Theory of Permutable Functions

/

^f J

i

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VANUXEM FOUNDATION

THE THEORY OF PERMUTABLE FUNCTIONS BY

VITO VOLTERRA PROFESSOR OF MATHEMATICAL PHYSICS IN THE UNIVERSITY OF ROME

LECTURES DELIVERED AT PRINCETON UNIVERSITY OCTOBER,

1912

PRINCETON UNIVERSITY PRESS PRINCETON LONDON HUMPHREY MILFORD OXFORD UNIVERSITY PRESS :

1915

Published April, 1915

LECTURE I

LECTURE I.

I

We shall begin with quite elementary and

general notions. First, let us recall the properties of a

This operation mutative, that

is

sum

both associative and com-

is,

= tt-r(i + ?) (a + l) + c and #

+i=I+

tf

.

Now we

can pass from a sum to an integral by a well-known limiting process. For the sake of simplicity, definition of

which

we

shall

use of the

Riemann: Given a function /(#)

defined over an interval ab,

is

divide the interval ab into h

make

......

hn

we

.

n parts

we

hi,

sub-

h,

Corresponding to every

Jt B

,

in-

then take some value f t of /(#) lying between the upper and lower limits of f(cc] onh,, and we form the sum terval h

l

4

THEORY OF PERMUTABLE FUNCTIONS

Now

suppose we allow h i9

become

hz,

&3 ,

h n to

Then if a unique sum regardless of

indefinitely small.

approached by the the way in which the subdivision of ab we have limit

is

Urn Si/j

Necessary and

7*

is

made,

= z

sufficient

for

the

known.

In

conditions

existence of this limit are well particular, if the function /(#)

is

continuous

over the interval ab or has at most a finite

number

of discontinuities, the limit

and hence

also the integral exists. 2.

Now let us

This operation that

is

form the product

is

associative

and commutative,

to say, (aV) c

= a (be)

and ab

= la

.

not worth our while to consider the operation which could be obtained from a product It

is

by a limiting process such

as the one

employed

LECTURE in defining

an

5

I

We

integral.

should be led to

logarithmic integration. 3.

cess

However, let us consider a limiting prowhich leads us to something more than

these elementary operations. Let us choose a set of numbers i,

s

= 1,

2,

g

.

.

,

which

m

isJ

where

be written in

may

an array

m lg

22

.... ....

mg2

....

mgg

mn m m m 2l

m

ffl

12

and numbers n is where ,

that

We

i9

$=

.

.

m$g

1 ? 2,

.

.

g

,

is,

n Ll

n 12

n gl

n g2

.

.

....

n lg

nffg

.

then consider the operation (1)

which we type.

shall call

XT,

^

w^ r

composition of the second

This operation

is

associative, for if

we

THEORY OF PERMUTABLE FUNCTIONS

6

also introduce a set of i,

s=I,

2

.

.

I

(j

?

numbers

jp l89

where

and write the sum

I

the expression which

we

thus obtain

is

equiv-

alent to either of the forms

which proves that the associative law

is

sat-

isfied.

Making use

we

shall

of the notation

have

which ma}r be written without the parenthesis, thus,

K

n > p\s

The commutative law

-

will in general not

satisfied. When it is, the quantities under consideration are called permutable, and we

be

have

LECTURE

=

(m, n) Lr

7

I

m) ir

(n,

.

We

can at once give an example involving permutable quantities. All that is necessary is

to consider (m, m) ir

(m, m,

and

m) ir

so on.

which majr be written (m 2 ) lr which may be written (m*) tl

And it is clear that k = (m k m h (m \ m

.

j

) ir

since the associative 4.

We

) ir

,

law

,

is satisfied.

shall consider also another operation

similar to the last,

namely s

(2)

I

2 m lh 7l

%

s

H-l

which will be called composition of the

first

The sum

type. (I) previously considered reduces to this one if we suppose that the num-

bers are zero unless the second subscript is In other words, we greater than the first.

have in

this case

m^ m

000 000

1B

.

.

.

... .

.

.

m lg

mg .

THEORY OF PERMUTABLE FUNCTIONS

S

Let us represent the sum

O

?

W]

This expression vanishes equal to

i

+

[[m, n]

we

shall

by

ts

s is less

if

Moreover,

1.

(2)

jp]

if

we

write

lg

have ,

which vanishes 2

+

2,

than or

and so

In general,

if

s

is

less

ia

,

than or equal to

on. it is

not true that

[m,n] l8

=

[n,i] t8

but when this condition

is

satisfied,

quantities are called permutable. guish this sort of pennutability

which we defined in section

we

the two

To

distin-

from that

shall

say that

the old are of types one In other words, if respectively.

and two

the

3,

new and

ff

(3)

2^

m th

n hg

we have pennutability whereas

9 = 2* n

lh

m hg

of the second type,

if

5-1

%h *+l

Sl

m ih n hs = 2^ i-rl

n ih

m hs

LECTURE

we have permutability Clearly, if we put

I

of the

first

type.

[^
,

an example of permutability type like the one mentioned in the

at once obtain

of the

first

last section,

Nevertheless,

other example which 1. suppose that n lh

shall give an-

of interest.

is

=

we

Then

Let us

the condition for

permutability becomes s-1

i

,s

2 h M =2fc M hs

.

th

1+1

Putting

s

= + i

i,

?%, H-l

we have

2,

1}l

and putting

i-rl

i+l

5

= +

+

MI, i+2

^ ^i+l,i+2 8,

2

?

we have

= ^i+l,

+m

+3

i+2, i+3

9

whence W^,i+2

and

so on.

Owing

= W+l, l+

to the above,

m rjr + g for all values of

8

r, s,

= m ltl+g

and

g.

From

lows that the matrix of the m*s ing type

:

we have

is

this

it

fol-

of the follow-

THEORY OF PERMUTABLE FUNCTIONS

10

#j

a>

^3

a

//

L

U

2

^i

...

tf^

.

.

.

a g _9

.

.

.

tt%

0000...^ ...

The law

for this matrix

m

is

may

wis

i

.

be expressed by

)

which puts into evidence the fact that the value of m, depends upon the difference between the two subscripts s

and

i.

Now

what do we find when we pass to the limit by a process analogous to the one employed in the integral calculus in going from a sum to an integral? We there passed from a set of quantities /i, f-2 /s, fn 5.

,

,

with single subscripts to a function f(oo] of one independent variable oc3 the variable taking the place of the subscripts. Here, we have instead a set of quantities m l8 with double subscripts;

hence,

we must

function of two variables

replace

them by a

LECTURE

11

I

where the two variables take the place of the double subscripts. Moreover, we also have another set of quantities n s which must be replaced by a different function of two variables (j>($,y).

2 m %h n hs must

Finally,

be replaced by

7l

We thus obtain two operations Composition of the

first

:

type:

Composition of the second type:

The

condition for permutability of the

first

is

type

,

y)

d

for permutability of the second type,

// a

The 6.

(x, f)

* (g, jj) d

=/ V

(*,

a

associative property

is

I)

/ & y)

always

*t

.

satisfied.

Let us begin by examining peraiuta-

THEORY OF PERMUTABLE FUNCTIONS

12

bility of the first type.

facts are here

The most important

summarized

:

All of the functions which can be ob-

1.

tained by the composition of permutable functions are permutable with one another and also with the original functions.

All of the functions which can be ob-

2.

tained by the addition or the subtraction of permutable functions are permutable with one

another and with the original functions. 'Now, let us see how the following problem

To

may

be solved:

tions

which are permutable with unity. can readily solve this problem if we

determine

all

the func-

We

re-

a

question which has already been answered. For before passing to the limit, we call

saw that

if

the functions

m

ls

were permutable

with unity, the condition MIS

was

satisfied.

= m,^

Now since,

in the limit, the sub-

scripts are replaced by the variables we are led to infer that

This

we can prove immediately.

x and y9

For

if

LECTURE

13

I

f"f(z, it

must follow that %

y

3

Hence $ and

x

J

l

'

JJ

.

/ are of the forms $>(y

00}

and

f(y OB) respectively. Moreover, all of the functions of the type

f(y

oo)

are permutable with one another; for

as can be verified at once.

type f(y

oo)

ever, it

we

it

functions of

form a group of permutable

functions which

have called

The

of especial interest.

is

the group of closed cycle.*

shall

We How-

not go into an examination of

here. 7.

We have used

representing

The

the

several different notations

operation

of

composition.

simplest scheme where no confusion with

multiplication write

is

f

liable to arise, is

merely to

or

*Leons sur les equations integrates et integro-differentielles. 1913. P. 150. Paris: Gauthier-Vfflars.

THEORY OF PERMUTABLE FUNCTIONS

14,

to represent the resultant

of

two function

/

and



.

of the composition But in a case where

confusion might arise, we star over the letters, thus

may

place a small

f'4--

We may also put the letters in square brackets [f,
we wrote [M ? n\ Jik to represent the composition of the quantities m th

just as in the above

and

m

To

lk

.

indicate the composition of / with itself,

the composition of the resultant thus obtained with f, and so on, we shall write *

*

Jf'2

respectively,

plication

is

J

and

Jf8 if

?

no confusion with multi-

liable to arise,

the small stars

we may even omit

and write

Jf2 ? JfS ? 8.

The notation

in certain cases

particular examination.

demands

Thus, to indicate the

product of a constant a by f, we write af; and if 6 is also a constant and a function which is

LECTURE permutable with

/,

then b

15

I

(f>

is

permutable with

af and the composition of the two gives us *

a 6f

Moreover,

if

* .



we have two polynomials made

up of permutable functions with constant coefficients, these polynomials will also be premutabie with one another, and to composition

all

that

the same rule which

is

is

necessary

effect their is

to apply

used when polynomials

are multiplied together. But if a and 5 are constants, then a

and &

+

<j>

will not in general be

with one another.

Nevertheless,

+

/

permutable

we

shall ex-

tend the definition so as to have in this case

= ab + a (a+f) (b + ) 9. Before going further, what takes place

let

us

consider

in the case of composition

of the second type. All that we have said above concerning permutability of the first type can be established for permutability of the second type barring the remarks on permutability with unity.

With

this exception, all of the properties just

mentioned may be extended to

this case at once.

THEORY OF PERMUTABLE FUNCTIONS

16

Let us pass

10.

in review

some of the most

interesting properties which can be derived from the operation of composition.

We

once more to the

shall return

finite case

and

consider the operation of composition for the saw above that if we comnumbers is

m m

We

.

j

s with the ris, the resultant thus posed the j obtained with the p s and so on, then after

s--I compositions, the resultant would be zero. In a like manner, all of the symbolic powers

m

of

i8

beginning with

+1 [m*"*"

] te

have a value

zero.

Having seen lytic

this, let

us consider any ana-

function 00

2n

- 7l

A ^n*

which converges within a certain let

circle,

and

us write

2, This

new

A n [m^ *. ls

expression

rational function of

#,

evidently an integral that is, a polynomial.

is

Moreover, this result may be generalized. Consider the analytic function

of

more than one

We

LECTURE

I

variable,,

and write

17

again obtain a polynomial.

Now, what does become when we pass by 11.

the

above

theorem

a limiting process to the composition of functions? This we proceed to investigate.

Consider

We

shall

prove that

expression

whatever

To

is

may

prove

always an

this

theorem, \

is

M

is finite,

some

\<

this

we

f (x,

y

z, )

.

notice that

M

n\<-Jn

R

quantity and where less than the radius of convergence of the is

finite

Moreover, let /x be a quantity which larger than the absolute value of /. Then

series. is

y)

entire function of

be the absolute value of

\A A

where

if f (x,

we

shall

have

THEORY OF PERMUTABLE FUNCTIONS

IS

H~&

= t# J

d

*

3

\

and

so on,

whence n Li

which proves that the all

of

*

J

values of 2 and

is

xY~

t

M

l

convergent for

series is

thus an entire function

z.

This theorem

may

also

Let

be generalized.

us consider the series

which represents the expansion of a function F (z l9 2 83,. .% e ) about a point, and which ,

converges If the absolute values of z^ z 2j z Bj z e do not exceed certain limits. Then the series

F=2 -n

o

Is

ij

.

.

.

^ 2

A

}i,

i

t

e

i

i

e

ff-

fi

f^

si

l

.

.

.

f

fe sf e

.

.

.

0je

o

always an entire function of g^ z z , % 2j

.

.

.z n

.

LECTURE

The proof

made

is

in the previous case.

/ e are each less

if fij /2j /3>

19

I

than

p,

Thus, in abso-

lute value, then

/e

1/1

...

t

l

and hence the theorem may be

verified im-

We

mediately. may also demonstrate another property besides the one just shown. Indeed, we have up to the present regarded the function

F (z^.

s3 ,

4,

and

.

x, y] as a function of z l9 z 2y

.z e)

.

.... z e , but

Regarded

y.

is

it

also a function of

as a function of these

variables, the function

oo

two

permutable with the This may be seen at

is

functions / x ____ f e once; for owing to the uniform convergence of the series, the operation of composition .

may

be performed term by term, and since is permutable with /i,

each term of the series /2> /a?

.

.

.

.

To sum

fe

5

so also

up,

... is

must be the sum.

we have

.,

the following theorem:

...

e

...

the expansion of an analytic function about

a point, then

THEORY OF PERMUTABLE FUNCTIONS

where

tions^ is tf/zd

fsj-.-.fe are

f2

permutable func.z e? an entire function of Zi, z^ z 35

fi,

.

x

as a function of

0?zd

y

with the functions f i? f 2 fs Now If in (z^ z 2j ---- z e Zi

=

z^

which 12.

=

Is

z

ze

=

convergent for

1,

all

.

permutable

fe.

,

F = ....

is

.

\

we

y] we put obtain a series oc,

values of the fs.

The theorems which we have been

de-

riving above suggest a method for investigating to a considerable extent the properties of

permutable functions and for carrying out the operations of composition. Thus, let us consider any analytic expression

ffr...*.) which can be expanded about the point %2

=

ZB

=

4

z 2j Z B , ---- z e In the series

nr .

__ If

we

z-i

g e =: o in powers of % 19 replace z^ z 2j z & , ---- z e

by fl9 /2 /8 ---- f e respectively, and write the operation of composition wherever we previously had multiplication, we shall ,

,

LECTURE always obtain a

I

which converges for all .... f e9 and which repre-

series

values of f l9 / 2 / 3 sents a function permutable with ,

,

f

e

We may represent

.

it

/i, / 2 , f $,....

by

Thus, every algebraic expression takes on a

new meaning for For example,

the operation of composition.

1+0 a series which converges within the unit But if we write cle.

is

cir-

we

obtain a series which converges for all values of / and which is permutable with /.

Consequently, a meaning has been ascribed to the expression on the left hand side of the equation.

Moreover,

and

if

we take two

expressions

THEORY OF PERilUTABLE FUNCTION'S

% > .,-y~-r/"-... *

,

make

then to

the composition of the

hand members,

It

Is

two

left-

only necessary to apply

the rules for finding their algebraic product

and we

shall

have

Hence, all the rules of ordinary algebra remain valid when we pass from the field of multiplication to the field of composition.

Some rived

of the consequences which can be de-

from

13.

this fact will

Now

let

be seen shortly.

us see what takes place for

the second type of composition.

Let (4)

jFte)

= _M_

be the ratio of two entire functions 1 4-

=

$

(z)

which are such that

<(0)

0.

Then we

shall



have an expansion

(z)

=

and

LECTURE

23

I

=A

lS + A 2 2* + AS # + F(s) which converges in general within a certain .

having the point z

circle

=

.

.

as center.

ISTow let us consider the expression FL*(Z)

We

shall

tient of 14.

=A

+A

h* %

prove that this

+

2

2

(m

)z

new

,

.

.

.

series is the

quo-

two entire functions.

Let us

write

first

= Si* + R <(*)

+

2

...

and determine a quantity

h,(*)=S m l

We say that For

<

is

l8

(3)

+

*

is

+

*(?)**

...

an entire function of

z.

^ be greater in absolute value than tn ls then have (3)

let

We

JMoreover, \B,

+

\tiz

\B, \p* ff #

converges for is

all

values of

|z? 8 2,

\(f/^+

the same reason, $(e)

then the i/f

= Ci s +

if

Cs

+

..

.

series

u (0)

=dm

is

z

an entire function.

+

.

.

.

and the theorem

proved.

For

is

+

Ct(m*) ts z

2

+

.

.

.

THEORY OF PERMUTABLE FUNCTIONS

54

Bearing these facts in mind, let us consider the system of algebraic linear equations

X + I ^ Xa = j*

d, *

t8

= 1, 2,

1

If

we

X

unknowns

replace the

ls

9)

by

-

F

ts

we

can verify without trouble that these equa-

But

tions are identically satisfied.

if

we

solve

X

for the unknowns is in the above system, the solution will be expressed as the quotients

of rational entire functions of

hence the quantities functions of It

is

X

19

\ft is

and

<j> is

and

are quotients of entire

z.

clear that the determinant

which con-

the denominator of these quotients cannot vanish identically, and hence the theostitutes

rem

is

proved. not difficult to generalize this. Thus instead of the quotient (4), let us write It

is

where

and

<^

variables

are entire functions of the

\j/

z 2j

z^

z e which vanish for

z^

= 3 = 2s =...** = 0. Zl 2

X7/^ JL

is

\

iCi

.

.

.

JC

g

\

"O

)

<^r

the expansion of

...

v* ^/

If

A

XI

^i.B.^fi
*

F about the point for which

LECTURE ^2

1

(?. Pis\*l

~

==

3

=

....

}=^^.,.Z,ri ^A %e)

?

where

m

n

,

the function

,

....

F (z

l9

fl>

q

z2 ,

Q,

z
y

^

S5

I

(m ll wit \m t%e

and l*

we

if r/

...(/

l

write

?' e e )ai...Z

e\ ?

l

il

are permutable, then % e & y) will be .

.

.

.

9

\

the quotient of two entire functions of z^ z& %$j .... % e .

To make

the proof in this case, it is only necessary to repeat the argument given above. may add that F^ is permutable with iS7

We

Wts,.

15.

m

?

Let us now pass

to the limit, that

is,

us consider permutable functions of the second type*. All of the theorems remain

let

valid.

In other words, we have the theorem

that

F&, ...*

|

where f^

x,y)

=2

.

.

.

24

lt

.

.

^tf

.

.

./>

^

.

.

.

*,S

f 2 , fs? ---- fe are permutable func-

tions of the second type, entire functions.

is

the quotient of two

Also,

* To indicate composition of the second type, we shall use of a double star thus:

make

^6

is

1z,

THEORY OF PERMUTABLE FUNCTION'S a function which

is

permutable with

/i,

/2 ,

..-./,.

We

study some applications of these fundamental theorems in the second lecture. shall

LECTURE

II

LECTURE I.

We

shall

II

begin by classifying integral

equations into several categories. us examine those which are linear. ones

plest

following

which

we run

across

First, let

The

sim-

are

the

:

(1)

known kind.,

as Volterras equation of the second

and

(!')

/ (y) +/o

/(*) F(x,y] dx

= i(y),

Fredholm's equation of the second kind.

We shall also consider certain other kinds further on.

Let us look

at equation

(

1

)

.

If

we

multi-

ply both sides by <& (y>z) and integrate with and we respect to y between the limits ,

obtain ,e}

dy

THEORY OF PERMUTABLE FUNCTIONS

30

now

If

(A) it

the function

*

be so chosen that

f^Fiz.tf

will follow that

|>, s)

which

is

f\y dy Far.

*)

l(*,

}

*

s)

reducible

fix) dx

by

=

A?)

J^ (y) Q(y, z) dy

(I) to ,

so that the difficulty has to the solution of

z) dy,

been narrowed down

(A).

In the symbols for

composition of the first kind, this equation be written as follows:

ma

(2)

If (!')

the

$0^) + F(x.y) +

J>(x,y)

=

.

we apply a similar argument to equation we find that the solution will be given in

form

where ** **

2.

Let us

be solved.

first see

If

we

how

write

equation (2)

may

LECTURE

we

31

II

shall obtain

__

the solution being valid

But suppose we

Then, in

this case,

equation

Hence we

(2)

shall

|

I.

write

we have an expansion which

converges for all values of fies

<

z

if

F and

which

satis-

by what we have proved.

have

and the integral equation (2) is solved. If we replace F (oc, y] by u F (at, y) in equation (2)

we

<(#, y]

which

+ u F(x, u] +

series is

always an entire function of u.

Turning to equation (2 by uF as in the previous

3.

F

obtain

then have

X

), let

case.

us replace

We

shall

THEORY OF PERMUTABLE FUNCTIONS

32

3>

u

=

F - uF

.

theorem of can be ex-

as a consequence of the last

and the

-

first

we have

lecture,

pressed as the quotient of

that

two

3>

entire functions

of u.

As

4.

soon

we have

as

the

stated

fun-

easy to see that they are only special cases of other

damental problems in

classes of

this

problems of a

form,

it is

much more

general

nature.

Indeed,

let

tion F(ZI, z z >

us consider any analytic funcz n ) whatsoever and write 3, .

.

.

.

the equation (3)

Furthermore,

JK^^-.-.^O. let

us suppose that this equation zn 3 2 0, and

is satisfied by Zi = z =

.

.

.

=

=

Si, .

.

us regard z n as a function dependent upon If the point z I %2 ^3 3? 2? 2n-i-

let

.

=

.

.

= zn =

is

not a

critical point,

velop z n as a power series in z i9 % 2

,

-

-

=

we may de% n-i and

the expansion will be convergent within region.

We

shall thus

.

.

.

have

== V

'

=

some

LECTURE

Now suppose we replace z

33

II

z2

i9

.

,

.

.

z n in equa-

.

tion (3) by the permutable functions /i, f 2) .... f n respectively and regard the operations .

as compositions of the first type. terms of our notation, we shall have

The equation which we have

.

Then, in

just found will

no longer be algebraic or transcendental but will be an integral equation, since the operation of composition is an operation of integration. Nor will the equation in general be linear as was equation (2), but of any degree

whatsoever. the its

unknown solution

function,

we

we regard

if

Nevertheless,

f

n

as

shall be able to find

by the same process which we used

in solving equation (8). Indeed, it necessary to replace z i9 s 2? 2 3? .... %

is

n

only

in the

by f^ f^ f z , ---- f n respectively and to treat the operations as operations of comIn this manner, we find position.

series (4)

(5)

An

/n = SS...Sl

l

...

v ^/ *.../,,V. l

interesting fact to be noticed

whereas the expansion (4)

is

is

that

in general con-

THEORY OF PERMU1ABLE FUNCTIONS

34

vergent over a limited region only, the solution (5)

valid for all values of f^ / 2 ,

Is

.

.

.

f n-i>

.

Evidently problems of integration are of a more complicated nature than algebraic or transcendental problems, yet

we have

the sur-

prising and interesting result that the solutions of the former are much more simple in

the sense that the regions over which they are valid

infinite.

is

We

may

also replace

z^

^

z.2 ,

% n-i

-

.... 3 w _i/ n _i res-

In equation (8)

by zj, Zsf 29 write and the equation pectively ^(*l/l, *8/8,

Then Jfn

'

-

-

*n-l

/-!,

fn)

=

.

the solution will be

= VZ

which

.

is

.

,

V 2,

J A.

^_

^^^ ^

-.11 .

.

.

A ... /_! ynl

%n-\ *_! fi

of the form

---*n-lK^) The series will be an entire function of z^ % 2j z^ .... zn^i and with respect to os and y y //ifel

-

?

,

f n (%ij

Z& 2 8 ,.

.

.

.s n

! j

#,

2/)

will be

permutable

with the functions / 15 / 2 / 3? .... / n _ x ,

5.

We

might

equations, as

also start with a

.

system of

LECTURE

II

35

(6)

which are

=

....

satisfied

=

s nt

0.

when

Now

^ =

let

^=z = 2

u2

us suppose that

= we

can define

(7)

Up = 2 2...

z

l

as implicit functions of * 15 ^ 2?

have no

critical

Then

we

if

where / Xj

*

l

*

^

..

n

____ a n which

=z =

____ % n point at z 2 write the integral equations

= o.

/ 2 ^ / 3j ---- / n are permutable functions, the solution of the system will be

THEORY OF PERMUTABLE FUNCTIONS

36

and the functions thus obtained functions of z ly z 2 r 3s .... z ,

/ij /2

for all values of

Moreover, these solutions

/*

/c>

tl

will be entire

will be pennutable with the given functions, All of the equations which we have been con-

sidering involve only integrals for which the limits of integration are say, they are equations

x and y;

with variable

Let us see what the situation

tions (6)5 (7) let

when

is

Returning to the

are constant.

we

shall

that

is

to

limits.

the limits

set of

equa-

suppose that the solutions

are quotients of entire functions.

Then

us examine the integral equations *

**

**

*

-

1

.

jy.?i /;,

where f l9 / 25 /3

.

.

s n fn

,


.

.

.

$p)

=

=o

f n are permutable functions of the second type. In these equations ,

the limits of integration are constant, since

we

are concerned with composition of the second type.

LECTURE

SI

II

now we put

If

the functions thus obtained satisfy the preceding system of equa-

1) tions,,

2) are quotients of entire functions, and 3)

have permutability of the second type

with the original functions / 15 / 2

Thus we

.... / n , , see that linear equations are only

a very special type of integral equations and that we can pass from their study to that of a

more general class. 6. Let us prove ties

certain important proper-

about functions which

may

be found by

a process like the one above outlined. More precisely, let us show what certain algebraic properties become when we pass from multiplication to composition.

We

shall

begin by

giving an example:

We

consider the exponential function

THEORY OF PERMUTABLE FUNCTIONS

38

z

_1 1

A

"~3

2 i

i

,

_L

rf-r^fTg-fi

Associated with

It is

...

an addition theorem

e ^^ = r ^ -

Suppose we put f(s)

= e*-l

^

.

We then have /(* + *i)

(8)

=/(^/(^J +/(^) +/(^i)

Keeping the above

.

in mind, let us write the

function

We

can see at once what the relation (8) Indeed, we have only to replace

becomes.

multiplication therefore have

that

by composition.

We

shall

is

In other words, the theorem of algebraic ad-

LECTURE

39

II

dition for the exponential function becomes for this new function a theorem of integral addi-

we have

tion as 7.

called it.*

To go from

the particular case to the

general involves no difficulty. Consequently, we may state the theorem: To every theorem of algebraic addition, there corresponds a the-

orem of

integral addition,

Thus, for example,, if we consider elliptic functions, we can pass from these to entire functions

by the process of

preceding

lecture.

To

II

of

for elliptic functions, there correspond

addition theorems for

manner,

let

the

the addition theorems

new functions.

us consider the

cr

new

In a like

function of

Suppose we examine for a moment the expansion of this function and replace Weierstrass.

u

in the expression

powers of tion.

by uf(x, y] where the

/ represent operations of composi-

The three-term equation

to a three-term equation for the *

Evans has studied

permutable

functions.

In a systematic

(Memorie


leads us

new

function

for

manner an Algebra of

Lincei,

S.V.

Vol.

1911; also Rendiconti del Circolo di Palermo, Vol. 1912.)

VIII,

XXXIV,

THEORY OF PERM UT ABLE FUNCTIONS

40

which

of the integral type since

is

we have

replaced products by compositions. 8.

What we

have said about compositions

of the first type may be repeated for compositions of the second. Returning once more to

the example involving the exponential function, let us put

This function

we

shall

W(s

- gl

is

also

an entire function and

have ;

x, y)

= W(z

\

x, y)

+

-

W(z,

\

z; y)

fr(*|*,y)fFk| a r,y),

or in other words,

+ S W( g \x,S) W(*\e,y)d. a It is

hardly necessary to prove that the above true in the general case. is

9.

A

similar theorem

may

also be stated

more than one variable hence, all the theorems about AbeHan functions may be carried over to the domain of integration by for the case of

a process like the one which

;

we have

indicated.

LECTURE Let us now return

10.

41

II

to linear Integral

As indicated, equations (1) and equations. Those of the first (!') are of the second kind. kind of the Volterra and Fredliolm types spectively may be written

re-

(9)

(9') 7

Leaving out of consideration equations which can only be attacked by methods

of a

different sort, let us consider equations

(9).

The

(9

)

may be reduced to equations of the second kind. For we can differentiate and latter

obtain

If

F (y,

y} does not vanish,

we can

by F(y,y) and get an equation

divide

of the second

kind.

If

F

(y,

equation

is

y}

is still

vanishes identically, the last of the first kind. But if

not zero, then by a second differentiation,

we

THEORY OF PERMUTABLE FUNCTIONS

42

shall get

an equation of the second kind, and

so on.

If F(jc,y) is such that F(x,os) $ 0, call it a function of the first order. If

=

and

(

x

)^0

,

we

shall call

it

we

shall

F (x, #)

a function of

sy'W

the second order,

and

order of the function

so on.

F (x

f

Hence,

if

the

y) in equation (9)

determinate, the equation can always be reduced to one of the second kind by a finite is

number solved

But

of differentiations,

and hence can be

by the method which we have the order of

minate.

F (x,

A case in point

y)

is

may

where

indicated.

not be deter-

F

(a?, OB)

is

in

general different from zero but vanishes for certain values of x. Then the nature of the

problem changes, and to solve it, new methods must be used. To develope these would lead

The solution of this question has been the goal of numerous enquiries. were the first to take up the matter and since us too far afield.

We

then Lalesco and others have studied

it.*

Instead of considering equation (9) which *See: Lalesco, Introduction d grates.

Paris:

Hermann,

1913.

la theorw des equations inteTroisi&me partie I.

LECTURE Is

of

the

first

kind,

43

II

we may

consider the

equation

where we can regard F and functions and as unknown.

*|/

as the

known

For we have a constant, when the



only to suppose that x is equation reduces at once to equation (9) If we take the equation of the first kind In .

this

that

we may

form,

is

also write

to say, the

problem Given a function

nature:

Is \jt

It

of the following

which

Is

the re-

F

and 4>, and sultant of the composition of of the factors the of one composition, given If for the moment to find the other factor 4>.

F

we were tion

to replace the operation of composithat of multiplication, the problem.

by would reduce operation;

to that of finding the inverse

that

problem which

is

we

dealing with a analogous to the problem of

is,

are

division.

Now

necessary to observe that certain conditions must be satisfied if the problem is it is

THEORY OF PERMUTABLE FUNCTIONS

44

to have finite solutions.

The order

of

$ must

F

be greater than the order of by at least to a higher when vanishes oc=y, $ unity. For Is of order m and ^ is of order than F. If

F

$ must be of order m-n. Moretwo cases may arise according as the

order n, then over,

functions

F and

are or are not pennutable Clearly In the latter case,

\b

with one another. <E>

cannot be pennutable with F, otherwise the

two would But if F and $

resultant of the composition of the

be permutable with either. * be permutable with JF

are pennutable, will

and

i/f'?

We shall prove that this property realized.

In

fact,

F^F =

ijf

is

actually

we have

F,

FF =

Hence

and

since this integral equation has but

one

solution,

<$>F

and the theorem II.

Is

=F

,

proved.

Furthermore,

when the problem

of

LECTURE

45

II

linear Integral equations of the first kind has

been put

in the

form

F <&==$ other problems suggest themselves at once. Thus, if F, and ^ are known functions, we

may

set the

problem of determining a quantity

such that (10)

or

PX+X3> = $;

again the following problem: given the

F

known

F4

2j F^ quantities J?lf calculate a quantity $ such that

(11)

F,

The above



+*F

are

2

-f

^3

*

and

,

F = 4

*|/

to

.

\ff

new equations which up

to the

present have never been studied and with which we shall now concern ourselves. First, let us consider equation

we can

write

X,

Let us put

(10) which

THEORY OF PERMLTABLE FUNCTIONS

46

Then we

shall

have

where we have put

F = l 1

From

the

first

c^ ,

c

-

3> ^2

.r

equation,

we

= c^ r

.

//

derive

and from the second

cy

where / and ^ are two kno\\Ta functions which one can obtain from F and and where

is

the

known

Then by subtracting second equation from the first, we have at

also a

once

function.

LECTURE

(.y>y)

^y

F\x.'x)

47

II

dx

and Integration by parts gives

Thus we

are led to the following result

:

To

solve the Integral equation (10) we must solve the problem which presents itself in the shape

of the last equation. This problem Is nothing more than the integration of an Integra-differential equation.

Indeed, equation (12)

of the type of an integral equation

is

both

and of a

differential equation.

The above problem admits of a solution, but we shall not go into details of the solution. The interesting point to notice Is that integro-differential equations arise in a great

THEORY OF PERM UT ABLE FUXCTIOXb

4S

We

have examined these variety of problems. equations in a number of forms and have made a particular study of the Integra-differential equations of the second order and of the ellip-

or hyperbolic types which arise In connection with certain problems of mathematical

tic

physics.*

The problem we were considering

is

of a

It Is of the first order, and different type. since two dependent variables appear, it cor-

responds to problems Involving partial derivaThe case of equation (II) may be

tives,

handled in a similar manner. 12.

We

wish to demonstrate certain inter-

esting results which are closely connected with the problems we have been discussing. Let us go back to equation (9). In certain

equation has a finite number of solutions, while in others the number of solutions Is infinite and the solutions Involve an arbitrary cases, this

function.

To

see this,

we need

only to consider the

equation *

Lemons sur

1913.

les fonctions

de lignes.

Paris: GantMer.Villars.

LECTURE

what conditions x = the only solution and under what conditions

and is

49

II

to determine under

solutions other than

exist.

To

simplify matters, we shall assume that and are of the first order the functions

F

and

shall determine

under what circumstances

the equation has a solution of the first order. Suppose we write our equation in the form

r

X J Jr>(*,|)x&;/)#+

Then by

X

differentiation with respect to y,

we

have ^

+f x

and when

as

=

x(*>

**&*)#

=

Q

y,

which gives us a necessary condition. Moreover, by suitable transformations of a simple sort, we are always led to the case where

= - *(*, *) =

(12)

F(x,

(120

^fos)^^,^

x)

where the subscripts

1

1

,

and 2 denote

,

partial

THEORY OF PERMUTABLE FUNCTIONS

SO

differentiation -with respect to

pectively; that

For we can

Then

we

if

we

x and y

res-

is

first

write

take

clearly see that the equation

(B) becomes

where

Hence we can suppose

at the outset that con-

ditions (12) are satisfied.

The above having been

established, equation

LECTURE (B)

may

If

we put

we

shall

/3

51

be written

have

But we can make use and

II

of the arbitrariness of a

to choose

=F

F\(x, x)

f

%(x, x)

=

Qf&x, x]

= ^2(37, x) =

,

which shows that we can always assume that 7

condition (I2 ) is satisfied. Now let us write

^= - /pj a;

We shall have

52

THEORY OF PERMUTABLE FUNCTIONS

whence we derive X'-''.

U)

=

?

^

/-^ ^ - A /,<-*: f)-, .

.

/

where

and therefore

(see

}\(z,

a:)

Lecture II,

=

<^ 2 (,r, .*)

=

1) .

Hence, integrating by parts, we have

where

and therefore

LECTURE

53

II

This integro-differential equation

may

be in-

tegrated.

If

we

write

we have (16)

where

$(x,y) is

= 6(v) -

an arbitrary function and v

~~ = %-ry

9

The

solution of the equation (16) is obtained the method of successive approximations. by Applications of the above will be brought

out in the next lecture.

LECTURE

III

Ill I.

We

of the

We

shall begin

work developed

with some applications in the last lecture.

have solved the problem of finding the ?/) which satisfies the equation

function x (#

JV(s,

|)

x &#) & +/*Xto ^ * *'^ /7* = (

on the hypothesis that order.

Now

and

'

are of the

first

suppose we put (*,!l)

Then

F

=

-F(x,i/)

.

the condition <(#, #)

+

jFfo x)

=

clearly satisfied, and hence we shall be able to calculate all of the functions x (^ y) which is

satisfy the relation

in

other words,

all

of the functions which

have permutability of type one with a given function.

However, in the last lecture (12) this problem was solved only in the special case where the given function is of the first order. If the function

breaks down.

is

of higher order, the

method

THEORY OF PERMUTABLE FUNCTIONS

5S

We

have seen that the problem

may

be re-

duced to the solution of an integro -differential equation of the first order. If the given function is of the second order, the integro-differential equation which we must solve is of the

second order and admits of a solution.

An

arbitrary function always enters in. As we increase the order of the given func-

problem becomes more and more complicated, hence we shall not go into details on tion, the

we

this question as

should be led too far

afield.

In the general case where the functions are analytic the question has been answered by

M.

Peres.*

2.

We

wish to present some of the prop-

of permutable functions. The very method which enables us to calculate all of the erties

functions that are permutable with a given function also leads us to the result that if

F

and

first

$

are permutable and

if

P

is

of the

we must have <(# X) = const.

order, then

*

,

F(x. x)

We shall give a rigorous proof of this * RendlcontI del Lincei.

1913-14.

fact.

LECTURE

We

59

III

write

Differentiating with respect to y,

= $(ar,^) F(y,y) +

*(*,

I)

we have

^(1,^)

^

,

differentiating this last expression with respect to x,

and

- *(a;, x] Ffc, Suppose we put x

y~]

=

= *i(^, y) that

is

Moreover,

if

we put

F(y,y}=f(i/},

we have

+ y.

F(y,

We

shall then

have

THEORY OF PERMUTABLE FUNCTIONS

60

and consequently whence the theorem. 3.

It

sion of

a simple matter to find the expanany function \b which is permutable

is

with a function

For b

where

c

F of the

is

order.

The expression

a constant. i/rU\^i

~c

will be of higher order

able with

first

the last theorem

F

F(x,t/)

l

than

F

and permut-

.

Now by one of the theorems which we proved in the last lecture,

we may

write

where *! will be permutable with F, there will be a constant c2

*!

and consequently we

$

c*2

such that

F = F$*

shall

have

= c F+ F + 2

l

Then

2

...

If this process can be carried on indefinitely,

LECTURE

we

shall

61

III

have under certain conditions an ex-

pansion of

We

in terms of F,

\ff

P

2

....

,

a short survey of the results which can be obtained by the intro4.

shall

duction of a

we can

give

new symbol.

If

we put

write

F= i" l

where F~ and

1

^,

O^ 1 are

merely symbols which do not represent functions but which may be they did. If the functions are permutable, we can write

treated as

and

If

If

=

-!$ -1

hence the symbols

=

$-1

^ and F~

permutable. Let us assume that

We wish to determine

l

we have

~1 ?

are themselves

pennutability.

THEORY OF PERMUTABLE FUNCTIONS

63

111

other words, let

We shall then have and owing to the property of permutability, /7 (! +

2

= )

(.F-f

whence !

and we

ma

+

= (/>r (J ^ - (/V*)!/'* T

1

2

write

rte rZ^ /or the sum of teo fractions be may applied. Thus we see that we can develop as it were an arithmetic for the symbol .F" 1 quite analthat

is,

ogous to the theory of fractions. 5.

We

have seen

are of the

first

<3>(.r ?

then a function such that

I)

(

type and .*|i

T)

that

= ^I(T,

(oc^y)

if

<1>

and

\jj

if .r)

may

?

always be found

LECTURE

Hence we can

And

write

by solving the equation

x * L

we

63

III

shall

= ix

l

,

have that

*

(2)

= x'- xf. 1

Therefore the two functions

-*

1

and

*

t|i

can

always be obtained the one from the other by a transformation through the functions x or x'-

In

particular,

if

$(x,x)

we

=

1,

shall always be able to find

xtX-

(3)

The even

if

relations (I)

X and

1

and

=

I-

(2)

may

and $ are permutable.

<3>

be obtained

In

this case,

do not belong to the group of functions which are permutable with the given

In

ones.

even

x'

if

tji

We

particular, equation is

(3)

may

hold

permutable with unity.

on permutable functions to a close by extending some 6.

shall bring these lectures

THEORY OF PERMUTABLE FUNCTIONS

64

of the results which were obtained in the first lecture.

11.)

(

A function which

depends upon all the values of a certain function f(cc) between the limits a (4)

and b admits of an expansion r6

-flo-rl *J

a

"

f(2\) FI(J\) th\

~b

Jb J a

F

j.\

2

(x^

f

]

(

j:

2

}

F*( ^ 1? .r 2

conditions

certain

provided

where

f(

cfa.2

th\

)

+

.

.

.

,

a


and

)

symmetric functions.

F

z

(oc^

are

x^

satisfied;

#3), etc., are

The expansion

in ques-

tion corresponds to Taylor's expansion (or to a power series) in ordinary analysis.*

With

these facts before us ? let

/

(#,

y

\

a)

be a set of permutable functions of type one, that is of such a sort that if a be given any 0,1 and a 2 the two functions thereby obtained will be permutable with one another.

two values

As an

,

we give /U"- V/ja)

example,

which has the above properties. *See:

Lecons sur

les

equations integrals

ferentielles.

*Paris: Gauthier-Villars.

Lecons sur

les

1913.

1913.

et

Chap.

integro-difI,

VIII.

Paris: Gauthier-Villars. fonctions de lignes. II. Lectures delivered at Clark University,

Chap. Worcester, Mass., 1912.

Third lecture,

IV.

LECTURE ,

III

65

us write

let

'

f /V
*/

f(%*/J\fi) <-%

=/lr,//,a,j8)

.

?

The function f($

J

a, j8) is

y)

permutable with

the original ones. Again let us write

/U%fW(^i a ^ and is

so on,

and let us suppose that the

convergent when

|

Then

tain quantity.

is

f(tf) if

\

we

less

series (4)

than a cer-

write the series

-

a

+ Jf

Jf a f(x^j\xl

^

converge no matter what the absolute

will

it

a

value of /(#, y

\

a)

may

be.

Moreover, let us consider the series "

*(f)

(5)

+

=/(f + Jf a /(2V)

fa *Jf

*^

)

Fifa,

f(^f( xt) Fsfci, x,\

}

I) dx,

dxidxt

+

...

a,

If the determinant of the linear integral

THEORY OF PERM UT ABLE FUNCTIONS

66

equation which

we

obtain by taking into con-

sideration only the first

we can

vanish,

*(l) from

derive / (

two terms does not fr)

as a function of

equation (5), provided

!

$()

|

does not exceed a certain value.*

But

let

us examine the series

is known, we can derive f(a},y form of a series which converges no matter what the absolute value of $ (#, y %}

Then

if

<

in the

\

may

be.

This

is

we have dewe have been

the latest theorem which

rived in the field of research

developing. "

sitr Us equations integrates et integro-diferentielles. XVI. Gauthier-VUlars. 1913. Chap. Ill,

Lecons

Paris:

138153