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M

CORNELL

Wfc2

UNIVERSITY LIBRARY

Hzot

MATHEMATICS

Cornell University Library

.

/QA 401.W62 1920 A course of modern analysis; an

introduct

3 1924 001 549 660

DATE DUE

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PRINTED

IN U S A

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Cornell University Library

The

original of this

book

is in

the Cornell University Library.

There are no known copyright

restrictions in

the United States on the use of the

text.

http://www.archive.org/details/cu31924001549660

A COURSE OF

MODERN ANALYSIS

CAMBRIDGE UNIVERSITY PRESS CLAY, Manager

C. F.

LONDON

FETTER LANE,

:

E.C. 4

NEW YORK THE MACMILLAN CO. :

BOMBAY CALCUTTA MACMILLAN AND CO., Ltd. MADRAS TORONTO THE MACMILLAN CO. OF :

CANADA,

Ltd.

TOKYO MARUZEN-KABUSHIKI-KAISHA :

ALL RIGHTS RESERVED

A COURSE OF

MODERN ANALYSIS AN INTRODUCTION TO THE GENERAL THEORY OF INFINITE PROCESSES AND OF ANALYTIC FUNCTIONS; WITH AN ACCOUNT OF THE PRINCIPAL TRANSCENDENTAL FUNCTIONS

BY

E. T.

WHITTAKER,

PROFESSOR OF MATHEMATICS IN

Sc.D., F.R.S. THE UNIVERSITY OF EDINBURGH

AND

G. N.

WATSON,

Sc.D., F.R.S.

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF BIRMINGHAM

THIRD EDITION

CAMBRIDGE AT THE UNIVERSITY J92Q

PRESS.

Hoi

First Edition 1902

Second Edition 1915 7%«Vo? Edition

1920

PBEFACE Advantage

has been taken of the preparation of the third edition of

add a chapter on Ellipsoidal Harmonics and Lame's Equation,

this

work

and

to rearrange the chapter

to

on Trigonometrical Series so that the parts

which are used in Applied Mathematics come at the beginning of the chapter.

A

number

of minor errors

have been corrected

and we have

endeavoured to make the references more complete.

Our thanks

are due to Miss

Wrinch

for

reading the greater part of the

proofs and to the staff of the University Press for

much

courtesy and con-

sideration during the progress of the printing.

E. T.

G. N. July, 1920.

W. W.

CONTENTS PART

I.

THE PROCESSES OF ANALYSIS

OHAPTBB

PAGE

I

Complex Numbers

II

The Theory

3

of Convergence

11

III

Continuous Functions and Uniform Convergence

IV

The Theory

V VI

of

41

Riemann Integration

61

The fundamental properties of Analytic Functions and Liouville's Theorems The Theory of Residues

;

;

Taylor's, Laurent's,

82

application to the evaluation of Definite Integrals

VII

The expansion

VIII

Asymptotic Expansions and Summable Series

of functions in Infinite Series

111

125 .

.

.

.

150

IX

Fourier Series and Trigonometrical Series

160

X

Linear Differential Equations

194

Integral Equations

211

XI

PART XII

The

THE TRANSCENDENTAL FUNCTIONS

II.

Gamma

Function

.

235

XIII

The Zeta Function

XIV

The Hypergeometric Function

281

Legendre Functions

302

The Confluent Hypergeometric Function

337

XV XVI XVII XVIII

265

355

The Equations

386

of Mathematical Physics

Mathieu Functions

XX

Elliptic Functions.

XXI

Riemann

Bessel Functions

XIX

XXII XXIII

of

404 General theorems and the Weierstrassian Functions

429

The Theta Functions

462

The Jacobian

491

Ellipsoidal

Elliptic Functions

Harmonics and Lame's Equation

APPENDIX LIST OF AUTHORS QUOTED GENERAL INDEX

536

579 591

595

[Note. The decimal system of paragraphing, introduced by Peano, is adopted in this work. The integral part of the decimal represents the number of the chapter and the fractional parts are arranged in each chapter in order of magnitude. Thus, e.g., on 9-7.] pp. 187, 188, § 9-632 precedes § 97 because 9632




y

(

x of the

of the Z-class, or given any

member x

of the -R-class, we can always find a smaller member of the such numbers being, for instance, y and y where y' is the same

-R-class,

'.

function of x' as y of If a section

is

x.

made

in which the .R-class has a least

Z-class has a greatest

number, which

it is

If a section

is

member

A

member A 3

lt

convenient to denote by the samef symbol

made, such that the

Z-class has no greatest

member,

,

or if the

the section determines a rational-real,

.R-class has

no

least

A

2

or

A

l

.

member and

the

determines an irrational-real

the section

number]..

numbers (defined by sections) we say that x is greater the Z-class defining x contains at least two§ members of the .R-class

If x, y are real

than y

if

defining

Let

y. a, fi,

...

be real numbers and

corresponding Z-classes while

The

.R-classes.

classes of

A B 2

,

which

2

,

let

A A x ,

A B it

1}

are any

... 2

,

...

...

be any members of the

members

of the corresponding

members

are respectively

will

be

denoted by the symbols (A,), (A 2),

Then the sum (written a + /3) of two real numbers a and ft is defined as number (rational or irrational) which is determined by the Z-class (A + B ) and the .R-class (A 2 + B ). the real x

2

x

It

is,

of course, necessary to prove that these classes determine a section of the rational

numbers.

It is evident that

classes (A x

+B

*

For

if

x ),

{A 2 + B2 )

A X + BX

I,

as

I

n—*

oo

.

If the sequence be such that, given an arbitrary number N (no matter how large), we can find n such that zn > N for all values of n greater than n we say that zn tends to infinity as n tends to infinity,' and we write |

|

'

,

|

|

|

zn -> oo

In the corresponding case when

x H —>

If a sequence of real

2"11.

when

n>

n

we

say that

is

numbers does not tend

to a limit or to oo or to

— oo

,

said to oscillate.

Definition of the phrase

'

of the order

of.'

and (zn ) are two sequences such that a number re exists such that whenever n > n where is independent of n, we say that fm is (Kn/zn) < of the order of zn and we write§ If

I

—xn >N

— oo

the sequence

'

.

|

(£"„) |

K

K

,

,

^

Zn=0{zn );

1J

thus If lim (£„/£„) *

A

=

(),

=

0a

we write %n = o(zn ).

definition equivalent to this

was

first

given by John Wallis in 1655.

[Opera,

i.

(1695),

p. 382.]

t The number zero is excluded from the class of positive numbers. % The arrow notation is due to Leathern, Gamb. Math. Tracts, No. § i.

This notation

(1909), p. 61.

is

due to Bachmann, Zahlentheorie (1894),

p. 401,

1.

and Landau, Primzahlen,

:

.

..

[CHAP.

THE PROCESSES OF ANALYSIS

12

II

The limit of an increasing sequence.

2'2.

Let (xn ) be a sequence of real numbers such that xn+1 ^scn for all values of n; then the sequence tends to a limit or else tends to infinity (and so it does not

oscillate).

Let x be any rational-real number

xn > *

(i)

the value of

Or If

xn —>

(ii) (ii)

for all values of

then either

;

n greater than some

number

n

depending on

x.

xn < x

every value of

for

n.

not the case for any value of x (no matter

is

how

large),

then

oo

But if values of x exist for which (ii) holds, we can divide the rational numbers into two classes, the Z-class consisting of those rational numbers x for which (i) holds and the _R-class of those rational numbers x for which (ii) holds.

This section defines a real number

a,

rational or irrational.

And if e be an arbitrary positive number, a — |e belongs to the Z-class which defines a, and so we can find n x such that xn ^a — £e whenever n >n 1 and a + ^e is a member of the .R-class and so xn n

lt

\a- xn \a.

A

Corollary.

Example

1.

For, given

e,

(i)

decreasing sequence tends to a limit or to

when

m>n,

I

\zm

(*,n

2.

— l\n

when


i(n + 3)->x; so the series

divergent; this result was noticed by Leibniz in 1673.

is

There are two general

problems which we are called upon to

classes of

investigate in connexion with the convergence of series

We may

(i)

by some formal process, e.g. that of equation by a series, and then to justify the

arrive at a series

solving a linear differential

process

it will

tained

is

usually have to be proved that the series thus formally ob-

Simple conditions

convergent.

such circumstances are obtained in

Given an expression

(ii)

§§

for

may be

S, it

establishing convergence in

231-261. possible to obtain a development

n

S—

Rn

2 um +

valid for all values of n

,

;

and, from the definition of a limit, CO

follows that, if

it

and

its

sum

we can prove

An

is S.

that

example of

R n —> 0,

this

then the series

S u m converges

m=l

problem occurs in

§ 5'4.

Infinite series were used* by Lord Brouncker in Phil. Trans, n. (1668), pp. 645-649, and the expressions convergent and divergent were introduced by James Gregory, Professor of Mathematics at Edinburgh, in the same year. Infinite series were used systematically by Newton in 1669, De analyst per aeqiiat. num. term, inf., and he investigated the convergence of hypergeometric series (§ 141) in 1704. But the great mathematicians of the eighteenth century used infinite series freely without, for the most part, considering the question of their convergence. Thus Euler gave the sum of the series

...

i 3 + z-,+ 1 + -z + l+z + z + z + 3 z' -

(a) v '

as zero, on the ground that 2

~

(c). v

error of course arises from the fact that the series (b) converges only

and the

For the history

2301.

I.

of researches

(1)

and

|

z |

>

1,

when

|

z

|

'

< 1,

so the series (a) never converges.

on convergence, see Pringsheim and Molk, Encyclopedie der unendlicken Reihen (Tubingen, 1889).

Keiff, Geschichte

Abel's inequality \.

Letfn >fn+1 > is

when

series (c) converges only

des Sci. Math.,

A

(ft)

l+- + \,+ ... = z z' z-\

and

The

+ 22 + ^ + ... = _±_

for

all integer values

the greatest of the

Kl>

K+

of

n.

Then

2,

a nfn

\a,+ a2 +

...

+ am

^ Af, where

sums

OjI,

\a 1

+ a i +a

...,

i \,

\.

* See also the note to § 2-7.

t Journal fur Math. Corollary

(i),

i. (1826), pp. 311-339. also appears in that memoir.

A

particular case of the theorem of § 2-31

/

/

THE THEORY OF CONVERGENCE

2'301, 2 ;31] For, writing a x

m

2 n=l

=s

— I

so,

.

.

.

+ an = sn we have ,

+ (*2 — si)f2 + \ s ~ sz)j3 + + \Sm — Sm _i)/m — + Sm _! \Jm-i —Jm) + smfm(/l / 2) + s2 (/2 ~Js) +

&n Jn = S\J\

Since/!

and

+ a^ +

2

i

s n— 1

s,

(/n-i

I

.

3

•

—

,/2

17

•••

•

.

•

are not negative,

~Jn) ^ -^ \Jn—i ~fn)

summing and using

.

we j

have,

alSO

S |

m

when n =

2, 3, ...

m ^ -Aj m

m,

y" j

,

we get

§ 1*4,

m

2 anfn < 4/. Corollary.

If oj,

a 2 •• >

wi> w2>

•• are an y numbers, real or complex,

2»Aki where

A

2 \w n + 1 p

is

the greatest of the sums

2 a„

-wn + \w„

(p = l,

\

2, ...

(Hardy.)

m).

n=l

2'31.

Let

Dirichlet's* test for convergence.

2 an
fn +i >

^

0,

given an arbitrary positive number

e,

we can

find

m

f

[CHAP.

THE PROCESSES OF ANALYSIS

18 Corollary

Taking a„ = (-)" _1 in Diriohlet's

(ii).

and lim /„ = 0, f

Example

-/2 +/s -ft + Shew that

1.

/„-*-0 steadily, 2 if

.

/n sin»i0

•

test,

it

II

/M ^/n +

follows that, if

i

converges.

• •

2 sin K0 -

(u n+1 /un )

\

= 1

and then




r,

l,

to zero, and,

by

§ 2-3,

2 un does not

K= l

converge.]

Example

1.

If

o

|

|l,

so the

than the corresponding terms of the

series are less

series

00

2 n

|

2

n_1 1

but this latter series

;

is

absolutely convergent, and so the given series con-

verges absolutely.]

A

2'37.

It |

w«+i

n is

obvious that

is

|

less

and the

,

if,

greater than \u n

is

—» oo

general theorem on series for which

The

\,

On

series is therefore divergent.

In

u

l±l

£ un

For, compare the series

|

vn ;

As

\n+\

2

|

c exists

_l} = _i_ c

tends to

=

1

such that

.

with the convergent series Xv n where ,

= An~

1

~^ c

(1+ic)

=(i+iy

=i-

]

'.-

+

«

nV'

1-ic, find

m such

that,

when n > m, ,

all

I

in which lim

...,

3

n-

and hence we can

By

-2±? j

we have 1+ * c

vn

»n

-

a positive number

Ern^I

a constant

-^

i

this case a further investigation is necessary.

will he absolutely convergent if

is

I

in § 2 36 that the series is absolutely con-

x

A

if

than unity and independent

is itself less

We shall now shew that* a series u + u + u +

and

the other hand,

that in which, as n increases,

critical case is

the value unity.

1.

n greater than some fixed value r, then the terms of the series do not tend to zero as

than some number which

vergent.

=

for all values of

(when n > r), we have shewn

of n

un+l

lim

v n+i

we can

a suitable choice of the constant A,

therefore secure that for

values of n we shall have \u n

As %vn

is

convergent,

X\un

\

is

\

11

1

1 ,

1

_1_

1 °"

°~4n

Sn=V2n-

for all positive integral values of

m

;

2 '5, 2 '51]

THE THEORY OE CONVERGENCE

27

Let the sum of the terms inside the rectangle, formed by the mj-ows of the ^r^t^j^olumns of this array of terms, be denoted by m> „.

first

/Sf

If a is

number S

any arbitrary positive number

exists such that, given

possible to find integers

m and n |

$m,

"

—O m and v > n, we say* that the double series of which general element is u p> „ converges to the sum S, and we write

SJ y-,y a . =

lim

that the given double series

Since

u^ v

is

lim Stolz' necessary

and

it

=

-

2 22)

§

Therefore, by § 2 22, -

ever

>m and

p,

z>

An

Corollary.

MiM

that

;

p.

is

+p=

v

moduli, then, given

e,

and

we can

But iSp^-S^nl $£p 3 -im ,

,

n

^ +I,

we

cr,

see that

and

also sufficient;

is

— $M>M
fi and g>ra>ju, t PlQ — tm n m>fi, q>n>fi; and this

find

p such

that,

and

so \Sp

-Sm,„\m, q>n;

which

consists of terms up

2

converges so long as the representative point of

the ring-shaped region bounded by the circles

|s|

=l

and

\z ]

z

= 2.

Power-Series^.

series of the type

+az+

a in

if

u m vn be taken

by multiplying the two

z4

z3

1+2+2-2 + 23 + 24+ -'

2*6.

is

to ST.

z

lies in

\,

Therefore, by § 252,

limit.

the double series, of which the general term their

,

I

which the

coefficients

proceeding according

to

a

x

,

a-,,

a2

,

a2 z2

a,, ...

+

a 3 zs

+

.

.

.,

are independent of

z, is

called a series

ascending powers of z, or briefly a power-series.

* Analyse Algebrique, Note vn. Analyse Algebrique, Ch. t The results of this section are due to Cauchy,

ix.

THE PKOCESSES OF ANALYSIS

30

We

now shew

shall

[CHAP.

II

that if a power-series converges for any value z of z, for all values of 2' whose representative points

will be absolutely convergent

it

are within a circle which passes through z and has For, if z be such a point,

an z n must tend such that

we have

n—>oo

to zero as

I

Thus I

z

|


+ 2nw 2 > 8 > for m and n, then we can find a positive number G. debut not on z, such that 1

|

j

integral values of

pending on

&

(z

S

(§

361

_I

how small

S

therefore

;

[

|

cor. (ii)) it is

but since |

—a

f(x)

is

|

may

[f(x)~

we can

be,

_1

a}

is

find values of

x

for

x'

which

not bounded in the range;

|

not continuous at some point or points of the

continuous at

points of the range,

all

ciprocal is continuous at all points of the range (§

those points at which f(x) = a therefore /(#) range the theorem is therefore proved. ;

=a

32 example)

at

its re-

except

some point of the

;

(i). The lower bound of a continuous function may be defined manner and a continuous function attains its lower bound.

Corollary in a similar

Corollary

;

(ii).

a closed region, |

A

363.

If f(z) be a function of a complex variable continuous in

f(z)

|

attains its

upper bound.

real function, of a real variable, continuous in a closed interval,

attains all values between its

upper and lower bounds.

Let M, m be the upper and lower bounds of f(x) then we can find numbers x, x, by § 362, such that /(a;) = M,f{x) = m let /u be any number such that m < fi< M. Given any positive number e, we can (by § 3'61) divide the range ;

;

(x,

x) into a finite number,

r,

of closed intervals such that

\fW*)-f{x^)\
)

j

,

THE PROCESSES OF ANALYSIS

60 6.

If the

two conditionally convergent 2 >n=\

where r and

s lie

and

between

7.

is

r+s

Shew that

be multiplied by

if

2

v

—— '-

n»

n=1

be multiplied together, and the product arranged as in

1,

and

Abel's result, shew that the necessary resulting series

sufficient condition for the

convergence of the

> 1.

(Cajori.)

1— ^ + \ — \-¥...

the series

itself

any number of times, the terms of the product being arranged as

in Abel's result, the resulting series converges. 8.

Shew that the jth power a^sin

is

Ill

series

- and

'

nr

[CHAP.

convergent whenever q

(1

(Cajori.)

of the series

#+a

- r)
y) dx \

•

Infinite integrals.

If lim

question

Similarly the other

of the repeated integrals is equal to the double integral.

repeated integral

44.

f(x,y)(dxdy),

I

a J a

I

(

/(«)

|

called

is

an

da;) exists,

we denote

it

by

f(x) dx; and the limit in

infinite integral f.

Examples.

r^ =

(1)

/" (

2)

i ini

xdx

p_na. (

.

/

Jtf^Wf= J™,

By

integrating

(,

1 1 \ = 1 " 8(6*+^) + 2^J &?' /•CO

(3)

Similarly

f{x)dx

f° of

a

is

we is

define

/

defined as

a matter of

by

parts,

f(x)dx

F

to

shew that

mean

f(x)dx+J

tK e~ t dt

/

lim

f(x)dx.

/

= n\.

f(x)dx, In this

if

(Euler.)

this limit exists

;

and

last definition the choice

indifference.

The upper bound of f{x, y) in the rectangle A m ^ is not less than the upper bound rectangle. oif(x, y) on that portion of the line x = which lies in the Math. Soc. xxxiv. (1902), p. 16, suggests the f This phrase, due to Hardy, Proc. London analogy between an infinite integral and an infinite series. *

$,

'

70 4'41.

THE PROCESSES OF ANALYSIS

[CHAP. IV

of continuous functions.

Conditions for con-

integrals

Infinite

vergence.

A necessary and

that, corresponding to

any positive number

e,

f(x) dx

convergence of

sufficient condition for the

a positive

number

X

is

should

fa"

dx
X.

obviously necessary

to prove that

;

it is sufficient,

suppose

ra+n it is satisfied

;

then, if n

^ X - a and n be

Sn =

a positive integer and

f(x), J

we have

|

Hence, by

e.

tends to a limit, S; and then,

Sn

§ 2'22,

Sn+P — Sn \
a + n,

a+n

S-

8-\

f{x) dx

+

f(x)dx

dx J a-\-n

X.

without difficulty on comparing

§§

that a necessary and sufficient condition that

converge uniformly in a given domain

number

e,

there exists a

number

2'22

and 3 31 with -

/ (x,

a)

that, corresponding to

is

dx should

any positive

X independent of a such that'

I'x''

f(x, a)dx for all values of a in

443.

the domain whenever x"

Tests for the convergence of

There are conditions to those given in

The


*' > X.

infinite integral.

the convergence of an infinite integral analogous

Chapter II

for

the convergence of an infinite

following tests are of special importance.

series.

THE THEORY OF RIEMANN INTEGRATION

4'41-4*43]

Absolutely convergent integrals.

(I)

may be shewn

It

71

f(x)dx

that a

J

certainly converges if

\f{x)\dx does so; and the former integral J

The proof

said to be absolutely convergent.

Example.

then

is

a

The comparison

similar to that of § 232.

is

and

If \f(x)\^.g(x)

test.

g{x)dx

/

converges, then

J i

/QO

f{x) dx converges absolutely. [Note. f°° I

It

was observed by Diriehlet* that

f{x)dx that f{x)-*-0

as ^-»-oo

:

the reader

not necessary for the convergence of

it is

may

see this

by considering the function

J a

f(x) =

{n^x^n + l-(n+l)- 2

f(x) = ( n + l)i(n+l-x){x-(n + l) + (n + l)-

where n takes

all

(n +

2 }

),

l-(n+l)- 2 £x^n + l),

integral values. /-n+1

/£ f(x)dx

increases with

and

|

f{x)dx=\{n + l)- i

/

;

whence

it

follows

It

o

without difficulty that

I

But when

f(x) dx converges.

x=n + \ -\

(re

+ 1) -2

f(x) = ^

,

;

J a

and

so fix) does not tend to zero.]

The Maclaurin-Cauchyf

(II) Too

If

test.

f(x)>0 and f(x)^0

steadily,

ao

f(x) dx and 2 f(n) converge

or diverge together.

M=l

J 1

/m+l m n

m=\

The

first

»+l

fn+J

f(x)dx^ 2

2 f(m)^\

and so

fix)dx>fim + l),

1

inequality shews that,

if

the series converges, the increasing sequence

fix)dx converges (§2-2) when n—-x> through

/

without

difficulty that

/

f(m).

m=2

1

f(x)dx converges when

integral values, xf

-*«>; also

if

and hence

it

follows

the integral diverges,

so does the series.

The second shews

that

if

the series diverges so does the integral, and

converges so does the series (§

(III)

Bertrand'sX

if

the integral

2'2).

test.

If

f(x)

= O^"

f(x)dx converges when

1

),

J tat,

\


dt

treatment of complex integration based on a different set of ideas and not making concerning the curve AB will be found in Watson's Complex Integration

many assumptions

and Oauchy's Theorem. This assumption will be J Cp. § 4-13 example 4.

J-

viz.

made throughout

the subsequent work.

:

THE PROCESSES OF ANALYSIS

78

By

§ 4'13

when

integral

[CHAP. IV

example 4, this definition is consistent with the definition of an AB happens to be part of the real axis.

f(z)dz=-j

f(z) dz, the paths of integration being the

same (but

in

opposite directions) in each integral. fz /

dz=Z-

zdz

jl

J zo

=\[{

x

t-yi +i (x t + yf)} dt

The fundamental theorem of complex integration.

4"61.

From

§ 4'13,

the reader will easily deduce the following theorem

Let a sequence of points be taken on a simple curve z Z; and let the first n of them, rearranged in order of magnitude of their parameters, be called z n m (z {n = z z n+1 w = Z)\ let their parameters be t^>, 4 mi >

,

.

and

when n > jV",

that,

whose parameter

tr*™

tr

lies

between

arbitrarily small

An

4 62. -

< tr

w

r= 0,

for

S, ,

tr+1

by taking n

upper limit

;

)

-

[*/(*)

the

'

length

to the

4"7.

Integration of infinite series. shall

f/(*)

now shew

dz

that

Mi

I

all

dt

cannot exceed Ml.

if

S (z) = u

r

vergent series of continuous functions of some region, then the series

(where

+i

0) dz+

I

(z) z,

+ u2 (z) +

sum

I

J c

S(z)dz.

. . .

is

for values of z

u2 (z)dz +

a uniformly con-

contained within

...,

the integrals are taken along some path

vergent, and has for

j

d¥

of the curve z Z.

'

is

We

|

value of a complex integral.

That

to say,

N such

I

M be the upper bound of the continuous function \f{z)

is

find

be any point

£. v >r

are r positive.

(Hardy, Messenger, xxxi. (1902),

r/I_I Jo \x

e

2

p. 177.)

of the integrals

-* +

+

r

_L\^ l-e*J r>

sin (.«+**)

**•

«•

J„

(Math. Trip. 1914.)

Shew that

7.

——

dx -^

/

exists.

x 2 (sin x)^

Shew that

8.

x~ ^esinx sin 2xdx converges

I

if

a >0, ra>0.

(Math. Trip. 1908.)

J a If a series

9.

in an interval,

# (z) = 2 (c„— c^ + i) sin

shew that g

7T

(z)

is

-.

sin

(2v

+ l)

tt?,

(in

which

c

=0), converges uniformly

x the derivative of the series/ (z) = 2

irz

C

— sin 2virz.

v== i

v

(Lerch, Ann. de V&c. norm. sup. (3) xil. (1895), p. 351.)

f

Shew that

K).

"...

f

f f^-^»

and

,

f

f "...

f

^dx

2

...dx n ^"n

converge 11.

a>\n and

when

a _1 + /3 _1 + ...+X _I

•••

fa2 —S 2

fb

+...+

|

I

{/(•»>

2/

+ A) —/(a

1

,

y)}dx, where the numbers

are so chosen as to exclude the discontinuities of f(x,

range of integration w. M. A.

+

;

a lt a 2

,

...

being the discontinuities off(x,

y).]

y+h) from

the

(B6cher.)

6

CHAPTEK V THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS TAYLOR'S, LAURENT'S AND LIOUVILLE'S THEOREMS

;

Property of the elementary functions.

51.

The reader

be already familiar with the term elementary function, as

will

used (in text-books on Algebra, Trigonometry, and the Differential Calculus) to

denote certain analytical expressions* depending on a variable

z,

the

symbols involved therein being those of elementary algebra together with exponentials, logarithms and the trigonometrical functions

;

examples of such

expressions are z2

e

,

z

log

,

arc sin z*.

2,

Such combinations of the elementary functions of analysis have in common

now be

a remarkable property, which will

Take

as

an example the function

Write Then,

e

J

e

investigated.

z .

=/0)-

1

if

z be a fixed point and if z be any other point,

f(J)-f(s) _e*-er 7

z

—

—z

/

z

" —z —

^ and since the

last series in -

brackets

z, it follows (§ 3 7) that, as z

—>z,

we have

-1 —z

e*-*

2!

}

z

+

+

3!

uniformly convergent for

is

all

values of

the quotient

/CO -/(«) z'-z tends to the limit

e

z ,

uniformly for

all

values of arg (/

— z).

This shews that the limit of

/CQ-/(*) z — z in this case independent of the path by which the point coincidence with z. is

It will

be found that this property

elementary functions *

(§

The reader

;

namely, that

will observe that this is

3 -1) in this work.

Thus

e.g.

x - iy and

is

if f(z)

z'

tends towards

many of the well-known be one of these functions and h be

shared by

not the sense in which the term function |

a

|

are functions of z

(

but are not elementary functions of the type under consideration.

= x + iy)

is

denned

in the sense of § 3-1,

FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS

-

5 l-5'12]

any complex number, the limiting value

exists and'is independent

The reader

will,

then lim*^

511.

^

of

of the mode in which h tends

however, easily prove that,

—

£-i-2 is not

83

to zero.

if f(z)=

x —iy, where z = x +

iy,

independent of the mode in which A->0.

Occasional failure of the property.

For each of the elementary functions, however, there will be certain points Thus it does not hold for

z at which this property will cease to hold good. the function \\{z — a) at the point z — a, since

a-^o

and **

= a. z = 0.

when

does not exist

at the point

h\z —

Similarly

z

a it

+h

z—

a

does not hold for the functions log z

These exceptional points are called singular points or singularities of the function f(z) under consideration

;

at other points f(z)

The property does not hold good

at

any point

is

said to be analytic.

for the function \z\.

Cauchy's* definition of an analytic function of a complex variable.

5"12.

The property considered definition of

in § 5'11 will be taken as the basis of the

an analytic function, which may be stated

as follows.

Let a two-dimensional region in the £-plane be given; and let u be a all points of the region. Let z, z + Sz be values of the variable z at two points, and u, u + Su the corresponding values function of z defined uniquely at

of

u.

Then, if, at any point z within the area, ^- tends to a limit when Sx—>0,

Sy—>0, independently (where which

is

Bz

= Sx + iSy),

u

monogenic or analytic^ at the point.

one-valued at

all

we say

points of the region,

is

said to be a function of z

If the function

is

analytic

that the function

is

and

analytic

throughout the region%.

We

shall frequently use the

word function alone to denote an analytic work will be almost exclusively '

'

function, as the functions studied in this

analytic functions. * See the

memoir cited in § 5 2. The words regular and holomorphic are sometimes used. A distinction has been made by Borel between monogenic and analytic functions in the case of functions with an infinite number of singularities. See § 5-51. t

p

'

'

X See

'

'

'

'

'

'

§ 5-2 cor. 2, footnote.

6—2

V

[CHAP.

THE PROCESSES OF ANALYSIS

84

In the foregoing definition, the function u has been defined only within a certain region in the z-plane. As will be seen subsequently, however, the function u can generally be defined for other values of z not included in this region; and (as in the case of the elementary functions already discussed)

may have

singularities, for

which the fundamental property no longer holds,

at certain points outside the limits of the region.

We

state the definition of analytic functionality in a

now

shall

more

arithmetical form.

Let f(z) be analytic at z, and let e be an arbitrary positive number; then we can find numbers I and 8, (8 depending on e) such that


- oo

= e-^

,

shew

! I

that, if

X

> 0,

e-^V^ = 2\-Je-^

J -oo

whose corners are

then 2 I

e~^dx.

J

Certain infinite integrals involving sines and cosines.

6 221.

Q (z) e

— oo

By

+ l) 3

shew that

satisfies

the conditions

(i), (ii)

also satisfies those conditions.

and

(iii)

of § 6'22,

and

m > 0,

then

;;

.

THE THEORY OF RESIDUES

6-221, 6-222]

Hence

means the sum and so If

(i)

e™x + Q(-x) e~mix dx

[Q (x)

Jo

of the residues of

Q (x)

an even function,

is

Q

I

Jo If

(ii)

Q (x)

is

Q (x)

I

The

(x) cos

i.e. if

2.R'

the upper half plane

Q (— x) = Q (x),

(mx) dx

=

(mx) dx

= ir%R

iriZR'.

an odd function,

Jo

6222.

(z) e miz at its poles in

Q

2mlR', where

equal to

is

}

115

sin

'

lemma*

Jordan's

results of § 6'221 are true if

condition Q(z)—>0 uniformly

when

Q (z) be subject to the less stringent ^a,rgz^ir as \z\—>oo in place of the

condition zQ(z)—»0 uniformly.

To prove

this

we

known

require a theorem

as Jordan's

lemma,

viz.

If Q(z)—*0 uniformly with regard to &rgz as \z\—> vihen 0^arg^^7r, and if Q (z) is analytic when both \z\> c (a constant) and $ arg z ^ it, then lira (

e

|

p^-OD \J

where

T is a

Given then,

if

p

e,

>

e

Q

= 0,

dz)

(z)

I

semicircle of radius p above the real axis with centre at the origin.

choose p p

so that

\Q(z)\


p

and

^ arg z
20/7T, whenf e miz

Q

(z)

^ 8^„

dz

and

tt,

(2e/7r)

[*""

so

pe-^^dd

Jo

=

(2e/7r).(7r/2m)


0, 6 '

> 0,

dx = =-e~ a

then

cos%ax — cos 26^

^

/:o '

.

2a

dx=TT

(b

— a)

(Take a contour consisting of a large semicircle of radius p, a small semicircle of 8, both having their centres at the origin, and the parts of the real axis joining their ends then make p-*~ oo 8-9-0.) radius

,

;

Example

3.

Shew

/;o -

Example

4.

that, if b

(^+W

Shew

> 0,

C0S

that, if £

mxdx = 1[

W-

> 0, o > 0,

5.

Shew

that, if

m

""";

sin

/,o

6.

Shew

362

-»2 -™ 6 (3& 2 + a2)}.

dr=^7re~ fa

.

a > 0, then

0,

fflic

ire

7r

-"

1 '1

/

2

2^~ ^^~V'K+ a

i(z 2 + a 2 ) 2

(Take the contour of example

Example

+

2 -*

(

then

x sin a#

'

'o /;

Example

then

m. ~^ 0,

2.)

that, if the real part of

z

be positive,

-«-

0, b

> 0, gaconfc*

I

snl ( a

i

_

dx ==£„. (e«_ 1). X

,

n

fop)

1

Shew that

5.

—

!»

asin2.r

1

^ ^ (-l-oo

.]

(Kronecker, Journal fur Math, cv.)

Shew

13.

that, if

m > 0,

then 1

f

Discuss the discontinuity of the integral at If

14.

r °°

mt

sin™

/:

dt

m = 0.

A + B + C+...=0 and a, b, c, ... are positive, shew that A cos ax + B cos bx + ... + K cos kx-dx = -A loga-.81og&- ...- K\o%k.

/:

(Wolstenholme.)

/ that, if

e x{k -j

+

ti)

r

dt taken round a rectangle indented at the origin, shew

k>0, ilim

/

p-j-oo J

,

.

-P K + tl

dt =

x

>

or

x> 0, x =

[This integral 16.

Shew

is

or

known

that, if

of § 6'222

-,-,-. J

-p k +

J

J ~p

example

—

dt,

'

2,

or its reflexion in the real

dt=2, lorO,

tl

x < 0. as Cauchy's discontinuous factor.]

< a < 2, b > 0, r > 0,

I

P

x < 0), that lim p-*«,k

according as

lim p-»-oo

and thence deduce, by using the contour axis (according as

vi+

x"- 1 sin

(JaTr

then

- =K"

rdx -bx) -s i x T" r

1

«" lr

-

—

THE PROCESSES OF ANALYSIS

124 Let

17.

>

*

and

n=— co

N

is

considering

an

integer,

e-»a rf = f

(C).

-3%(

r

By

2

let

e -^

/

round a rectangle whose corners are ±(iV+|)±i, where

-:dz

and making N-*- oo shew that ,

g-3%(

fee— i

By expanding

Hence, by putting

(This result ;

is

t

=\

,

e 2 " fe

respectively

and integrating

example 3 that

§ 6-22

J -•>

(irty

420

Q—&nt

C&-H

these integrands in powers of e~ iwiz

term-by-term, deduce from

p.

[CHAP. VI

shew that

due to Poisson, Journal de VEcole polytechnique, xn. (cahier xix), (1823),

see also Jacobi, Journal fur Math, xxxvi. (1848), p. 109 [Oes. Werke, n. (1882),

p. 188].) 18.

Shew

that, if

OO, e -n*irt-2nKat

2

=t -i

7i=-oo

(Poisson, (1828), pp.

Mem.de

403-404

e

vaH

Jl+2 t

I'

Acad, des

[Oes.

Werke,

Math. cxi. (1893), pp. 234-253

;

2

Sci. vi. (1827), p. I.

g^V' COS 27l7TCtl

n=i

592; Jacobi, Journal fur Math. in. and Landsberg, Journal fur

(1881), pp. 264-265]

see also § 21-51.)

.

J

;

"

CHAPTER

VII

THE EXPANSION OF FUNCTIONS

A formula

7*1.

due

IN INFINITE SERIES

Darboux*.

to

Let f(z) be analytic at all points of the straight (t) be any polynomial of degree n in t.

let

line joining

a to

z,

and

Then

(-) m (z

t

-r

^t^

if

we have by

1,

- a) m 0("-™)

differentiation

(t)f ""» (a

"" m=l

+

t

(z

- a))

= -(z-a)

) + . F(a 2o))+... F(x)dx=a>\lF(a) ^ + -„, K „, + ^ V 2

^

/*

,

^

Rn =

This

last

'

.-^—(t) {M)\Jo fa

formula

-

?,

{F* m -v (a

2 i^ " 2

J (

If f(z)

n

Example

(a

-

+ ra) - F^~» (a)} + R n

+ ma +

,

a» oo

to)}

R m™

/_^™*

1

+ a>, a +

+

(2n+i)

term tends to zero as

last

—a

3

^7

obtain an infinite series tor f (2) If

{/"""('W^W}

i%lv ^/m>!

m=l

-

-/' (a)}

{/' («)

—

(

+ 2

[CHAP. VII

be an odd function of , /

7?

2,

9 5A2m-2

F(x)

is

analytic at

+ ra>.

shew that 9271

~2n+

1

("I

Shew, by integrating by parts, that the remainder after n terms of the

2.

expansion of - 2 cot -

2

may

be written in the form

_

/

yt+i z 2« + i

(2m)

1

!

sin 2

/"i /-1 I

02t>

(0 cos

(zt) dt.

(Math. Trip. 1904.) 7'3.

We

Biirmann's theorem*. shall

next consider several theorems which have for their object

the

expansion of one function in powers of another function. *

Memoires de

I'Institut,

(1902), pp. 151-153.

11.

(1799), p. 13.

See also Dixon, Proc. London Math. Soc. xxxiv.

7 '3]

a

is

THE EXPANSION OP FUNCTIONS IN INFINITE SERIES

Let

Suppose

also

that

(a)

cj>'

=j=

(a)

= b.

Then

0.

Taylor's

theorem

the

furnishes

expansion

$(,)-& = 0'(a)(*-a) + and

if it is

*^(*-a)» +

...

J

we obtain

legitimate to revert this series

which expresses z as an analytic function of the variable { (z) — b], for sufficiently small values of z — a If then f(z) be analytic near z = a, it follows that f{z) is an analytic function of {

(*)-&} cj>(t)-b

.*(*)-&.

27ri(m+l)

m— 1

Therefore, writing

/(,) -/(a)

"

(t- a)»+*

Jv

"2 { * m=l

+

(

6}

l7

"

-E

da"

Example

1.

da- L/

I

- b} m +"

W W Wt

J

"

[/' («)

{* («)}"]

'HO-v WViQdtdS '

7"

J

4>(t)-bj

a Jy

4>(t)-oo we may

If the last integral tends to zero as

an

1)

J y { 1,

.

/

sum 22

,

V+

2i

4 ll +xi ) \1 +«"/

42

1. .~4 2.4 2 6 \l .

member

the second

denote the

residues at

ir

ir

*

Let z = x + iy and

when O^x^tt;

analytic except at a certain

z,

be a rectangle whose corners are*

— ip,

ir

— ip,

cot (a r

— z),

in order.

—

-

Consider

;

I

f(t) cot

taken round this rectangle the residue and the residue at z is f(z). ;

(t

— z) dt

of the integrand at a r

is cr

DA and GB cancel on account of the periodicity and as p—>x>, the integrand on AB tends uniformly to I'i, while as p'—*co the integrand on CD tends uniformly to — li; therefore Also the integrals along

of the integrand;

n

1

2 * If

(I'

-

I)

=/0) +%

any of the poles are on x = tr,

taken so large that a\

,

a

(>£)

= 0.

*-»-oo

t-»-0

Therefore

t Lecons sur

an analytic function

-

- e~*

= 2

(zt) dt

§ 5 32.

integrate

=

e~ l

/,(» les

= 2 am zm + Rn

,

m=0

series divergentes (1901), p. 94.

See also the memoirs there cited.

.

7

-

8,

THE EXPANSION OF FUNCTIONS IN INFINITE SEEIES

7*81]

141

/•no

Rn
r, we

let

ze~ tz (t)dt,

I

,

\

Jo

m=o 00

where 4>(t)=

2

a n t n /(n

!)

;

this result

may be

obtained in the same way as

re=0

that of

§ 7-8.

Modify this by writing e~

l

=1-f

(t)

,

= # (f)

;

then

Jo

Now

if t

= u + iv

and

if t

be confined to the strip

— ir
?/.

That

is

to

I

THE PROCESSES OF ANALYSIS

144

M tends to a limit to zero

(§

271)

n-»oo

as

r+S

8

1+ ^and

,

so

[CHAP. VII

e

)

Rn |-»0

|

(n

if

+ 2/e

1/m

(?-+8)

?

tends

but

;

rn+l

»

2l/m> m=l

J

—xr = log(n + fj

l),

s r by §4-43 (n), and (n + 2) (n+ l)-»- ->0 when 8 >0; therefore re—>oo and so, when R (js) > r, we have the convergent expansion

Rn -+0

as

,

'W-^ + JTl + (^ + l)(^ + 2) + Example

1.

(^

(z

+ l)(z + 2)...(z + » + l) J C

2.

+

l)(0

+ 2)...(^ + n)

+ --

Obtain the same expansion by using the results 1

Example

+

-'-

Z-t

=

f1

1

—

M"(l-«Vcfo,

,-

m!J

J0

J C

Obtain the expansion 1^

_1

«y

2

a„ =

where

I

(1

«

a,

«2

2(2+1)

2(2+1) (2+2)

-(a)+/P»-)(*)}

^

,

(z-a) 2 " + 1

+

!

[S (^l)Lo'

Prove that

6.

W -f(*i)=Ci

fc - %)/'

«+c

2 (* - *i) /"

2

- C4 fo-*,)*/* (%) + ... + (-)" fe-z^ in the series plus signs

?—5) n '

If

7.

.»!

(z 2

— zi)";

ascending powers of

and # g are

values of z such that

(^.)

1

-

integers,

x 1 ^R

Cn

is

- ^i) 3 /"'

,

provided that

all

m = n are omitted from the summation.

terms for which

(Math. Trip. 1895.) 19.

Sum

the series 1

n=-q where the value n =

is

\(

&- -a — n

)

nj

omitted, and/), q are positive integers to be increased without

limit.

(Math. Trip. 1896.) 20.

If

F(x) = elo

mreo ^ m' )dx

shew that

,

F(x) = e* n=1

J, and that the function thus defined

^(-^ =

iV

\HJ

satisfies

^,

the relations

1

22

i?, (.r)^(l-^)

Z3

+ «=* + p + ^+-=-

Further,if

shew that

i^ (x)

=e

when

1

= 2sin^.

f2

f//

logfl-*)?,

/

2m

-«-$«'* oo

where

n—> oo

as

,

and the

series

2 A m x~m

does not converge.

m=0

We

now

consider what meaning, if any, can be attached to the

a non-convergent

we wish

That

series.

to formulate definite

'

to say, given the

,

,

of

,

CO

an

2

if

an converges, and such that

S

exists

»=0

re=0

when

sum

a1 a2 ..., rules by which we can obtain from them a is

CO

8= 2

number S such that

numbers a

'

this series does not converge.

8'41.

We

Borel's\ method of summation.

have seen

7-81) that

(§

r">

oo

2 an zn = n=0 fj,

Since

^

that

a>„




n,

we have j

— v~\

+ ... + an (l 1

— 2v~

l ,

...

+ an+1 {l

)

+ ...+a„(l -

J is

a positive decreasing sequence,

it

from Abel's inequality (§ 2 301) that -

n

0^(1-3+0^(1 - -~) +

---

+a

i

l

-

v

-^r)\t/

^erf, lim b^ „ = 1, and

V

2 aw

It follows

is

=&

CO

A series

when n

2

a n be a

series

which

is s

summable (C (

an = the series

^

1).

Then if

(l/n),

a n converges.

n=l * Bromwich, Infinite Series, § 122. t Comptes Bendus, cxlix. (1910), pp. 18-21. i Proc. London Math. Soc. (2), vm. (1910), pp. 302-304. indebted to Mr Littlewood.

For the proof nere given, we

are

;

8-431-8-5] Let

sn

SUMMABLE SERIES

= a, +

a2 +

...

+

a„

2 an

then since

;

157 is

summable (0

1),

we have

«=i

+ s2 +

«!

. .

+ sn = n

.

+

[s

o (1)),

QO

where

s is

the

sum (G 1)

of

X an

.

B=l

Let

and

sm

let

this notation, it

independent of

n,

and

an

Suppose

...

=n

.

2, ... n),

+tn = an

sufficient to

is

if

(m=l,

*m,

+ t2 +

t1

With is

-s =

o (1),

shew

.

that, if

then tn —*

as

j

an < \

n —>

oo

Kn~\ where

K

.

first that a 1} a 2 ... are real. Then, if tn does not tend to zero, some positive number h such that there are an unlimited number of the numbers tn which satisfy either (i) tn >h or (ii) t n h.

there

,

is

Then, when r

= 0,

1, 2, |

an+r < K/n. I

Now diagram.

plot the points

Since

Pr

whose coordinates are

tn+r+1 — tn+r =a n+r + lt

(r,

tn+r )

the slope of the line

in a Cartesian

Pr Pr+1

is

less

than 6 = arc tan (K/n). Therefore the points

P P P P P ,

Draw

rectangles as

shewn

,

is

to say O'n+k

... lt ...

lie

above the

which

— °"n— = tn + *n+i + l

+

•

2

The reader

hypothesis.

which

will be

= h — % tan 6. x — h cot

0,

of these rectangles

and the axes

£»+*; ,

n.

by using arguments employed in dealing with the former

will see that the latter hypothesis involves a contradiction

of a precisely similar character to those

y

left oi

bounded by y = h — x tan

> %h? cot 6 = lh K*

line

on the

lie

The area

in the figure.

exceeds the area of the triangle that

2,

1;

Let Pi be the last of the points so that k ^ h cot 0.

.

THE PROCESSES OF ANALYSIS

158 But

cr, l+i

|

— «

this is called the sine series.

Thus the

series f?

l

+ £

^a 1

opposite in sign

—

^

/(*) cos

O^x ^l; ^x^—

when

Z 2

,

7

n7rx

•

b n sin

—j-

K=l

J

same sum when

Ae i/ie

nirx j-

»=1

=

\la n

where

an cos

'

\^n=j

eft,

f(t) sin

^

m + am m .

m

:

where am", bm" are the Fourier constants of/" (x) and 7rA m '=

1 sinm^{/'(fcr -0)-/'(^v + 0)}, r=l w

wBm = - 2 cosmhlf (K-0)-f (kr + 0)} '

- cos Therefore

Now

~ ~m~

as rn—>oc,

D % ^n ~ "^ ~ „

/

/|

-ft-m

ttm

we

ii

-

Irf

mir {/'

j> 1J m m ~~m

l '

(tt

- 0) -/' (- tt + 0)}.

-°-m

" h ^Tft

m*

m

A

.

'

2

'

see that

A m = 0(l), '

Bm

=0(l),

"

and, since the integrands involved in a m and

b m " are

bounded,

it is

evident

that

a m "=0(l),

Hence

if

A m = 0, Bm = 0,

6

m" =

0(l).

the Fourier series for f(x) converges absolutely

and uniformly, by § 3'34.

The necessary and

m

sufficient conditions that

A m = Bm =

for all values of

are that

f(kr -0)=f(kr + 0), that

is

to say

9'31.

The

/(TT

- 0) =/(- TT + 0),

that*/(#) should be continuous

for all values of x.

Differentiation of Fourier series. result of differentiating °°

1

5 2

a

+ 2 m=l

(a m cos

mx +- 6m sin m#)

QO

term by term *

2 {mbm cos mx — mam sin mx}.

is

Of course /(a;)

is

also subject to the conditions stated at the beginning

|7r/(a: + O), v

.

sin 23a

7tJ

7

.,

and write x + 28 in place of

is

obvious that,

m + (m - 1)

(X

7/1

if

we

write X for

ri' sin

2

m5

,

then

..

„„.

(\2

the second line, then

e*(*-') in

"

.

.

7/1

.

,,

_x

-r~-^f(x-26)d0^> W(«-0).

+ X" 2 ) + + (X»'-l + X l m ) = (i-x)- 1 {x 1 - m +x2-™+...+x-i + i-x-x2 -...-xm } = (1 - X)"2 {X1 "™ - 2X + \m+1 } = (X im - X~*m 2/(X* - X" *) 2

+ X-i) + (m - 2)

in

f(x-26)dd.

* See § 8-43.

t It

t

get

^— + 2d)d0 + sm 7r f(x

A + 2 Sn (x) = »=i

we

.

)

-

FOURIER SERIES

9 "4]

Now,

we

if

171

integrate the equation

1 sin 2 ra6 „ Q a 2 sin 2

we

.

-

_

.

= Am + (m ,

„

.

1) cos

„.

+

20

...

,

,

.

+ cos 2 (m - 1)/ 0, \

>

find that

(i'sin'mfl ,„ sin2

,

2

Jo

and

we have

so

to prove that

[^ sin 2 m6>

1

sm

mJo where

2

w

p

(0) stands in turn for each of the

/(a:

Now, given an

+

two functions

-f(x +0), f( x -2d)-f(x-0).

28)

number

arbitrary positive

we can choose

e,

8 so that*

\4>(0)\(0)d0 ^x '

sin2

«S

6>

mj

sin 2iQ »

the convergence of

|/(0

I

„

dd+

fK

1

m

'

^-— \4>(0)\d0 msin |6jj S r v ,

,

.

7/)

2

'

'

entails the convergence of

|'di

(0)

|

—

^v ta \${0)\d0

.

sin 2

ra'jj

•

'

2

mj

Now

\$(d)\de+-\ rv 7

sin 22/3

|

X^)

it

follows that

jB,

and

(m+J)S g i

nM

m+h)£

U du has an upper

then clear that

it is

sin 8

But

it

has been seen that this

< (48 +

(8)

d0

=

1)

e

that

0.

a sufficient condition for the limit of

is

Sm (a)

to be \ {f(oc + 0) +f(x - 0)} ; and we have therefore established the convergence of a Fourier series in the circumstances enunciated in § 9 42. -

Note.

The reader should observe that

in either proof of the convergence of a Fourier

mean-value theorem is required but to prove the summability of the series, the first mean-value theorem is adequate. It should also be observed that, while restrictions are laid upon f(t) throughout the range ( - w, tt) in establishing the summability series the second

;

at any point x, the only additional restriction necessary to ensure convergence is a re-

on the behaviour of the function in the immediate neighbourhood of the point x. that the convergence depends only on the behaviour of the function in the immediate neighbourhood of x (provided that the function has an integral which is striction

The

fact

absolutely convergent) was noticed by

Eiemann and has been emphasised by

Lebesgue,

Series Trigonome'triques, p. 60. It is

in

obvious that the condition t that x should be an interior point of an interval total fluctuation is merely a sufficient condition for the con-

which f{t) has limited

vergence of the Fourier series

;

and

lim

The reader

may

be replaced by any condition which makes

fh" sin(2TO + l)oo

2

{/(«)}

dar

|i a„ 2

•)

(o»'

+

:

m~l

(.,

2

^-7r-!ia + 2 2

(a n 2

2

+6

2 re

)

«=i

'

I

the theorem stated.

Corollary.

Parseval's theorem*.

at the beginning of this section,

and

Iif(x), F(x) both satisfy the conditions laid on f{x) if

A n Bn ,

by subtracting the pair of equations which

be the Fourier constants of F(x),

may

that

J"

f (x)F(x)dx =77 ha A +

follows

{(a n

devoted to

series

±A n + (b n ±BnT}~j )*

i

2 (a n A n + b n B n )\

Riemann's theory of trigonometrical

The theory

it

be combined in the one form

jljf( X)±F(x)}Zdz=^(a ±A o r+i

9'6.

+ 2

.

{/(*)} is

m-1

(, jt

cfe

(.»=0

6„')f

it

This

—

J ) 2

2 ^„(>H

I

?r

=

—n (« =

.

series.

of Dirichlet concerning Fourier series

is

which represent given functions.

Important advances in the theory were

made by Riemann, who considered

properties of functions defined by a series

°°

l

the typef ~a

of

lim (a n cos nx

+ 2

(a n cos nx

+ b n sin nx) = 0.

Riemann's theorem J that *

Mem. par

divers savans,

i.

if

We

+

Throughout

two trigonometrical

(1805), pp. 639-648.

is

due

to G. Cantor,

it

is

assumed that

series

up

to

converge and are equal

Parseval, of course,

assumed the

permissi-

term-by-term.

§§ 9-6-9-632 the letters a„, 6„

t The proof given

where

shall give the propositions leading

bility of integrating the trigonometrical series t

b n sin nx),

do not necessarily denote Fourier constants. Journal filr Math, lxxii. (1870), pp. 130-142.

9*6, 9

-

;

:

.

TRIGONOMETRICAL SERIES

6l]

183

at all points of the range (-7r*7r) with the possible exception of a finite

number

of points, corresponding coefficients in the two series are equal.

Riemann's associated function.

9'61.

CO

*

Let the sum of the at

any point x where

it

+ 2

^a

series

OO

+ bn sin nx) = A + 2 A n (x),

(a n cos nx

»=1 converges, be denoted by f(x).

F(x) = \A

Let

«?-t

n-*A n {x).

Then, if the series defining f{x) converges at all the series defining

To obtain

F(x) converges for

this result

Cantor's lemma*.

If lim

For take two points

'points,

all real values

of any finite interval,

of x.

we need the following Lemma due An

x+b

x,

n=l

{x)

=

Then, given

of the interval.

we can

e,

Cantor

to

a^x ^.b,

for all values of x such that

find

then an -*-0,

n such that t,

when n > na a n cos nx 4- b n sin nx \0.

and

by

so,

for all

§ 3'34,

the

CO

\A

- 2

A

n~' n (x)=F(x) converges absolutely and uniformly for all »=i real values of * therefore, (§ 3-32), F{x) is continuous for all real values of x. series

t>

x

l

i

;

Properties of Riemann's associated function ; Riemann's first lemma.

962. It is

now

possible to prove Riemann's first

_,

=

,

(x, a)

fcr

F(x +

2a)

-

lemma

that if

+ F(x-2a)-2F(x) >

4a2 QO

i/tew

lim (r(«,

=/(«), provided that 2 4„(«) converges for

a)

the value of

x

under consideration.

F(x ±

Since the series defining F(x); rearrange them

2a) converge absolutely,

+ 2a) + cos n (x — 2a) — 2 cos nx = — 4 sin sin w (x + 2a) + sin n (x — 2a) — 2 sin na; = — 4 sin

cos n (x

it is

we may

and, observing that

;

2

2

na cos nx, ?ia

sin n«,

evident that

n

/

\

G(x, a) It will

now be shewn

/sinna\ 2

v =A + 2 .


CO

oo+ 2

m cosm£+/3 m cosm£),

(o

when - n dt—

4>it,

\\

u)\duldt+

\4>it,

it,u)\dtdu

j

j

\

W