, 143 generating functions for, 137,139 graphs of, 141 indefinite integrals involving, 142 infmite products for, 188 integral representations for, 143 modified [see Modified Bessel functions] of first kind of order n, 136, 137 of order half an odd integer, 138 of second kind of order n, 136, 137 orthogonal series for, 144, 145 recurrence formulas for, 137 tables of, 244-249 zeros of, 250 Bessel’s differential equation, 106, 136 general solution of, 106, 137 transformed, 106 Bessel’s modified differential equation, 138 general solution of, 139 Beta funetion, 103 relationship of to gamma function, 103 Biharmonic operator, 120 in curvilinear coordinates, 125 Binomial coefficients, 3 properties of, 4 table of values for, 236, 237 Binomial distribution, 189 Binomial formula, 2 Binomial series, 2, 110 Bipolar coordinates, 128, 129 Laplaeian in, 128 Branch, principal, 17 Briggsian logarithms, 23 Cardioid, 41, 42, 44 Cassini, ovals of, 44 Catalan’s constant, 181 Catenary, 41 Cauchy or Euler differential equation, 105 Cauchy-Sehwarz inequality, 185 for integrals, 186 Cauchy’s form of remainder in Taylor series, Chain rule for. derivatives, 53 Characteristic, 194 Chebyshev polynomials, 157-159 generating functions for, 157, 158 of first kind, 157 of second kind, 158 orthogonality of, 158, 159 orthogonal series for, 158, 159 recursion formulas for, 158, 159 relationships involving, 159 special, 157, 158 special values of, 157, 159 Chebyshev’s differential equation, 157 general solution of, 159
Bernoulli
115
115
2
6
5
110
2
6
Chebyshev’s inequality, Chi square distribution, percentile values for,
6 186 189 259
Circle, area of, 6 equation of, 37 involute of, 43 perimeter of, 6 sector of [sec Sector of circle] segment of [sec Segment of cirele] Cissoid of Diocles, 45 Common antilogarithms, 23, 195, 204, 205 sample problems involving, 195 table of, 204, 205 Common logarithms, 23, 194, 202, 203 computations using, 196 sample problems involving, 194 table of, 202, 203 Commutative law, for dot products, 118 for vector addition, 117 Complement, 20 Complementary error function, 183 Complex conjugate, 21 Complex inversion formula, 161 Complex numbers, 21, 22, 25 addition of, 21 amplitude of, 22 conjugate, 21 definitions involving, 21 division of, 21, 25 graphs of, 22 imaginary part of, 21 logarithms of, 25 modulus of, 22 multiplication of, 21, 25 polar form of, 22, 25 real part of, 21 roots of, 22, 25 subtraction of, 21 vector representation of, 22 Components of a veetor, 117 Component vectors, 117 Compound amount, table of, 240 Cone, elliptic, 51 right circular [sec Right circular cane] Confocal ellipses, 127 ellipsoidal coordinates, 130 hyperbolas, 127 parabolas, 126 paraboloidal coordinates, 130 Conical coordinates, 129 Laplacian in, 129 Conics, 3’7 [see aZso Ellipse, Parabola, Hyperbola] Conjugate, complex, 21 Constant of integration, 57 Convergence, interval of, 110 of Fourier series, 131 Convergence faetors, table of, 192 Coordinate curves, 124 system, 11 Coordinates, curvilinear, cylindrical, 49, 126 polar, 22, 36 rectangular, 36, 117
124-130
INDEX
Coordinates, curvilinear (cent.) rotation of, 36, 49 special orthogonal, 126-130 spherical, 50, 126 transformation of, 36, 48, 49 translation of, 36, 49 Cosine integral, 184 Fresnel, 184 table of values for, 251 Cosines, law of for plane triangles, 19 law of for spherical triangles, 19 Counterclockwise, 11 Cross or vector product, 118 Cube, duplication of, 45 Cube roots, table of, 238, 239 Cubes, table of, 238, 239 Cubic equation, solution of, 32 Curl, 120 in curvilinear coordinates, 125 Curtate cycloid, 42 Curves, coordinate, 124 special plane, 40-45 Curvilinear coordinates, 124, 125 orthogonal, 124-130 Cyeloid, 40, 42 curtate, 42 prolate, 42 Cylinder, elliptic, 51 lateral surface area of, 8, 9 volume of, 8, 9 Cylindrical coordinates, 49, 126 Laplacian in, 126
Definite integrals, 94-100 approximate formulas for, 95 definition of, 94 general formulas involving, 94, 95 table of, 95-100 Degrees, 1, 199, 200 conversion of to radians, 199, 200, 223 relationship of to radians, 12, 199, 200 Del operator, 119 miscellaneous formulas involving, 120 Delta function, 170 DeMoivre’s theorem, 22, 25 Derivatives, 53-56 [sec aZso Differentiation] anti-, 57 chain rule for, 53 definition of, 53 higher, 55 of elliptic functions, 181 of exponential and logarithmie functions, of hyperbolic and inverse hyperbolic functions, 54, 55 of trigonometrie and irlverse trigonometric functions, 54 of vectors, 119 partial, 56 Descartes, folium of, 43 Differential equations, solutions of basic, Differentials, 55 rules for, 56 Differentiation, 53 [sec aZso Derivatives]
64
104-106
INDEX
Differentiation (cent.) general rules for, 53 of integrals, 95 Diocles, cissoid of, 45 Direction cosines, 46, 47 numbers, 46, 48 Directrix, 37 Discriminant, 32 Distance, between two points in a plane, 34 between two points in space, 46 from a point to a line, 35 from a point to a plane, 48 Distributions, probability, 189 Distributive law, 117 for dot products, 118 Divergence, 119 in curvilinear coordinates, 125 Divergence theorem, 123 Dot or scalar .product, 117, 118 Double angle formulas, for hyperbolic functions, for trigonometric functions, 16 Double integrals, 122 Duplication formula for gamma functions, 102 Duplication of cube, 45
Envelope, Epicycloid,
27
Eccentricity, definition of, 37 of ellipse, 38 of hyperbola, 39 of parabola, 37 Ellipse, 7, 37, 38 area of, 7 eccentricity of, 38 equation of, 37, 38 evolute of, 44 focus of, 38 perimeter of, 7 semi-major and-minor axes of, 7, 38 Ellipses, confocal, 127 Ellipsoid, equation of, 51 volume of, 10 Elliptic cane, 51 cylinder, 51 paraboloid, 52 Elliptic cylindrical coordinates, 127 Laplacian in, 127 Elliptic functions, 179-182 [sec uZso Elliptic integrals] addition formulas for, 180 derivatives of, 181 identities involving, 181 integrals of, 182 Jacobi’s, 180 periods of, 181 series expansions for, 181 special values of, 182 Elliptic integrals, 179,180 [see aZso Elliptie amplitude of, 179 Landen’s transformation for, 180 Legendre’s relation for, 182
267
Equation of line, 34 general, 35 in parametric form, 47 in standard form, 47 intercept form for, 34 normal form for, 35 perpendicular to plane, 48 Equation of plane, general, 47 intercept form for, 47 normal form for, 48 passing through three points, 47 Errer function, 183 complementary, 183 table of values of, 257 Euler numbers, 114, 115 definition of, 114 relationship of, to Bernoulli numbers, 115 series involving, 115 table of first few, 114 Euler or Cauchy differential equation, 105 Euler-Maclaurin summation formula, 109 Euler’s constant, 1 Euler’s identities, 24 Evolute of an ellipse, 44 Exact differential equation, 104 Exponential functions, 23-25, 200 periodicity of, 24 relationship of to trigonometric functions, 24 sample problems involving calculation of, 200 series for, 111 table of, 226, 227 Exponential integral, table of values for, Exponents,
189
95th and 99th percentile Factorial n, 3 table of values Factors,
values
for,
for, 234
2
Focus, of conic, 37 of ellipse, 38 of hyperbola, 39 of parabola, 38 Folium
of Descartes,
43
Fourier series, 131-135 complex form of, 131 convergence of, 131 definition of, 131 I’arseval’s identity for, special, functions]
131
132-135
Fourier transforms, 174-178 convolution theorem for, 175 cosine, 176 definition of, 175 I’arseval’s identity sine, 175 table of, 176-178 Fourier’s
table of values
Fresnel
254, 255
183 251
23
F distribution,
of the first kind, 179 of the second kind, 179 of the third kind, 179, 180 for,
44 42
integral
for,
theorem,
sine and cosine
175
174
integrals,
184
260, 261
268 Frullani’s integral, Frustrum of right area of, 9 volume of, 9
INDEX
100 circular
cane, lateral
surface
Gamma function, 1, 101, 102 asymptotic expansions for, 102 definition of, 101, 102 derivatives of, 102 duplication formula for, 102 for negative values, 101 graph of, 101 infinite product for, 102, 188 recursion formula for, 101 relationship of to beta function, 103 relationships involving, 102 special values for, 101 table of values for, 235 Gaussian plane, 22 Gauss’ theorem, 123 Generalized integration by parts, 59 Generating functions, 13’7, 139, 146, 149, 151, 153, 155,157,158 Geometric formulas, 5-10 Geometric mean, 185 Geometric series, 107 arithmetic-, 107 Gradient, 119 in curvilinear coordinates, 125 Green’s first and second identities, 124 Green’s theorem, 123 Half angle formulas, for hyperbolic functions, for trigonometric functions, 16 Half rectified sine wave function, 172 Hankel functions, 138 Harmonie mean, 185 Heaviside’s unit function, 173 Hermite polynomials, 151, 152 addition formulas for, 152 generating function for, 151 orthogonal series for, 152 orthogonality of, 152 recurrence’formulas for, 151 Rodrigue’s formula for, 151 special, 151 special results involving, 152 Hermite’s differential equation, 151 Higher derivatives, 55 Leibnitz rule for, 55 Holder’s inequality, 185 for integrals, 186 Homogeneous differential equation, 104 linear second order, 105 Hyperbola, 37, 39 asymptotes of, 39 eccentricity of, 39 equation of, 37 focus of, 39 length of major and minor Hyperbolas, confocal, 127 Hyperbolic functions, 26-31 addition formulas for, 27
axes of, 39
27
Hyperbolic functions (cont.) definition of, 26 double angle formulas for, 27 graphs of, 29 half angle formulas for, 27 inverse [sec Inverse hyperbolic functions] multiple angle formulas for, 27 of negative arguments, 26 periodicity of, 31 powers of, 28 relationship of to trigonometric functions, 31 relationships among, 26, 28 sample problems for calculation of, 200, 201 series for, 112 sum, difference and product of, 28 table of values for, 228-233 Hyperbolic paraboloid, 52 Hyperboloid, of one sheet, 51 of two sheets, 52 Hypergeometric differential equation, 160 distribution, 189 Hypergeometric functions, 160 miscellaneous properties of, 160 special cases of, 160 Hypocycloid, general, 42 with four cusps, 40 Imaginary
part of a complex
Imaginary unit, 21 Improper integrals, Indefinite integrals, definition of, 57
number,
21
94 57-93
table of, 60-93 transformation of, 59, 60 Inequalities, 185, 186 Infinite products, 102, 188 series [sec Series] Initial point of a vector, 116 Integral calculus, fundamental theorem of, 94 Integrals, definite [SM Definite integrals] double, 122 improper, 94 indefinite [SW Indefinite integrals] involving vectors, 121 line [sec Line integrals] multiple, 122, 125 Integration, 57 [SM also Integrals] constants of, 57 general rules of, 57-59 Integration by parts, 57 generalized, 59 Intercepts, 34, 47 lnterest, 201, 240-243 Interpolation, 195 Interval of convergence, 110 Inverse hyperbolic functions, 29-31 definition of, 29 expressed in terms of logarithmic functions, graphs of, 30 principal values for, 29 relationship of to inverse trigonometric functions, relationships
31 between,
30
29
INDEX
Inverse Laplace transforms, 161 Inverse trigonometric functions, 17-19 definition of, 17 graphs of, 18,19 principal values for, 17 relations between, 18 relationship of to inverse hyperbolic functions, 31 Involute of a circle, 43 Jacobian, 125 Jacobi’s elliptic functions, 180 Ker and Kei functions, 140, 141 definition of, 140 differential equation for, 141 graphs of, 141 Lagrange form of remainder in Taylor series, 110 Laguerre polynomials, 153, 154 associated [sec Associated Laguerre polynomials] generating function for, 153 orthogonal series for, 154 orthogonality of, 154 recurrence formulas for, 153 Rodrigue’s formula for, 153 special, 153 Laguerre’s associated differential equation, 155 Laguerre’s differential equation, 153 Landen’s transformation, 180 Laplace transforms, 161-173 complex inversion formula for, 161 definition of, 161 inverse, 161 table of, 162-173 Laplacian, 120 in curvilinear coordinates, 125 Legendre functions, 146-148 [sec uZso Legendre polynomials] associated [sec Associated Legendre functions] of the second kind, 148 Legendre poiynomials, 146, 147 [sec uZso Legendre functions] generating function for, 146 orthogonal series of, 147 orthogonality of, 147 recurrence formulas for, 147 Rodrigue’s formula for, 146 special, 146 special results involving, 147 table of values for, 252, 253 Legendre’s associated differential equation, 149 general solution of, 150 Legendre’s differential equation, 106, 146 general solution of, 148 Legendre’s relation for elliptic integrals, 182 Leibnitz’s rule, for differentiation of integrals, 95 for higher derivatives of products, 55 Lemniscate, 40, 44 Limacon of Pascal, 41, 44 Line, equation of [see Equation of line] integrals [see Line integrals] slope of, 34
269 Linear first order differential equation, 104 second order differential equation, 105 Line integrals, 121, 122 definition of, 121 independence of path of, 121, 122 properties of, 121 Logarithmic functions, 23-25 [see uZso Logarithms] series for, 111 Logarithms, 23 [sec aZso Logarithmic functions] antilogarithms and [see Antilogarithms] base of, 23 Briggsian, 23 change of base of, 24 characteristic of, 194 common [sec Common logarithms] mantissa of, 194 natural, 24 of compiex numbers, 25 of trigonometric functions, 216-221 Maclaurin series, 110 Mantissa, 194 Mean value theorem, for definite integrals, 94 generalized, 95 Minkowski’s inequality, 186 for integrals, 186 Modified Bessel functions, 138,139 differential equation for, 138 generating function for, 139 graphs of, 141 of order half an odd integer, 140 recurrence formulas for, 139 Modulus, of a complex number, 22 Moments of inertia, special, 190, 191 Multinomial formula, 4 Multiple angle formulas, for hyperbolic functions, 27 for trigonometric functions, 16 Multiple integrals, 122 transformation of, 125 Napierian logarithms, 24, 196 tables of, 224, 225 Napier’s rules, 20 Natural logarithms and antilogarithms, 24, 196 tables of, 224-227 Neumann’s function, 136 Nonhomogeneous equation, linear second order, 105 Normal, outward drawn or positive, 123 unit, 122 Normal curve, areas under, 257 ordinates of, 256 Normal distribution, 189 Normal form, equation of line in, 35 equation of plane in, 48 Nul1 function, 170 Nul1 vector, 116 Numbers, complex [sec Complex numbers] Oblate spheroidal coordinates, 128 Laplacian in, 128 Orthogonal curvilinear coordinates, 124-i30 formulas involving, 125
2
7
Orthogonality and orthogonal series, 14’7, 150, 152, 154,156,158,159 Ovals of Cassini, 44
0
INDEX
144, 145,
Parabola, 37, 38 eccentricity of, 37 equation of, 37, 38 focus of, 38 segment of [sec Segment of parabola] Parabolas, confocal, 126 Parabolic cylindrical coordinates, 126 Laplacian in, 126 Parabolic formula for definite integrals, 95 Paraboloid elliptic, 52 hyperbolic, 52 Paraboloid of revolution, volume of, 10 Paraboloidal coordinates, 127 Laplaeian in, 127 Parallel, condition for lines to be, 35 Parallelepiped, rectangular [see Rectangular parallelepiped] volume of, 8 Parallelogram, area of, 5 perimeter of, 5 Parallelogram law for veetor addition, 116 Parseval’s identity, for Fourier transforms, 175 for Fourier series, 131 Partial derivatives, 56 Partial fraction expansions, 187 Pascal, limacon of, 41, 44 Pascal’s triangle, 4, 236 Perpendicular, condition for lines to be, 35 Plane, equation of [see Equation of plane] Plane analytic geometry, formulas from, 34-39 Plane triangle, area of, 5, 35 law of cosines for, 19 law of sines for, 19 law of tangents for, 19 perimeter of, 5 radius of circle circumscribing, 6 radius of circle inscribed in, 6 relationships between sides and angles of, 19 Poisson distribution, 189 Poisson summation formula, 109 Polar coordinates, 22, 36 transformation from rectangular to, 36 Polar form, expressed as an exponential, 25 multiplication and division in, 22 of a complex number, 22, 25 operations in, 25 Polygon, regular [sec Regular polygon] Power, 23 Power series, 110 reversion of, 113 Present value, of an amount, 241 of an annuity, 243 Principal branch, 17 Principal values, for inverse hyperbolic functions, for inverse trigonometric functions, 17, 18 Probability distributions, 189 Products, infinite, 102, 188 special, 2 Prolate cycloid, 42
Prolate spheroidal coordinates, Laplacian in, 128 Pulse function, 173 Pyramid, volume of, 9
128
Quadrants, 11 Quadratic equation, solution of, 32 Quartic equation, solution of, 33 Radians, 1, 12, 199, 200 relationship of to degrees, 12, 199, 200 table for conversion of, 222 Random numbers, table of, 262 Real part of a complex number, 21 Reciprocals, table of, 238, 239 Rectangle, area of, 5 perimeter of, 5 Rectangular coordinate system, 117 Rectangular coordinates, transformation of to polar coordinatee 36 Rectangular formula for definite integrals, 95 Rectangular parallelepiped, volume of, 8 surface area of, 8 Rectified sine wave function, 172 half, 172 Recurrence or recursion formulas, 101,137, 139, 147,149, 151, 153, 156, 158, 159 Regular polygon, area of, 6 cireumscribing a circle, 7 inscribed in a cirele, 7 perimeter of, 6 Reversion of power series, 113 Riemann zeta function, 184 Right circular cane, frustrum of [sec Frustrum of right circular
cane]
lateral surface area of, 9 volume of, 9 Right-handed system, 118 Rodrigue’s formulas, 146, 151, 153 Roots, of complex numbers, 22, 25 table of square and cube, 238, 239 Rose, three- and four-leaved, 41 Rotation of coordinates, in a plane, 36 in space, 49 Saw tooth wave function, 1’72 Scalar or dot product, 117,118 Scalars, 116 Scale factors, 124 Schwarz inequality [see Cauchy-Sehwarz Sector of circle, arc length of, 6 area of, 6 Segment of circle, area of, 7 Segment of parabola, area of, 7
29
inequality]
arc length of, 7 Separation of variables, 104 Series, arithmetic, 107 arithmetic-geometric, 107 binomial, 2, 110 Fourier [sec Fourier series] geometric, 107 of powers of positive integers, 10’7, 108 of reciprocals of powers of positive integers, 108, 109
I
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a
N
D
E
2
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