The use of transforms in connection with difference-differential equations and related topics

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1^121 t e ^ l ç n on

U.. , CF vrL.in _L.J__. Il

J I T L I I^ T E L nL J ^ -r iF F o^iV ^T I^ L E\U GIC^L ^LF ( ï o be T re r e n t e c^ l 'o i ’ o.n F . e c .

JGNbhCTÎ

FC. 1 J _ . "

l a ^ r e e by . . o r i r i... J e.Die e . ;

The t h e o r y c i u i i f e r e n o ^ - ^ i i ' i ' e l e n t i r 1 e ç u n t i e n s i s e r w ith s v e c is l re ie ie n c e

to t h e v i ’cbleh; o i f i n d i n g o s o l u t i o n ly

o f t r a n s f o r m s , one t h e d i s c u s s i o n i s c o n f i n é e , nr i n l y t o t h e 1: equation,

o n l y b r i e f r e f e r e n c e s b e i n g rnace to t h e n o n - l i n e s r i

The q u e s t i o n o f s i m u l e e x p o n e n t i a l s o l u t i o n s i s c c n s i i first.

F o l i o .'inn t h i s

ch ro n o lo g ically , p r o c e e d i n g to p r e s e n t d ay.

t h e r u b l i s h a c . ma, t e r i 9.1 i s d e a l t u i th

b e g i n n i n g -.vith a r a i e r by .ch.mic.t i n 19 11,

r

d i s c u s s i o n o f t h e m ai n c o n t r i b u t i o n s ir o m rnui

r e tw ee n 1911 and 1921 a num ber o f German m -.the m s

s tu d ie d these eq u atio n s in co n sid e rab le

d etail,

--nd some o f ti

a r e shown t o ha ve u s e d m e th o d s b a s e d on t r a n s f o r m s . by E o c h n e r end T i t c h m a r s h ,

b o t h o f whom made n e f i n i t e u s e o f -

F o u r i e r t r a n s f o r m i n t h e i r work on t h e s u b j e c t , p a p e r s by W r i g h t a r e c o n s i d e r e d , the L a p la c e t r a n s f o r m method. in the e a r l i e r p a p e r s ,

They ri-e

enn l i n ^ ’l l y

sc

w h ich c l e a r l y e x h i b i t t h e voi

,,r i g h t a v o i d s c e r t a i n

'ssum ptic

a n d h i s r e s u l t s a r e s e e n to be by f a r ■

i m p o r t a n t an d f a r - r e a c h i n g . The d i s s e r t a t i o n

c o n c l u d e s w i t h a. r e f e r e n c e

t o t h e wc]

t h e s e e q u a t i o n s ha ve be e n t r e a . t e a i n p r a c t i c a l p r o b l e m s o f va; type s .

D i s s e r t a t i o n s u b m i t t e d f o r t h e D e g r e e o f M .S c . i n th e U n i v e r s i t y o f London.

THE

USE

OF

TRAÎTSPORMS

DIFFERENCE -DIFFERENT IAL

IN

EQUATIONS

CONNECTION and

By D o r i s Maude Jame s R oyal H olloway C o l l e g e .

M arch, 1950,

WITH

RELATED TOPICS.

ProQuest Number: 10096352

All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, th ese will be noted. Also, if material had to be removed, a note will indicate the deletion.

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CONTSÏTTS

Section I. II.

Pag e In tro d u ctio n

...

...

.. .

The T r a n s c e n d e n t a l E q u a t i o n

1 4

III.

Schmidt

..................

11

IV.

Schürer

..................

26

H ilh

..................

34

...

41

..................

55

V. VI. V II. V III. IX. X. XI.

H oheisel Boohner

T itchm arsh

...

W right

78

A pplications C onclusion

67

102 ...

113

1.

THE

USE

0¥

TRAlISPORIi/IS

DIFtmSHCE^DIPFEREFJIAL

I.

IÎT

EQ.UATIQHS

CONKECTIOU an d

WITH

RELATED TOPICS.

IHTRODUCTIOH. The s t u d y o f d i f f e r e n c e - d i f f e r e n t i a l e q u a t i o n s h a s

te e n pursued in c o n s id e ra b le d e t a i l d uring th e p re se n t c e n t u r y , a n d much i n f o r m a t i o n a b o u t t h e s e e q u a t i o n s h a s b e e n o b t a i n e d by t h e u s e o f t r a n s f o r m s an d s i m i l a r operators. b y S c h m id t

The f i r s t

p a p e r o f i m p o r t a n c e was p u b l i s h e d

(29)* i n 1911.

H is method o f f i n d i n g a

s o l u t i o n i n v o lv e s th e use o f a fo rm u la w hich i s

seen to

be e q u i v a l e n t t o t h e i n v e r s i o n f o r m u l a o f a t r a n s f o r m . From 1911 o n w a r d s t h e continuously, W right

study of th e s u b je c t h a s developed

c u l m i n a t i n g i n t h e r i g o r o u s d i s c u s s i o n s by

( 4 2 - 4 - 7 ) , p u b l i s h e d i n t h e l a s t fe w y e a r s .

His

wor k i s b a s e d a l m o s t e n t i r e l y on t h e u s e o f t r a n s f o r m s . By a d i f f e r e n c e - d i f f e r e n t i a l e q u a t i o n i s m e a n t h e r e an e q u a t i o n o f th e form * R e f e r e n c e s o f th e form ( l ) ,

(2),

...

B i b l i o g r a p h y , t h o s e o f t h e f o r m ( 1 *1 ) , ( 2 *1 ) ,

...

are to the e q u atio n s.

are to the (1-2),

...

2.

■■ ' , where t h e

■'■ ■>

4r^ a r e i n d e p e n d e n t o f

i s t h e unknown f u n c t i o n .

■ ■ -, ^ ^ ”' t > n - K ) j - o X , and

^ (xj

Of s u c h e q u a t i o n s , t h e t y p e

f i r s t d i s c u s s e d was t h e l i n e a r e q u a t i o n , VW j^Z.O

-#1 'V - *

w h e r e e a c h t e r m c o n t a i n s o n l y one f u n c t i o n and t h e f u n c t i o n s

and

i^Cpc)

^

a r e known.

I

s h a l l be m a i n l y c o n c e r n e d h e r e w i t h t h e l i n e a r e q u a t i o n , m a k i n g o n l y b r i e f comments on t h e n o n - l i n e a r e q u a t i o n , sin ce the th e o ry of t h a t type i s s t i l l being developed. Some o f t h e m e t h o d s u s e d f o r t h e s o l u t i o n o f l i n e a r d i f f e r e n t i a l e q u a t i o n s may be a d a p t e d t o t h e s o lu tio n of lin e a r d if f e r e n c e - d if f e r e n tia l equations, a l t h o u g h t h e a n a l y s i s i s u s u a l l y more c o m p l i c a t e d . F o r e x a m p l e , when s i m p l e e x p o n e n t i a l s o l u t i o n s a r e c o n s i d e r e d , t h e u s u a l a u x i l i a r y e q u a t i o n i s f o u n d t o be a tra n scen d en tal equation.

W ith t h e developm ent

o f t h e O p e r a t i o n a l C a l c u l u s , h o w e v e r , a m eth od o f s o l v i n g d i f f e r e n t i a l e q u a t i o n s by L a p l a c e t r a n s f o r m s was e v o l v e d , a n d t h i s may be a p p l i e d t o d i f f e r e n c e d i f f e r e n t i a l eq u atio n s w ith co nsiderable

success.

3. I n c o n s i d e r i n g t h e t r a n s f o r m m e t h o d a number o f problem s a re found to a r i s e an d , i n p a r t i c u l a r ,

it

is

seen th a t the o rd e r a t i n f i n i t y of a s o lu tio n i s of g r e a t im portance to th e v a l i d i t y of th e m ethod. of the f i r s t

One

ste p s in a rig o ro u s approach i s the proof

o f an e x i s t e n c e theorem s t a t i n g c o n d i t i o n s u n d er w hich the eq u atio n has s o lu tio n s of a c e r ta in ty p e .

The

asym ptotic behaviour of s o lu tio n s under c e r t a i n c o n d itio n s i s a ls o of i n t e r e s t , to g e th e r w ith the q u e stio n of o b t a i n i n g an a c t u a l s o l u t i o n i n c e r t a i n sim ple c a s e s . It

seems c o n v e n i e n t i n d i s c u s s i n g l i n e a r

d i f f e r e n c e - d i f f e r e n t i a l eq uations to follow a c h r o n o l o g i c a l scheme, b e g i n n i n g w i t h Schm idt*s work i n 1911, but f i r s t th e t r a n s c e n d e n t a l e q u a t i o n , a l r e a d y m entioned, i s

considered.

I s h o u l d l i k e t o a c k n o w l e d g e my i n d e b t e d n e s s t o M i s s B.U-. Y a t e s f o r t h e v a l u a b l e h e l p w h i c h I h a v e r e c e i v e d from h e r i n f r e q u e n t d i s c u s s i o n s .

4.

II.

THE TRAFSCmOENTAl EQ.ÜATIOU. C e r ta in p o i n t s o f n o t a t i o n w hich a r e used th ro u g h o u t

t h e d i s s e r t a t i o n w i l l be s t a t e d h e r e so t h a t r e p e t i t i o n may be a v o i d e d . The num ber

C is a p o sitiv e

c o n s ta n tw hich

i s not

a l w a y s t h e same a t e a c h o c c u r r e n c e , w h i l e ...

are p o sitiv e

c o n s t a n t s e a c h o f w h i c h h a s t h e same

value a t each o c c u rre n c e .

The n u m b e r s

r e p r e s e n t a r b i t r a r y c o n s t a n t s , an d

S

Ayi,

f^

&«2/ " V

i s any sm all

p o s i t i v e number. The g e n e r a l l i n e a r (x) e ^

(1 . 2 ) w here I t is also

e q u a t i o n i s t a k e n i n t h e form W

supposed t h a t

and

0 z

4 -Ir,

>/ \, -n y/ I ,

' ' ' 4

.

The

l i n e a r e q u a tio n w hich i s c o n s id e r e d i n g r e a t e s t d e t a i l i s t h a t w i t h c o n s t a n t c o e f f i c i e n t s , namely an e q u a t i o n o f th e form m

where t h e numbers

A

d

a r e r e a l o r c o m p le x c o n s t a n t s .

T h is i s r e f e r r e d t o a s th e non-homogeneous e q u a t i o n , and the eq u atio n ^ f^z 0

(x-h lr ) - - O -y Z.0

iji-A j

5. a s t h e homogeneous e q u a t i o n . As i n t h e it

case of l i n e a r d i f f e r e n t i a l eq u a tio n s

i s c l e a r t h a t t h e most g e n e r a l s o l u t i o n o f ( 2 . 1 )

i s g i v e n h y a d d i n g a n y p a r t i c u l a r s o l u t i o n o f ( 2 «1 ) to the general s o lu tio n of ( 2 .2 ). g en eral s o lu tio n of (2 . 2 ) i s

By a n a l o g y , t h e

sometimes c a l l e d th e

complementary f u n c t i o n . O ccasionally i t operator

i s c o n v e n ie n t to use th e

A d e f i n e d by

/V * 0 r- o i n w h i c h c a s e e q u a t i o n ( 2 *2 ) may he w r i t t e n i n t h e

fo r m

The number J Â

i s a c o m p l e x q u a n t i t y g i v e n by

r (^-4- y t" w h e r e o" a n d

otherw ise s ta t e d .

f

are r e a l ,

unless i t

It is

seen im m ediately t h a t

' ( x ) - K “

th e n t h i s i n t e g r a l i s a l s o of o rd e r \x \^ . The s e c o n d t e r m Cv) in the ex p ressio n fo r ^ I x ) > h o w e v e r , r e d u c e s by

23 . C a u c h y ' s R e s i d u e T he ore m t o t h e f o r m A %/ where th e

du ^

E,A

and th e n

— awd.

re s u lt is also true fo r lx/ ^

for

it

of the zeros of

n/ & o,

tit)

.

Thus

i s of

i s the g r e a te s t m u l t i p l i c i t y ^

where

in te g e r le s s th an or equal to /x/

seen t h a t t h i s

,

, the

t h e form

is

^ a /14 i

On t h e o t h e r h a n d , i f ^

j J

sura v a n i s h e s

is of order |x /^ f o r

By c o n s i d e r i n g

fo r large

-n)

are c o n sta n ts.

ITow i f ^{ t ) h a s no r e a l z e r o s t h i s

order

(vzo,f^ -

fu n ctio n s ù(^

^ /"xj

a re of

i s zero or a p o s itiv e

C'p-i) •

I t then follow s,

, that

ir(u)

- u) Mi ^ é

* I ^ /tr{u)l +

I ir ( - u ) \ J da.

ixi

i I xl T herefore since

trfuj i s o f o r d e r

lul^

as

/u(-7x?»

it

i 8 seen t h a t

0

L

( 1^1

/«jx j )

rt-i) U < -')

24. a n d “by c o n s i d e r i n g

and

a n d 'V =. n ,

found to h o ld a l s o f o r F in ally ,

these r e s u lts are

S ch m id t's r e f e r e n c e to a w id er c l a s s of

so lu tio n s of exponential order i s i n t e r e s t i n g , as in a l l t h e l a t e r w ork on d i f f e r e n c e - d i f f e r e n t i a l e q u a t i o n s it

i s s o l u t i o n s o f t h i s t y p e w h ich a r e most i m p o r t a n t .

I f t h e c o n d i t i o n on irix) i s r e l a x e d , the o r ig i n a l o rd er c o n d itio n ,

it

trOO =■ where of ^

i n s t e a d of

sa tisfie s

1^1 "-y

J

J

i s p o s it i v e , th en the i n t e g r a l s in the d e f i n it i o n w i l l converge a b s o l u t e l y and u n i f o r m l y , p r o v id e d < y

that

in term s of ^ be o f o r d e r 6 \

so t h a t ,

^

.

.

A p a r t i c u l a r s o l u t i o n c a n t h e n Toe f o r m e d p r e c i s e l y a s b e f o r e , b u t t h i s tim e i t w i l l

A)^ I a s This

w h ich Schmidt u s e s

ix I ( X>

f o r some p o s i t i v e c o n s t a n t

c a n be shown by a m e t h o d s i m i l a r t o t h a t in the o r ig in a lcase.

I f th e g en eral s o lu tio n of the d i f f e r e n c e - d i f f e r e n t i a l equation i s a lso to s a ti s f y t h i s l a s t order c o n d itio n , i t i s c l e a r t h a t th e o n ly sim ple e x p o n e n t i a l s o l u t i o n s w hich may be i n c l u d e d i n

t h e com plem entary f u n c t i o n a r e th o se

which a r e th e m se lv e s o f o r d e r o th e r w ords, the only p o s s ib le where

€

as

i xl

s o l u t i o n s o f t h e form

t" i s a r o o t o f ( 3 * 3 ) , a r e t h o s e f o r w h i c h

,

In g

25. but t h i s I n e q u a lity d e fin e s a s t r i p of f i n i t e w idth p a r a l l e l to th e r e a l e a r l i e r ^(t) such s t r i p .

t'-a x is ,

has only a f i n i t e

and a s h as been m entioned number o f z e r o s i n a n y

Th u s t h e c o m p l e m e n t a r y f u n c t i o n w i l l

c o n s is t of a f i n i t e

number o f t e r m s o n l y ,

so t h a t a g a i n

no q u e s t i o n o f t h e c o n v e r g e n c e o f a n i n f i n i t e a r i se s .

series

26.

IV.

SCHUHER. I n 1912 and 1913 S c h u r e r p u b l i s h e d t h r e e p a p e r s ,

(30),

(31) and ( 3 2 ) , d e a l i n g w i t h a d i f f e r e n c e -

d i f f e r e n t i a l equation of the sim plest p o ssib le ty p e. The e q u a t i o n , o f t h e f o r m

^ ' ( x + >) ~

^

(^'0

)

i s one w h i c h f r e q u e n t l y o c c u r s i n p r a c t i c a l p r o b l e m s , a s w i l l be s e e n i n S e c t i o n X. It S ch u rer does not use tra n s fo rm s to f in d a so lu tio n .

I n s t e a d he c o n n e c t s e q u a t i o n ( 4 * 1 ) w i t h a n

i n t e g r a l e q u a t i o n o f t h e t y p e s t u d i e d by H e r g l o t z i n a p a p e r (17) p u b l i s h e d i n 1908, and he t h e n u s e s th e s o lu tio n obtained f o r th e l a t t e r .

By t h i s m ea ns a

s o l u t i o n o f (4*1) i s f o u n d , w hich i s o f p r e c i s e l y th e same f o r m a s t h a t g i v e n by W r i g h t ’ s t r a n s f o r m m e t h o d . I n o r d e r t o co mpare t h e s e r e s u l t s l a t e r ,

it

is

c o n v e n ie n t t o o u t l i n e S c h u re r* s argument h e r e , H e r g l o t z d i s c u s s e s t h e hom o g e n e o u s i n t e g r a l e q u a t i o n o f th e form

Jo T h i s i s a w e l l - k n o w n i n t e g r a l e q u a t i o n , an d i t

is

i n t e r e s t i n g t o n o t i c e t h a t i t may i t s e l f be s o l v e d

27. s t r a i g h t f o r w a r d l y “by means o f t r a n s f o r m s .

T his i s done,

f o r e x a m p l e , hy T i t c h m a r s h ( 3 6 ) . H e r g l o t z , h o w e v e r , p o i n t s o u t t h a t i t h a s a. s o l u t i o n o f t h e form - c p rovided

i s a ro o t of

X if)

-

f e ^

K ( m ) Mi

-

I.

He t h e n p r o v e s h y a m e t h o d o f c o n t o u r i n t e g r a t i o n , t o g e t h e r w ith an a p p l i c a t i o n of F o u r i e r ' s I n t e g r a l F o r m u l a , t h e . t a n y a r b i t r a r y f u n c t i o n l h ) - i ‘' o f

e

I

t h e e x p a n s i o n h o l d i n g f o r t h e w h o le i n t e r v a l except a t the point So f a r t h i s

K =. X0 +

r ^

€'

c X c %,-f ,4%

,

e x p a n sio n h o ld s f o r any f u n c t i o n w hich

s a t i s f i e s t h e n e c e s s a r y c o n d i t i o n s , b u t H i l b n e x t shows

3 8. by d i f f e r e n t i a t i o n t h a t i f

such a f u n c tio n i s a s o lu t i o n

of e q u a tio n (5 .3 ) th e n the c o e f f i c i e n t independent of

X’j, .

Thus hy r e p e a t e d l y t a k i n g t h e o v e r i n t e r v a l s o f l e n g t h s -(r,

expansion f o r d i f f e r e n t and

K;^iXo) i s

a s o l u t i o n o f ( 5 . 3 ) i s found i n t h e form o f

a s e r i e s w hich c o n v erg es u n if o r m ly i n any f i x e d i n t e r v a l of

X .

F u rth er,

s u c h a s e r i e s c a n he shown t o he

unique. H ilh a lso o b ta in s th e r e s u l t of S e c tio n I I f o r a d o u b l e z e r o o f T'/'ij

hy u s in g a l i m i t i n g p r o c e s s .

H i s n e x t s t e p i s t o e x t e n d h i s r e s u l t s t o more com plicated e q u a tio n s .

He f i r s t

considers a d iffe re n c e

e q u a t i o n o f o r d e r wt , t h e n t h e p a r t i c u l a r e q u a t i o n o f form (5*1) f o r w h ic h

^

equa,tion o f form ( 5 « l ) . yiyij-h X(^)

e.

and f i n a l l y th e g e n e r a l

By p u t t i n g

'Tf^)

i n t h e form

21 s u f f i c i e n t nu mber o f wgtys, he o b t a i n s a n

expansion in th e case of the d iff e r e n c e equ atio n in p r e c i s e l y t h e same way a s a b o v e .

For th e d if f e r e n c e -

d i f f e r e n t i a l e q u a t i o n s , h o w e v e r , a f u r t h e r p o i n t m ust be considered.

I n t h e g e n e r a l c a s e an e x p a n s i o n o f t h e form

-

1

/A~0

i

/V, 6 0

^

Û/k,

W

e

/

5i* is

o b t a i n e d , which, i s v a l i d

in the i n t e r v a l

except a t the p o in ts t o show t h a t

^

order

i s independent of

^(xj

if

is a

s o l u t i o n , and t h a t t h e s e r i e s c o n v e r g e s u n i f o r m l y i n X , two more c o n d i t i o n s a r e f o u n d

every fixed i n te r v a l of t o be n e c e s s a r y . must have w

If n eith er

0. ^^ n o r

i s zero X ,

continuous d e r iv a tiv e s f o r a l l r e a l

w h i l e i f one o f

or

i s zero

\f ^

m ust h a v e c o n t i n u o u s

d e r iv a tiv e s of every order f o r a l l r e a l

x •

d ifferen tiatin g

i s e a s i l y shown

th at

w .r.t.

(5*1) h a s a s o l u t i o n o f

converges un ifo rm ly i n every As H i l b p o i n t s o u t ,

it

th e form (5*6) w hich fix ed

in te rv a l of

X .

Schurer*s p a r t i c u l a r eq u atio n

^ Vx+/j =.

Ix).

i s o f t h e f o r m ( 5 - 1 ) , b u t o n l y one i n t e r v a l i s i n v o l v e d , an d so i t

T h e n , by

(}(,, %«+ ' j

does not d i s p l a y th e r e a l

d i f f i c u l t y of o b ta in in g a s o lu tio n

w hich i s v a l i d over •

a number o f i n t e r v a l s

Itis

o f i n t e r e s t t o n o t i c e , h o w e v e r , t h a t on p u t t i n g

in

Yh -

/y

A ^ ^

- //

^oo =

ac, ' 0 ^

^,0 -

- h

(5 «6 ) we o b t a i n Jo'*'!

K>,(x.)-

- Aj

e

.

^ij^) df^

-^

40. T he n s i n c e t o he z e r o ,

Independent of

, we may c h o o s e

Xo

so t h a t

A lso, in t h i s case y(h) ^ yàZ and t h e r e f o r e

(5*6) g iv e s

}U) -w hich i s of th e Schurer,

^

h

+1

same f o r m a s t h e s o l u t i o n o b t a i n e d hy

,

4 1.

VI,

HOHEISEL. I t was n o t u n t i l f o u r y e a r s l a t e r ,

in 1922, t h a t

H o h e i s e l p u b l i s h e d a p a p e r ( 2 0 ) on l i n e a r d i f f e r e n c e d i f f e r e n t i a l e q u a tio n s w ith polynom ial c o e f f i c i e n t s ,

in

w h i c h he u s e s r e s u l t s o b t a i n e d b y S c h m i d t f o r t h e c o n s t a n t c o e f f i c i e n t e q u a t i o n a n d r e t a i n s t h e same o r d e r c o n d i t i o n on a s o l u t i o n .

H i s m e t h o d , h o w e v e r , i s more

d i r e c t l y r e l a t e d t o W r i g h t * s t r a n s f o r m m e t h o d an d i s th erefo re of in te re s t h ere. H oheisel f i r s t

i n d i c a t e s how h i s e q u a t i o n i s

connected w ith t h a t of Schm idt.

He c o n s i d e r s a n e q u a t i o n

o f t h e form T , l x ) ÿ ‘' " ' U x - L ) + - ■ • + 7^(x) a r e p o l y n o m i a l s i n

where t h e degree

z

.

If

and th e o t h e r c o e f f i c i e n t s

le s s th an or equal to

^

, it

is

?tfx) i s o f are of degree

seen th a t

Llj) a w h er e

4^ ^

^0

are o p e ra to rs w ith constant

c o e f f i c i e n t s o f t h e f o r m d i s c u s s e d by S c h m i d t . It

is c le a r th a t re a l zeros of

w i l l be o f

im portance i n a c o n s id e r a t i o n of th e s o l u t i o n s of (6 * 1 ), j u s t as i n th e c o rre s p o n d in g case f o r pure d i f f e r e n t i a l

42. equations.

T h e re fo re , in c o n fin in g h im se lf to

e q u a tio n s i n which th e c o e f f i c i e n t s a r e l i n e a r f u n c t io n s of

X , H o h e is e l s t i l l has to d i s t i n g u i s h between th e T^fx)

case f o r w hich

h a s a r e a l z e r o , and t h e case f o r

w h i c h i t h a s no r e a l z e r o .

I n t h e f o r m e r c a s e he

c o n s i d e r s an e q u a t i o n of t h e form L-i^ ) H. where

S

f

( jrX-h

K-)

a r e r e a l , p o i n t i n g o u t t h a t t h i s may he

r e d u c e d h y r e a l l i n e a r t r a n s f o r m a t i o n s t o t h e f orm L(^)

z

oc^Hxj ^

'=^xrioc)^

(^'^)

I n t h e l a t t e r c a s e he c o n s i d e r s = (X-h-J w h er e

*■

ÿ - l x t l)~ - ^ ( x )

a r e r e a l and n o n - z e r o ,

(( i )

so t h a t t h e

c o e f f i c i e n t o f t h e h i g h e s t d e r i v a t i v e h a s no r e a l z e r o . The ho mog en eo us e q u a t i o n a s s o c i a t e d w i t h ( 6 * 2 ) i s o f t h e form L{0

=

^

/I, ( f )

f-

^

whe r e

A, Ijf-I =

}j- 'faJ -h

^ (x -n )

and f\ 0 l ÿ ! As u s u a l i t

z

follow s t h a t

^ f

0 '

V

43. where ^ yS -h ^ ^

t

l^)^

the fu n c tio n s

i ^

^ c

^

having only a f i n i t e

number o f

z e r o s i n an y s t r i p o f f i n i t e w i d t h p a r a l l e l t o t h e im aginary a x is . w hich l i e

M oreover, i t

i s e a s i l y seen t h a t zero s

on t h e i m a g i n a r y a x i s o c c u r o n l y f o r p a r t i c u l a r

values of a

and

excluded, i t

i s p o s s i b l e to choose ^

7, IS) o r

lie

, so t h a t

i f th ese values are so t h a t no z e r o s o f

i n the s t r i p

I ^ K . H o h e i s e l s k e t c h e s f o r m a l l y a m e th o d o f s o l v i n g ( 6 * 2 ) a n d ( 6 * 4 ) , l e a v i n g a s i d e f o r t h e moment t h e q u e s t i o n o f th e convergence and o r d e r o f t h e s o l u t i o n o b t a i n e d .

He

assumes t h a t

~J where t h e c o n t o u r

^

m t U ) di> ^

, and t h e f u n c t i o n

found by s u b s t i t u t i n g t h i s

in

(é-r)

nr('i) a r e t o be

the eq u atio n .

This

i s eq u iv alen t to the s o lu tio n of d i f f e r e n t i a l equations by L a p l a c e I n t e g r a l s .

By s u b s t i t u t i n g i n

(6*4) i t

is

seen th a t ^

.

C

\

ch ir r

M rU}di

3i

r e

.

xe

fir(A)cli

+

g e

44. where

Th us f o r ^( xj first

t o he a s o l u t i o n o f ( 6 * 4 ) i t

th a t th e contour

i n te g r a te d term i n

^

is necessary

s h o u l d he c h o s e n so t h a t t h e

( 6 * 6 ) v a n i s h e s , an d s e c o n d l y t h a t

ur(-^)

s h o u l d he a s o l u t i o n o f =■ 0

.

V

This i s a l i n e a r d i f f e r e n t i a l e q u a tio n , i t s

so lu tio n

h e i n g o f th e form

r

C . _L-

g_

7, f i )

A s o l u t i o n o f ( 6 « 4 ) i s t h e n g i v e n hy s u b s t i t u t i n g t h i s value of

atU)

fu n ctio n

in

(6*5),

As H o h e i s e l p o i n t s o u t , t h e

i s a L aplace tr a n s f o r m i f th e co n to u r

i s c h o s e n t o he a l i n e p a r a l l e l t o t h e i m a g i n a r y a x i s . I n o r d e r t o e x t e n d t h i s m e th o d t o e q u a t i o n ( 6 * 2 ) H o h e ise l uses F o u r i e r ' s I n t e g r a l Form ula.

Af J • i6

and, p u ttin g

J ^

, it

/ 4)

^ is

This g iv es

xrfuj U seen fo rm a lly t h a t r6t> e-^A T i4^)cU .

J -«

45. The e q u a t i o n c o r r e s p o n d i n g t o If

^

(6 .6 ) i s th en con sid ered .

i s ta k e n as th e im ag in ary a x i s i t s e l f th e

i n t e g r a t e d t e r m v a n i s h e s , an d f o r ^ ( x ) t o he a s o l u t i o n of (6 .2 ) i t

is necessary th a t

f T hus

y

-

f

e

^ ,

m u s t he a r o o t o f

-

-Since

-

^

is

Ij'

sim ply a p a ra m e te r h e r e , i t

fin d a so lu tio n

of the

^U )cU . is

s u f f ic ie n t to

sim ple e q u a t i o n ( ( ■?)

= e and t h e n t a k e /à f-i)

-

f

ir /

^

as a s o lu tio n of (6*8).

Then th e s o l u t i o n o f ( 6 .2 ) i s

g i v e n hy s u b s t i t u t i n g f o r

ur(\)

in (6*5), as b e fo re .

H o w ever, when he comes t o j u s t i f y t h e s e f o r m a l m e th o d s, H o h e i s e l d i s c o v e r s t h a t a c o n v e rg e n c e f a c t o r must he i n t r o d u c e d i n t o t h e s o l u t i o n , and s o , i n s t e a d o f ( 6 * 9 ) , he c o n s i d e r s t h e e q u a t i o n =■ C where

^ 7 X,

•

1^)

The e f f e c t o f t h e e x t r a f a c t o r i s

a n n u l l e d l a t e r i n t h e same way a s f o r S c h m i d t ' s e q u a t i o n .

liNow ( 6 . 1 0 ) i s a n o t h e r l i n e a r d i f f e r e n t i a l e q u a t i o n and its

s o l u t i o n i s o f t h e form

H o h e i s e l c h o o s e s t h e tvm s o l u t i o n s , (Jj by

^ '± y ^

I

the p a th of

th e im aginary a x is -oTj

and ^

and

^

say, given

i n t e g r a t i o n beingta k e n along

in each

case.

It

can

be shown t h a t

a r e a b s o l u t e l y c o n v e r g e n t , and t h a t ,

if

th en th e se i n t e g r a ls are a ls o a b s o lu te ly convergent, uniform ly convergent w ith re sp e c t to x once u n d e r t h e i n t e g r a l s i g n . t h e r e f o r e be o p e r a t e d on by

&nd d i f f e r e n t i a b l e

The f u n c t i o n

L

/(I-

f ^

may

, a n d by e m p l o y i n g t h e

f o l l o w i n g d i f f e r e n t i a t i o n d e v ic e a s o l u t i o n o f (6*2) can be o b t a i n e d .

The o p e r a t i o n

is introduced,

an d i n s t e a d

of c o n sid e rin g

t h e form —

L_

r^

/ ^ ( xa) V.x,

H oheisel c o n s id e rs th e fu n c tio n

^~aO

a so lu tio n of

47.

L

T h i s may "be o p e r a t e d on by

L [ w h er e

)

- W x; +

,

X but v a r ie s w ith each

i s independent of

d iff e r e n t fu n ctio n

, and i t i s found t h a t

,

Thus

— i^(x) JfO À

is

a s o lu tio n of

Î- xrCx)

Ll j f )

o r, i n o th e r w ords, i t an a d d i t i v e c o n s t a n t .

-h

i s a s o l u t i o n o f ( 6 . 2 ) a p a r t from In f a c t,

th e problem o f s o lv in g

(6*2) h a s b een re d u c e d t o t h a t o f s o l v i n g U ^ l - // b u t i t w i l l be shown l a t e r t h a t t h i s e q u a t i o n i s i n s o l u b l e i n H o h e i s e l *8 s e n s e .

F i r s t , however, sin ce th e s o l u t i o n

o f ( 6 «13) o b t a i n e d a b o v e i s i n t h e f o rm o f a n i n f i n i t e in teg ra l,

its

c o n v e r g e n c e m u s t be d i s c u s s e d , and t o

d isc o v e r w hether i t sense i t s

i s a p ro p er s o lu t i o n i n H oheisel*s

o r d e r a t i n f i n i t y m u s t a l s o be c o n s i d e r e d .

To

d e a l w i t h b o th o f t h e s e problem s H o h e is e l t r a n s f e r s th e paths of i n t e g r a ti o n ,

i n ( 6 . 1 1 ) an d ( 6 * 1 2 ) , f r o m t h e

im aginary a x is to a p a r a l l e l l i n e th ro u g h th e p o in t w here it

^ .

From t h e d e f i n i t i o n o f

îC g i v e n ab o v e

is

c l e a r t h a t th e v alu e of th e i n t e g r a l i s unchanged. “ T'a I n t h e new f o rm a f a c t o r €, i s p resen t in the in teg ran d an d i t th at

c a n be shown, by c h o o s i n g t h e s i g n o f

\\I(x)

Y

properly,

i s a b s o l u t e l y a n d u n i f o r m l y c o n v e r g e n t , and

48. may "be d i f f e r e n t i a t e d o n c e u n d e r t h e i n t e g r a l s i g n . In f a c t s tra ig h tfo rw a r d i n e q u a l i t i e s give the r e s u l t

|w(x)| < c

\e

o U +c v-^eO

H ere t h e f i r s t

t e r m on t h e r i g h t i s o f t h e f o r m ( 3 * 1 5 )

d i s c u s s e d hy S ch m id t. if

F tom h i s r e s u l t s i t f o l l o w s t h a t

\r{Ui) i s o f o r d e r

of order

(3C|*

j u i "7

as —y oo .

as

M oreover th e second te rm

on t h e r i g h t i s i n d e p e n d e n t o f m u s t he o f o r d e r

% , so t h a t

as

A)

p o s i t i v e c o n s t a n t o r z e r o an d

, th e n t h i s term i s

00

itself

, where ^

jh >/ o( ,

is a

T h is r e s u l t can

he made more p r e c i s e , h o w e v e r , hy u s i n g t h e f a c t t h a t Wl>^) i s a s o lu tio n of (6«13). (SII4 he w r i t t e n i n t h e f o r m —

/) (iVj = -

i.e.

w h e r e '^(x)

A, I w )

-

tr(x) -

y tx )

T h i s e q u a t i o n may

/)

_

4T/,Ctr

IX}

as

^

,

i s a fu n ctio n of order

00,

How t h i s i s i t s e l f a d i f f e r e n c e - d i f f e r e n t i a l e q u a t i o n o f S ch m id t's

t y p e , and XU)

so hy h i s

r e s u l t s i t follow s th a t W (x) s

as stage

lxl"7

00

•

h a s no p u r e l y i m a g i n a r y z e r o s ,

0 (lD(j^ ' )

j

W Uo() s. O ( i x

T h i s a r g u m e n t c a n he r e p e a t e d u n t i l

) the

4 9. is reached.

The f u n c t i o n

'VW c a n n o t be r e d u c e d t o a

low er o r d e r th a n t h i s , however, because of th e term irCx) , so t h a t t h i s i s t h e f i n a l o r d e r r e s u l t

involving

s a t i s f i e d by th e s o l u t i o n

of (6 .1 3 ).

H oheisel next d isc u s s e s th e

same p r o b l e m s f o r a

s o l u t i o n o f ( 6 . 3 ) an d i t s a s s o c i a t e d hom ogeneous e q u a t i o n . The m e t h o d s u s e d a r e s i m i l a r , b u t t h e q u e s t i o n o f c o n v e r g e n c e i s f o u n d t o be much s i m p l e r b e c a u s e o f t h e new f o r m o f

% ^

.

In t h i s

case th e f a c to r

^ is

s u f f i c i e n t t o e n s u r e c o n v e r g e n c e , a n d o n l y one s o l u t i o n of the e q u a tio n corresponding to considered.

M oreover a s o l u t i o n o f ( 6 .3 ) i t s e l f i s

obtained w ithout th e a d d itio n of the th is

( 6 . 1 0 ) n e e d be

constant

, and

s o l u t i o n s a t i s f i e s t h e same o r d e r c o n d i t i o n a s a b o v e . The o r d e r o f a s o l u t i o n o f t h e hom o g en eo u s e q u a t i o n

( 6 . 4 ) h a s y e t t o be c o n s i d e r e d , and t o d e a l w i t h t h i s H o h e i s e l n e x t shows t h a t i f of (6 .4 ) then

y.Yarj

i s a continuous s o lu tio n

is also

c o n tin u o u s , even a t

^=-0.

By w r i t i n g t h e e q u a t i o n i n t h e f o r m /I,//;

^ *

,.e, it

is

order the l a s t

=

th at IXj^

as

TW y

i s of o rd e r / %f

A) .

if

^ (x ) i s of

Thus by S c h m i d t 's r e s u l t s

e q u a t i o n must have a s o l u t i o n

^(x)

f o r w hich

50 .

o O x i " " ) , y '(x) = 0( 1X1*"; ,

yx) In t h i s

c a s e t h e p r o c e s s may be r e p e a t e d a n y number o f

t i m e s and i t

follow s th a t

t^(x) = 0( 1X1- ' ; , f o r an y

0(IXI“ ),

C , no m a t t e r how l a r g e ,

H o h e i s e l p r o c e e d s t o u s e a m e th o d s i m i l a r t o t h a t em p lo y ed l a t e r by W r i g h t i n h i s s o l u t i o n by L a p l a c e t r a n s f o r m s , t h e o n l y d i f f e r e n c e b e i n g t h a t he u s e s a tw o-sided in s te a d of a o ne-sided tra n sfo rm . th at

i s a s o l u t i o n o f (6*4) o f th e c o r r e c t o r d e r a t

i n f i n i t y an d he d e f i n e s From ( 6 * 1 5 ) i t

by t h e e q u a t i o n j f ( x ) ebc.

I e J , af

f o l l o w s t h a t t h i s i n t e g r a l , an d a l l i t s

d e r i v a t i v e s , w i l l converge i f It

He s u p p o s e s

is therefore ju s tif ia b le x

and i n t e g r a t e w . r . t .

e

over

i s p u re ly im aginary.

to m u ltip ly f - * , oo)

L(^)

( 6 . 4 ) by giving

- 0.

T h i s i n t e g r a l may be w r i t t e n i n t e r m s o f a n d s i m p l i f i e d , and i t i s f o u n d t o r e d u c e t o - 0 .

V

IhU3

m -)

■

«

a n d /Ij,

€

mm0^9C

5ian d t h i s

i s "bounded on t h e i m a g i n a r y

th erefo re

0^

-y

It

follow s th a t I X*

as

^V-axis,

00

I

^

along the im aginary a x is .

On t h e o t h e r

hand fix)

=

e

" " 'J '

^

J - to eo

I.L and i t

^

y

c a n he shown hy e l e m e n t a r y a r g u m e n t s t h a t

i s c o n tin u o u s and o f o r d e r ex ists.

y *'i^) ^

, so t h a t

T h i s me an s t h a t on t h e i m a g i n a r y a x i s

\ h ‘ Pi X) I

} , i n w h i c h ^

Schm idt’s p a p e r.

i s r e p l a c e d by

/T" , a s i n

The r e a s o n f o r u s i n g t h i s f o r m i s

a g a i n due t o t h e f a c t t h a t o n l y t h e r e a l z e r o s o f are of i n t e r e s t ,

since in

(7 .1 )

is a real,

not a

complex, v a r i a b l e . I f now iriX) then i t

is

v it-) C

^

seen from (7*2) t h a t

( 2 . 1 ) i f an d o n l y i f i(t)

J tf

- >> i s as o lu tio n

of

satisfies ipLt)

-

V(t-)

T h i s means t h a t

(t) j f(t)

and f u r t h e r , t h a t

y (;(] , g i v e n by

Must

i//^i

be a

fu n ctio n of

^

/p(t-

Iftx}

must be d i f f e r e n t i a b l e

*/t t i m e s i n

Before d is c u s s in g th e s e c o n d itio n s i n g r e a t e r d e t a i l t h e hom o g e n e o u s e q u a t i o n

(2 .2 ) i s

co n sidered.

In th is

52. case

( 7 * 3 ) ‘becomes ^Lt)

it)

=

o ,

' { ( t ) h a s no r e a l z e r o s ,

Thus i f

he z e r o f o r a l l r e a l

y /x^ -

O •

F u rth er, i t

is

clear

m ust i n a n y c a s e h a z e r o a l m o s t e v e r y w h e r e .

I t th e r e f o r e fo llo w s a g a in t h a t the only p o s s ib le i s the zero s o lu tio n .

so lu tio n

The u s u a l s i m p l e

e x p o n e n t i a l s o l u t i o n s g i v e n hy r e a l z e r o s o f

are

om itted here because

t h e y do n o t b e l o n g t o

and

are not s o lu tio n s in

Bochner*s r e s t r i c t e d

sense.

so

R e t u r n i n g t o t h e non-homogeneous e q u a t i o n ( 2 . 1 ) , tit)

i s next supposed t h a t A , of m u l t i p l i c i t i e s

resp ectiv ely . the zero

h ,

- ' '/

f ',,

"

fi,

The n

i n a sm all i n t e r v a l

'

it

*i n c l u d i n g

i s a s s u m e d th %t

{■• ( t - H ^ ( t )

i s n o n - z e r o a n d d i f f e r e n t i a b l e an y number

of tim es in t h i s i n t e r v a l .

it

' ' '/ /a,

, h u t no o t h e r s , =

they

has zeros a t

f>,

■lltj where

must

i s a r e g u l a r f u n c t i o n whose z e r o s a r e i s o l a t e d , (^[tj

so t h a t

^(t)

follow s th a t

, an d c o n s e q u e n t l y t h a t t h e o n l y

s o lu tio n of (2 .2 ) i s t h a t '(it)

it

Thus, in

/ ^

j

6 0.

ï l ÿ

The n i f

._ . J -

,rr>

(2»1) has a s o l u t i o n

jffx) o f f o r m ( 7 . 4 ) i t

can

he s e e n f o r m a l l y t h a t t h e f u n c t i o n

must b e l o n g t o ^

fo r each

^

.

Bochner proves t h i s

r e s u l t r ig o r o u s ly , using th e tech n iq u e of m u l t i p l i c a t o r s . He a l s o shows t h a t a n e q u i v a l e n t c o n d i t i o n i s t h a t t h e fu n ctio n

s h o u l d he i n t e g r a b l e

tim es in

In o rder to consider s u f f i c i e n t to have a s o l u t i o n

. conditions fo r

^ f x) i t m u s t he r e m e m b e r e d t h a t t h e

so lu tio n

i t s e l f i s t o he d i f f e r e n t i a b l e

From ( 7 . 4 ) i t

th e re fo re fo llo w s th a t the fu n c tio n s

must b e l o n g t o

(2.1)

, a n d t h i s i s so i f

it

-/v t i m e s i n

,

c a n be p r o v e d

th a t the fu n c tio n s

,

are general m u ltip lic a to r s .

(■V- 0,

(f.y)

61. Bochner p o i n t s out t h a t t h e r e a re c o n s id e r a b le sim plifications i f ,

in ste a d of the general equation

(2*1), th e e q u a tio n u'*> i x ) ^ is considered.

f-

i. j\x.O ‘y '^O

y

ii-^)

-r/fj

The c o r r e s p o n d i n g f u n c t i o n

'((t) i s t h e n

o f t h e same f o r m a s t h a t d i s c u s s e d by S c h m i d t , and h a s only a f i n i t e

num ber o f r e a l z e r o s .

Bochner supposes

f i r s t t h a t iOr) h a s no r e a l z e r o s , a n d i t proved t h a t

h ^ (t)

m e th o d i s q u i t e

is

then e a s ily

is a general m u ltip lic a to r.

straig h tfo rw ard ,

i n v o l v i n g a lemma i n

w h i c h he p r o v e s t h a t t h e f u n c t i o n s in teg rab le for a l l

HJt}

are ab so lu tely

, to g e th e r w ith the g e n e ra l

p ro p e rtie s of m u l t i p l ic a to r s .

I t t h e r e f o r e fo llow s t h a t

(7 * 8 ) h a s a s o l u t i o n i f an d o n l y i f

i^'lt belongs to

His

V(t)

By t h e t h e o r y o f r e s u l t a n t s , t h e

s o l u t i o n i s o f t h e form

it where k ( i )

■-

I

(It)

62. Secondly, Bochner supposes t h a t

has a f i n i t e

number o f r e a l z e r o s the

order to obtain

same r e s u l t s h e r e , he i n t r o d u c e s a s e r i e s o f g e n e r a l Py^it)

m u ltip licato rs

» each of which has th e v a lu e u n i t y

i n a s m a l l i n t e r v a l a b o u t t h e c o r r e s p o n d i n g z e r o , an d vanishes outside a s li g h t l y la r g e r i n t e r v a l . considering a m u ltip lic a to r of

it

i s p o s s i b l e t o c h a n g e '{(t)

g i v e n by

i n th e neighbourhood of each

zero in to a non-vanishing fu n c tio n the r e s u l t .

T h e n by

fjt)

w ithout a f f e c tin g

The p r o o f t h e n f o l l o w s i n t h e same way a s

before. A p a r t i c u l a r case of an e q u a tio n of t h i s type i s a pure d i f f e r e n t i a l e q u a tio n . R e t u r n i n g t o th e g e n e r a l e q u a t i o n ( 2 * 1 ), Bochner i s a g a i n c o n c e r n e d w i t h t h e r e a l z e r o s o f {ft) t w h i c h may now be i n f i n i t e

i n number.

In order to lim it him self to

a f i n i t e n u m b e r , he c o n s i d e r s t h e f u n c t i o n w h i c h corresponds to n o tation th is

, a s g i v e n by ( 2 * 6 ) . Is ■