The differential-difference equation $\alpha x f'(x) + f(x-1) = 0$

The differential-difference equation $\alpha x f'(x) + f(x-1) = 0$

Citation for published version (APA):

Beenakker, J. J. A. (1966). The differential-difference equation $\alpha x f'(x) + f(x-1) = 0$. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR36041

DOI:

10.6100/IR36041

Document status and date: Published: 01/01/1966

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THE DIFFERENTIAL-DIFFERENCE

EQUATION

THE DIFFERENTIAL-DIFFERENCE

EQUATION

axf'(x)

+

f(x-1)=0

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCH.E HOGESCHOOL TE EINDHOVEN OP GEZAG VAN DE RECTOR MAGNIFICUS DR. K. POSTHUMUS, HOOGLERAAR IN DE AFDELING DER SCHEIKUNDIGE TECHNOLOGIE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP DINSDAG 22 NOVE:MBER 1966 TE 16 UUR

DOOR

JOZEF JOHANNES ANIONlUS BEENAKKER

GEBOREN TE

•s-

HEER ARENOSKERKE

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR

CONTENTS

General introduetion

7

Chapter I P.reliminary iuvestigations

10

1.1

Introduetion

10

1.2

!llle a.d.joint equation

12

1.3

!lbe Green tunetion Ga(y,x)

13

1;.4

A solution expreseed in terms of its

14

ini tial values

1.5

An integral representation of the Green

16

funotion

Chapter II SJ"stems of special solutions

23

2.1

Introduetion

23

2.2

!llle oase a > 0

23

2.3

!llle oase ex

<

0

33

Chapter III !lbe oase a > 0

35

3·1

Introduetion

35

3.2

!llle saddle points of cp(z)

36

3·3

'.lhe behaviour of Fn (x) for large values

39

of lxl

3·4

!lbe &s1Jl1Ptotio behaviour of ~(x)

51

3.5

Some resul te

57

3.6

!lbe biorthogon&li ty-

59

Chapter IV !llle series E Fn (x)Hn (y)

62

4·1

Dltroduotion

62

4.2

The behaviour of Fn (x) and ~(x) for

62

Chapter V

Series expansion of the Green funotion

70

5.1

lhtroduotion

₇₀

5·2 SOme preliminary resul ts

70

5·3 'lhe case y

>

0

74

5·4 'lhe case y • 0

78

Chapter VI

Series ezpansion of arbitrary solutions

79

6.1

lhtroduotion

79

6.2 'lhe case

x >

y

+

1

79

6.3 'lhe case y

<

x .;

y

+

1

80

Chapter

vn

'lhe a.symptotic beha.viour of the solutions

85

7·1

Introduotien

85

7.2 Some a.eymptotic rela.tions

85

7.3 Some theorema

BB

7·4

~e

sta.bility of the solutions

95

Oha.pter VIII 'lhe oase ex

<

0

98

8.1

Introduetion

98

B.2 'lhe beha.viour of the speoia.l solutions

98

for

lxl .. oo

8.3 The biorthogona.lity

104

B.4 'lhe series E f

₂

n+l

(x)h₂

n+t (y)

105

8.5 Series expa.nsions of arbitrary eolutions

106

8.,6 'lhe a.symptotio beha.viour of the solutions

107

Oha.pter IX

A parking problem

111

Heferenoes

115

GENERAL INTRODUCTION

Analytic number theory is a souree of a great variety of highly interesting asymptotic problems. The subject of this thesis arises from a problem studied by saveral mathematicians, such as A.A. Buchstab [10], N.G. de Bruijn [6],

v.

Ramaswami [13], and others. They discuss the number, W(x,y), of positive integers .;;x which are free from prime factors > y (x >

o,

y ;;. 2). P.:l.rt of the prob-lem of the asymptotio behaviour of w(x,y) for x and y tending to infinity has been solved. If u is a fixed positive number, and

u

if :x:

=

y , then we have

(1) lim y-u W(yu,y) = p(u), y-oo

where p(u) is continuous for u >0 and satisfies the equation

(2) up'(u) + p(u-1)

=

0 (u > 1)

together wi th the ini tial condition p(u) .. 1 (0 ~u .;;; 1 ). In 1951 N.G. de Bruijn

[7]

determined the behaviour of p(u), for large values of u, by means of techniques which are frequently

u~ed in the theor,y of linear differential equations.

Another problem, analogous to the one forinulated above, deals with the number of positive integere .;;;x which have no prime fac-tors < y (:x: > O, y ;;. 2). I f we denote this number by ll(:x:,y), then

we have for any fi:x:ed value of u > 1 (see

[3 ])

(3) lim y -u •log Y·li (yu ) ,y .. u -1 P(u -1 ).

The function p(u) is oontinuous for u >0, and uniquely defined by

(4) _(o

"u "'

_{1 ).}

It was proved in

(3]

that lim u-1p(u) = e-y, where y is Euler's constant.

In the present thesis the more generel differential-difference equa.tion

(5) a.xf1_{(x) + f(x-1) • 0}

will be investigated both for positive and for negative values of the parameter a.. We shall study (5) for positive vatues pf x, and in particular the behaviour of the solutions if x tènds to infini-ty.

Since there is very close similarity between the theory of linear differential-difference equations and that of linear differentlal equations, many of the methode useful for deriving information ahoutthe solutions of linear differentlal equations {suoh as Green funotion, adjoint equation, biorthogonal system, Iaplace transform and saddle-point analysis) oan be extended in such a way as to be helpful in analysing linear differential-difference equdtions. N.G. de Bruijn, in a paper (8] essentially based up9n these meth-ode, treated the equation F1 _{(x) • eett+j3 F(x -1 ).}

~e s~ll

_deal

with the equation (5) in a aimilar way.

In this thesis an infinite set of analytic solutions will be con-structed. These solutions constitute a fundamental system, i.e. they are linearly independent, and any other solution OljJ1 be ex-p~essed, in a unique way, in terms of these by fini~e or infinite linear oombinationa.

From this representation the asymptotio behaviour of a solution oa.n be deduced, expreseed in terms of the in i tial values of that 8 solution.

Chapter 1 contains some prelimi:naey- inTeatiptions. 'l'he ini tial value problem is f'ormula.ted, a.nd the oonoepta of' ed.joint equatiOD., inner product a.nd Green funotion are introduoed.

In ohapter 2, sets of a.:nalytio solutions of' the equation (5) a.nd of the adjoint equation respeotively, are oonstructed by mea.ns of' the Iapla.ce transforma In order to find these solutions we have to deal with the oases

«

> 0 a.nd u < 0 sepa.rately.

The next five cha.pters are devoted to a. discussion of the equa.tion with

«

a fixed positive number.

In cha.pter 3 saddle-point a.nalysis is a.pplied to the La.pla.oe inte-grale repreaenting the special solutions, in order to obtain in-formation concerning the a.symptotio behaviour of these functions for large va.lues of lxl. The principal result in this ohapter is theorem ).2, whioh statea tha.t the sets of analytic solutions of the equation (5) a.nd of the adjoint equa.tion form a. biorthogonàl system.

In oha.pter

5

we prove tha.t the Green function can be expreseed in terms of the funotions of the biorthogona.l system in the form of a.n infinite series (the converganee of whioh is investigated in oha.pter

4 ).

This result can be extended in a simple way to arbitrary solutions (ohapter 6).

The discuesion of the asjmptotio properties of the solutions is the subject of the next oha.pter.

The oase « < 0 is treated in ohapter

a.

A number of details are omitted here aince no new arguments are needed.

Thia thesis ende with a chapter on the a.pplication of the theory to a. certain spa.ce-filling problem.

CHAPTER I

PRELIMINARY INVESTIGATIONS

1. 1 Introduetion

In this thesis we deal with the linear differential-difference equation

( 1 .1 ) a x f 1 _{(x) + f(x -1) = 0}

where a is a constant, either positive or negative.

We shal~ study (1.1) for positive values of x, and in particular the behaviour of the solutions if x tends to infinity.

The following examples illustrate that this linear functional equa-tion has a great variety of soluequa-tions, one of which can be singled out by specifying its values over an interval of length one. Elcample 1

Let g(x) be a continucue function for 1 <EO x <EO 2. We, tryl to find a

function f(x) which is continuous for x ;;. 1, which :equa:).s g(x) for

' I

1 <EO x <EO 2, and which is a salution of (1.1) for alli x > 2.

Since the values of f 1_{(x) for 2 <x <EO 3 are determined by (1.1)}

and f(x) is required to be continuous at x

=

2, we find

x-1

f(x)

= g(2) -

~

J

Î(;

₁) dt (2 ... x <EO 3 ).

1

Since f(x) is now known for 2 <EO x <EO

3,

equation (1.1) determines

f(x) for

3

<EO x <EO

4.

We can preeeed in this fashion, extending the

definition of f(x) from one interval to the next.

The salution so obtained bas a continuous first derivative for x> 2.

Example 2

Suppose tha.t f'(x) • 0 (-1 <x< 0), f'(O) • 1. We try to continue

f'(x) toa oontinuous solution of' (1.1) for x~ 0.

We have f'' (x) .. 0 (0 < x< 1) and theref'ore, beoause of the oon-tinuity to the ~ight at x • 0 : f(x) = 1 (0 <x< 1 ). Ir (1.1) is to be satisfied for 1 < x< 2, we must therefore .have f • (x) .. -1 /ax

(1 < x < 2) and so f (x) • 1 ..

~

log x in that interval. By

repeat-ing this prooess, we oan continue f(x) as far as we please. This

funotion satisfies (1.1) for x> 0 exoept at x • 1. Equation (1.1) is satisfied at x • 1 only in the sense of left-ha.nd and right-hand limi ts.

A boundary condition which prescribes the solution in an initial interval of values of x, from which the solution can be continued, will be oa.lled an initial condition. Henoeforth, we shall ordina-rily impose initial conditions on the solutions of (1.1).

We now introduce the following definition.

Definition 1.1

Assume y ~

o.

let f (x) ba a real- or complex-valued function

de-fined for x ~y-1. Suppose, moreover, tha.t the ratio f (x)/ (x + 1 )

is bounded in y - 1 <x < Y• Then f(x) is said to be "a solution

of (1.1) for x~

r',

if it is oontinuous for x~ y, piecewise

con-tinuous for y- 1 <x< y, and satisfies the equation (1.1) for

all x > y except · those for which x - 1 is a point of discontinui ty. It bas to be understood that f(x) is oontinuoua to the right at x = Y•

Evidently, any function f(x) that is continuoua by parts in y - 1 < x < y, and for whioh the quotient f (x)/ (x+ 1 ) is bounded in that interval, can be oontinued uniqualytoa solution of (1.1)

for x ~ y. In the points of disoontinuity the values of f(x) are

irrelevant exoept at x .. Y•

We notice that a solution for x ~y bas n successive continuous

f .2 The adjoint equation

In the theory of linear differential equations a oertain part

is played by what is called the adjoint equation.

An

adjoint for

differential-differe~oe equations was introduoed at an early date by

R.E.

Borden

[2].

The concept is also used by

N.G.

de Bruijn

[8].

We oonsider the linear differential-differenoe operator L of the

form

Lf(x) • f'(x) +.Lf(x-1)

ax

and the · adjoint operator L* defined by

L* b (x) • - h 1 (x) + h tx + 1

j .

a: x +1 'lhen we have, formally,

b (1.4)

I

{h(x) Lf(x) - f(x) L*h(x)}dx •

q~(b)

-

q~(a)

a where x {1.5)

q~{x)

= f{x)h{x)-

~

I

f{t\h};+1) dt. x-1

If f and h satisfy Lf • 0 and L*h • 0 respeotively, then, by

(1.4),

q~(b)

•

q~(a)

and so

q~(x)

is independent

of~.

Sinoe,

obvious-1

ly, there are oonneotions between the solutions o~ L f • 0 and

L*h •

o,

we shall also oonsider the equation L*h .. O, henoeforth

to be called the adjoint equation, in order to get informtion about the solutions of(1.1). The adjoint equation reads

(1.6) a:{x + 1)h•{x)- h(x+1)

=

0.

It is not hard to see that h (x) ean be taken equal to an arbi trary

pieoewise oontinuous tunetion over an initial cloéed interval of

i

length one entirely to the leftof x • -1. Onoe this has be done,

the rttnotion h {x) ean be oontinued backwarde in a unique way to a

equation (1.1 ) I! the ini tia.l interval is taken

to

the right

ot

the point x • -1, then it ma.y happen tha.t the prooase of

oontinna.-tion will oome to a.n end a.t x • -1 beoa.use of the possible

singu-la.ri ty of h • (x) a.t this point. Sinoe, however, we a.re ma.inly

in-terested in the equation ( 1 •

6)

in order to deduce something a.bout

the solutions of 0.1) for positive vs.lues of x, it suffices to

oonsider (1.6) for x> -1 onlyo This mea.ns that we oa.n confine

our-selves to the following definition.

Def'ini tion 1. 2

Assume

y

>

o.

A rea.l- or oomplex-va.lued tunetion h(x), defined for

-1 <x c

:r

+ 1, ie ea.id to be "a. solution of (1.6) for -1 <x .s; y", i f it is oontinuous for -1 < x< y, continucue by pa.rts for y .s; x< y + 1, and sa.tisfies the equa.tion (1.6) for a.ll -1 <x< y

exoept those for whioh x +1 is a. point of disoontinuity. I t ha.s to

be understood tha.t h(x) is oontinuoue to the left a.t x • y.

Slippose now tha.t f(x) is a. eolution of (1.1) for x> a., a.nd h(x) is a. solution of (1.6) for -1 <x< b, where 0 ca.< b. '!hen the expreesion f(x)h(x)-

~

x

I

f(t) h(t +1) dt t +1 x-1

is a. oonsta.nt for a.ll x in the olosed interval [a., b]. This

invari-ant of the f'unctions f a.nd h will henceforth be oa.lled the inner

product {f,h}, a.s it is bilinea.r over the linea.r solution spa.oee

of (1.1) a.nd (1.6).

1.3 '!he Green f'unotion G« (y,x)

We now introduce a. speoia.l salution of the equa.tion (1.1 ), to be oa.lled the Green f'unotion, which ie of major importa.noe in the following cha.pters. Let 'IJ be a. rea.l non-nega.tive number. '!ben we denote by Ga ('IJ,x) the solution of (1.1) for x> 1Jj subject to the initia.l condition

(1.8)

Moreover, we dei'ine Ga (Tl ,x) • 0 i'or x < T} - 1. By the process of continuatien described in seo. 1.1, we oan continue Gcx(T},x)

indef-ini tely to the right" By this prooess we might establish a

for-mule. whioh gives the value of Gcx(T},x) in any interval

Tl + n < x < Tl + n + 1 (n • o, 1, •••• ). At any .ra. te, suoh a for-mule. would not be pe.rtioule.rly helpf'ul in finding the behe.viour of this funotion e.s x beoomes indei'initely large. For this ree.son

.we shall derive in seotion

1.5

an integral represente.tion bymeans

of the I.aplaoe t.ra.nsi'orm.

We now prooeed to the adjoint eq\lation. We e.saume t to be a non-nega.tive number. T.he Green i'unction G*(x,t) is the solution of

ex

(1.6) i'or -1 <x< t, subject to the initie.l condition G*(x,t) "' 0

ex

(x>

t);

It is not dii'i'ioul t to prove that Ga (tJ

,t)

= GJ'{tJ

,t)

i'or 1) >

o,

t >

o.

T.his is trivial ii' t Cl). Now suppose t > 11• Then we ini'er i'rom

(1.

7) the.t the inner product

(1.10) {G«(T},x), G;(x,t)} •

1

Jx

Gcx(1J,t) G*(t +1,t)

Ga ('l),x) G;(x,t) -

;x

t

~

₁ dt

x-1

is dei'ined and constant throughout the interval 11"'" x < t •

rr

we insart the vs.lues x .. T} and x "' t respeotively, the integral in the e.bove expresaion vaniahea both times e.nd the result oonstitu-tes . the prooi'.

In this way we have shown the.t the Green i'unotion G«(y,x), whioh

is a solution of ( 1.1 ) wi th respect to x, is a solution of (1. 6)

with respect to Y• We notioe the.t Gcx(y,x) .. 1 i'or y

c

x < y + 1

(y > o).

1.4 A solution expreseed in termsof its initie.l vs.lues

T.he Green i'unotion ene.bles us to express e.rbitre.ry solutions

oan be done we use the well-known method of superposition. We

con-sider a. f'unotion f(x) whioh is defined a.nd ha.s a. oontinuous first

derivative for y- 1 .,.; x.,.; y (y > o). In this interval f(x) oa.n

be a.pproxima.ted by the finite sum of Green funotions n f(t₀) Ga:(t₀,x) + ~

1

{f(\:) - f(\:_. 1) }Ga(tk,x) with y - 1 • t < t < ••••• < t • y a.ndt_ .t_

.1

(k=1,2, ••• ,n), o 1 n l [ -k-1 n n large.

Sinoe f(x) ha.s a. oontinuous differentia.l ooeffioient we ma.y even wri te (y - 1

<

x

<

y)

(1.11 )

y

f(x) • f(y-1) Ga:(y-1,x) +

J

ft(t) Ga(t,x)dt.

y-1

'lhis rela.tion expressas f(x) as a.n 11_{infinite linea.r oombina.tion} 11

of Green f'unotions. We oa.n continue f(x) in the usua.l we.y a.s a.

so-lution of (1.1) for x> y. Sinoe the equa.tion is linea.r a.nd

homo-geneoua, we ma.y hope tha.t the right•ha.nd side of (1.11) a.lso

re-presente f(x) for a.ll x> Y• In order to get rid of the

differen-tia.l ooeffioient ft(t), we integra.te by pa.rta a.nd obta.in for the right-ha.nd member of (1.11), provided tha.t x> y

1 Jy f(t) G (t +1 ,x)

f(y) Ga:(y,x)

-;x

t'

_{+ 1} dt •

y-1

This seems to be the expreasion we were looking for. We reoognise it a.s a. special oase of the expression (1.7). It is ea.sy to get rid of the differentia.bili ty oondi tion impoaed upon f. We have the following theorem:

Theorem 1.1

Assume y>

o.

Any funotion f(x) which is a. solution of (1.1) for

x > y,

oa.n

be expreseed in terms of its initia.l vs.lues by the re-la.tion, whioh holde for all x> y,

1

--

a: _J

Y f(t) Ga:(t+1,x)

t + 1 dt.

IToof

Let f; be a.ny number ;;. y. Th en G~{x, f;) is a. solution of ( 1 • 6) for -1 <x .e;

t•

'lhe inner product

1 x f(t)G*(t+1,f;)

{f(x), G;(x,f;)}- f(x) G;(x,f;)-

«

I

tex+1 dt

x-1

is defined and constant throughout the interval y

J..

x

<

f;. If

x • f;, the integral in ( 1 • 1

3)

vs.nishes and we find f(f;) • {f(x), G:(x,f;)} •

1 Yf(t)G*(t+,,t).

f(y) G:(y,t) - ;

J

tex+1

I

ldt •

y-1

The proof now olea.rly follows from the fa.ot tlla.t Gex a.nd G: are

identioe.l. for non-nega.tive va.lues of the a.rguments. The theorem is important a.s it ena.bles us to derive certa.in properties of

a.rbi-tra.ry solutions from simila.r properties of the Green funotion.

1.5 An integral representa.tion of the Green funotion

Using the Ie.pla.oe tra.nsform, we sha.ll deri ve in this section

an integral representa.tion of the Green function. rif f(x) is

de-i

fined for x;;..

o,

then we write

00

(1.15) F(t) •

J

~-xt

f(x)dx

for those t for Which the integral converges a.bsolutely. F is

oa.ll ed the I.a.pla.oe tra.nsform of f. Wi th ( 1 .15) we oa.n tra.nsform the

linea.r funotiona.l equa.tion (1.1) into a. linea.r differentia.l

equa.-tion invalving F(t) and its deriva.tive F'(t) only.

In order

to

esta.blish the oonvergenoe of the a.bove. integral i f ·

f(x) • Gex(y,x), we must ha.ve some a. priori estima.te of the order of magnitude of Gex(y,x) for large va.lues of x. First we a.ssume 16 ex< o. Since G (y,x) • 1 i f y <x .e; y + 1, we infer from (1.1)

that this funotion inorea.ses for x

>

y + 1. Henoe we ha.ve

(x> y

+ 1 ),

e.nd thus

(1.16) (x ;;. y + 1 ).

This upper bound is pretty close to the asymptotio beha.viour as

will be proved later on. The above relation still holde i f « >

o.

In this oase, however, the upper 'bound will turn out to be a very

rough one. It follows from (1.16) tha.t the Iaplaoe transform of

the Green funotion exists for Re ( t) >

o.

Moreover,

co

F(t) •

I

e-xt G« (y,x)dx is e.n e.nalytio funotion in the right

0

half-ple.ne R (t) >

o.

Now we sha.ll eva.luate F(t). For oonvenience

e

we repeat the definition of G (y,x): ex

(x< y),

G

(y,y)

= 1 ex (y;;.O),

.!..

G (y,x) • - j_ G (y,x•1) dx « «X « (x > y, x

r

y + 1 ). -xt

I:t we mul tiply the equation by e a.nd integra.te from y to co, we

get

CIC) CIC)

(1.18)

I

e-xt ! G« (y,x)dx • ..

~

I

e : t Ga (y,x -1 )dx •

y y~

The integral on the left oe.n be expreseed in terms of F(t) by

mea.ns of integra.tion by parts

(1.19)

co

I

e · x t ! Gex(y,x)dx • .. e-yt + tF(t).

y

We denote the right-he.nd side of (1.18) by q~(t). Then

CIC)

(1. 20) _q~'t ( ) _==-1 I • x t _{e G y,x-1dx=-e} ( ) 1 - t ( _{F t ) .}

ex

«

«

18

Hence, F(t) sa.tisfies the linear differentie.! equa.tion

FOr convenience, we shall deal with the cases y

=

0 a.nd y > 0 separately.

(i)

y

=

o.

The general solution of (1.21) is given by t

(1. 22) F(t) = c exp

{~

J

e-ss-1 ds }tt/a. -1

0

where cis a.n arbitrary constant. The function

t

1

/~

-1ib defined

by t 1

/a ""'

= exp { ( 1

/a

_{-1 )log t}; log t denotes the}

~rinc~pal

_value

of the logarithm of t, i.e. the value whose imaginary part lies between - n and n.

'!he constant c can be determined by evalua.ting, in two different ways, the asymptotic beha.viour of F(t) for large positive values of t. First we deduce from

00

F(t) •

J

e-xt Gcx(O,x)dx •

J

e-xtdx +

J

ëxtGcx(O,x)dx

0 0 1

that

(1. 23) F(t) ,..., 1/t (t ... 00 ) .

Secondly, we have by formula (1.22)

1 00 00 c 1

J

e-s -1 F(t) "'- exp{-_{t a} - -_s 0 ds

+~Je;

ds

-~

Je:s ds} (t>1). 1 t Since 00

I

7

-s ds

=

r ,

1

where r denotes Euler1_{s constant, it follows that}

(1.24)

(t

> 1 ).

Oomparing (1.23) and (1.24), we find c .. er/a.. We oa.n state the result au the following theorem.

00

( ) I

e-xt

The I.aplace tra.neform F t = Ga(O,x)dx of the Green

func-0

tion Ga.(O,x) ie given by

t (1. 25) F(t) .. exp {y/a + 1/a

I

e-8

8- 1 ds}t 1

/a-1

0

where y ie Euler 'a constant.

(R _e

(t)

>

o),

We notice that the right-ha.nd side of (1.25) is a.n a.na.lytic func-tion in the complex t-pla.ne cut a.long th,e negative real a.xis from the origin to infinity and so it provides the ana.lytic continua-tion of F(t) in tha.t region.

(ii)

y

>

o.

The general solution of (1.21) is now given by

00 u (1. 26) F(t) • F (t) • [c+yJexp{-yu-1 Ie-8 - 1ds}ü1/adu], o a s t 0

where o is an a.rbi trary constant and F ( t) is defined by

0 t

F

(t) .. t

1/a -1 exp {

1

I

e-s -1 ds} •

o a s .

0

The mul ti-valued functions t 1/a -1 and u-t

/a

are defined by exp{(1/u -1 )logt} and exp(-1/a log u) respectively, all loga-rithms having their prinoipal values. Finally, we note that the pathof integration (t, oo) should not interseet the negative real

20

The constant o can be determined by the metbod used previously in

the oase y •

o.

Flrst we deduoe from

that (1.27) co F(t) •

J

e-xt Gcx(y,x)d:x: • y

f

+1 -xt -xt • e d:x: + Je Gcx(y,x)d:x: y y+1 (t > 1 ).

Next, putting u • t + v (t > 1) in (1.26), we obtain

-yt "" -t v -s (1.28) F(t) • cF

0(t) + y T

J

exp{-yv ..

e«

l

Js,+tds}dv.

0

:o

In oase (i) we proved that the first term of the right -hand side

-y/cx t

of (1.28) is c T < 1 +O(e- )). Sinoe the seoond term is

-yt t

7 ( 1 +O(e- )), it follows that

(t

> 1 ).

Comparing the asymptotio formulae (1.27) and (1.29) we lfind o = 0.

We state the result as the following theorem.

Theerem

1.3

Assume y >

o.

Then the ls.plaoe transferm F(t) ""

I

e·xt Gcx(y,x)dx 0

of the Green funotion G«(y,x), is given by

co u

F(t)

=

yF

0(t)

J

exp

{-yu-~

I

@-ss .. 1 ds}u·t/« du,

t 0 where t

1/.

J

-s F ( t ) • t « -1 e:x:p

{l

L....=-1.

ds } • o ex s 0

It should be remarked that under the oonditions previously

mention-ad the right-hand side of

(1.

30} is ari a.nalytio function in the

complex t-plane cut along the negative real a.xis from 0 to - oo ,

and 80 it provides the analytic continuation of F(t) in that

re-gion.

Sinoe the Green funotion is oontinuous for x > y and of bounded

variation on any finite interval, the inversion formula for the

Iaplace transform yields (see [

11 ] ,

§

17)

P.+ip

(1.31)

Gu(y,x)·

2

~

1

lim

J

ext F(t)dt (x> y >0),

p-oo ~-ip

where

p

is an arbitrary positive number.

Replacing t by -t we find by means of

(1.25)

and

(1.30)

'lheorem 1• 4

Iet ~ be a.ny positive number. Then we have for x > 0

Gu(o,x) • .I -~+ip t eYt«

I

1

Ies-1

1/."'"'

-2 1 _1t lim - 0 0 exp{-xt +-a: - 8- d s } (-t) a: dt. p -~-ip 0

'lhe mul ti-valued function ( -t) 1 /a: -1 is defined by

exp{ (1/a: -1 )log(-t)}, where log(-t) denotes the principal value of

the logari thm of -t. 'lheorem 1 •

5

Iet ~ be any positive number. Then we have for x > y > 0

Gu{y,x)•

-~+ip t

2~

limoo

I

A(t)exp {-xt

+~

I

ess-1 ds}(-t)1/a:..., dt,

p- -~-ip 0 where (1 0 34)

t

V S A{t) = y

I

exp{yv-~

I

6 8 -1 ds}(-v)-1/a:dv. -oo 0 · 21

22

'lhe mu1 u-valued tunotions

<

-t

>

1

I a. ..

aru1 c-v

r'

la.

are defined in the usual way. Jlbreover, the pa.th of integra.tion

(-oo,t)

should not interseet the positive rea.l axis.

We note tha.t the integra.nds in (1.32) a.nd (1.33) are a.na.lytio tunotions in the complex t-plane out a.long the positive rea.l a.xis trom the origin to infinity.

CHAPTER 11

SYSTEMS OF SPECIAL SOLUTIONS

2.1 Introduetion

It is well known that any function satisfying a linear homo -geneaus differential equation can be wri tten as a linear combina-tion of a finite number of particular solucombina-tions. In the present thesis we shall see that the solutions of the differential-diffe-rence equation (1.1) can likewise be written as sums of particu-lar solutions. In this connection, however, our equation bas to be regarded as a differential equation of ~ order. In the follow· ing seotions we shall first construct infinite sets of analytic solutions of ( 1.1 ) and ( 1. 6) respect i vely, where we have to dis-tinguish between positive and negative values of a. It is not dif-ficult to see that the case a< 0 can be reduced to the case a> o. The reasoning is as follows. The substitution

h(x)

=

f(-x-1) transforma the adjoint equation (1~6) into the equation -a x f' (x)+ f (x -1) =

o.

If h (x) is a salution of (1.6 ),

then f (x)

=

h(-x -1) satisfies the linear funotional equation (1.1) where a is ·replaoed by -a. Conversely, if f(x) is a solu-tion of (1.1 ), then h (x) = f ( -x -1 ) satisfies the adjoint aqua-tion where a is replaced by - a.

2.2 The case a > 0

We try to solve (1.1) by a Iaplaoe integral f(x)=Je-xzg(z)dz.

VI

I:ntegrating by parts, we ha~e, formally,

(2.1) x f' (x)

=

-x Ja-xz z g(z)dz =

w

= e-xzzg(z)

1-

Je-xz{zg1_(z)+g(z)}dz.

Suppose now that the path W and the function g(z) have been oho • sen in such a way that the integrated term vanishes, then f(x)

satisfies (1.1) i f

(2.2) Je•xz{a zg•(z)

+(a

-ez)g(z)} dz • 0.

w

Tbe integrand is identically zero if g(z) is a solution of the

linear differential equation azg1_{(z) + (a-ez)g(z) •}

o.

_Clearly,

z

{1 J.es-1 d }

expä"

- s - s ' 0

z

1

/a -

1

having i ts principal value, is an analytic solution of this '

equation throughout the z-plane cut along the negative real axis from the origin to -co. Since this function tends to zero very rapidly as z tends to infini ty along the half ~linea z =

.:t,

ni + t

(t

>

0), we choose W as a curve starting at -lti +co and tending to

1t i +co , avoiding the negative real axis. For this choice of g and

W, the integrated term in (2.1) vanishes, and the same thing is

true for the contours W + 2n n i, where n is any integer (W + 2n ni

is the ourve described by z +2n ni i f z describes w). Now, one oan

easily verify that the functions Fn (x), n = O,

.:t,

1,

.:t,

2, ••• , de-fined by (2.;)

z

i S' F (x) = - 1-

J

z 1

la-'~

exp {-xz +

1

J

L=-1

ds} dz n 2ni ex s

w

+2nlti 0

are holomorphic and satisfy (1 .1 ) for all complex values of x ..

Because of the discontinuity of the integrand on the negative

rea.l axis we must demand that the pa.th W + 2nni. a.voids this

ha.lf-line•

Now let w₁ and W

2 be contours starting at the crigin and tending

to ni +co and -'!ti+ co respectively, both contours avoiding the

neg-ative rea.l a.xis. For this choice of the path of integration the

integrated term in (2.1) vanishes in either case. ~e functions

z s

F (x) "' -₊ 1-

J

z 1

la.""'

exp {-xz + 1

J

L:..1

de }dz, 2n i

a.

s

w,

0 and z a 1

J

1/. ""' 1

J

e 1 (2.5) _{F_(x) • 2rc} i z

a.

exp {-xz +ä --.,;- ds}dz

w

2

o

are entire funotions aa.tiafying ( 1.1 ) for all va.luea of x.

It should be noted tha.t

(2.6) F (x) .. F+(x) - F (x).

0

-In wha.t follows we aha.ll alao use the salution F(x) where

Still another salution of (1e1), in the half-plane R (x)< 0, ia

e 0

z

(2.8) N(x) •

2

~

1

J

(-z)1/a-1 exp {-xz

+~

Je 8 6 -1 ds }dz. -~ 0

The mul ti-valued funotion ( -z) 1 /a -1 ia made defini te by choosing

(-z) 1

/a -

1 real on the negative real axis.

Dealing similarly wi tb the a.d.joint equation, WA find that the

funotions (n • O, .:!: 1, .:!: 2, •• , • ) z Yf

~(x)"'

.1

Jz·1/a-'1exp{(x+1)z . 1 J e 6 - 1ds}dz 2n., a

a.

a W + (2n+1 )ni 0

are analytio everywhere and satisfy (1.6) throlJ#out the whole

-1/

tr

-1

complex X""Plane. '!he mul ti-va.lusd funotion z a is defined in

the usual way. Beoause of the discontinuity of the integrand on the nega.tive real axis the path of integration, whioh leads from

'.loreover, it can easily be verified tha.t the function z

(2.1 0) =-

~

J<-z(

1

1a

-1

ex:p{(x+1)z-~

J

ess_1 ds}dz

r

o

is an analytic solution of ( 1. 6) in the whole complex x-plane.

The contour

r

leads from= to 1 in the lower half-plane, eneirales

the origin once in the negative direction, and returns to co in the

upper half-plane. The function (-z

f

1

/a -

1 is defiri.ed by its

prin-cipal value.

We add to the eet {W

2n+1, W0} the functions W+(x) and W_(x) which

are solutions of the adjoint equation for R _e (x) > -1 only. We have

(2.11)

--a z s

f

_z

-1/.'

_a -1 _exp {( _{x + 1} ) _{z -}

_ä

1

~

_-

e -1 } 8- ds dz p1 and (2.12) z s W (x) _{_} = -

.1

_a:

f

z-1/ " -1 exp {(x +1 )z-

.!.f

_a

~

_s ds }dz _' p2 0 P

1 being a curve lying entirely in the upper half•plane Im(z) > 0

and leading from z = - co to z = oo , while P

2 is the complex

con-jugate of P₁• /.

(. )-1

la

-1 - 1 . a-1

Since the principal values of -z and z i are related

by ( )- 1/a -1 ni/a - 1

/a

-1 -z = - e z (I (z) >

o),

m (I (z)<O), _m we have (2.14) W x) ( = - e n:.i/a W x ( ) + e -n:.i/a W x ( ) 0

+

-

(R e (x) > -1).

Omitting the proof, we now state the following lemma about the

Lemma 2.1

Assume

a:

>

o.

:F\Jrther, suppoee tha.t n is a. fi:x:ed integer, a.nd tbat & is a.ny positive number

<

n/2.

J:f l~(z)- (2n+1)nl <n/2-e, u•Re(z), thenforu""00

z

(2.15)

zt/a:exp

{~Jes

s-1 ds} • 0

{exp ( -

2~u

sine

)l •

0

I f

I

Im(z) - 2n n

I

< n/2 - e, u"' Re(z), then for u .... oo

(2.16) z-1

/a

exp

{-~

Jze 8 8 -1 ds} • O{exp(- ₂: : sine)}. 0

The two constante implied in the o-ayrnbols depend on

a:,

n only.

Furthermore, the two functione occurring on the left-band side of the a.bove relations, are bounded in the part of the half - pla.ne Re(z)..:; 0 where Jzl ~ 1.

It follows from Cauchy's theorem and lemma 2.1 tha.t a. vertical

shift of the path W + 2nn i over a. distance les a tha.n n/2 ha.s no influence upon the va.lue of the integra.l on the right -hand side of (2.3). Needless to say, the new pa.th should not interseet the cut along the negative rea.l a.xis. Moreover, we notice tha.t a

ver-tical shift of the contour in (2.9) over a dista.nce leas tha.n n/2

does not affect the va.lue of the integral.

Since the equation (1.1) is linea.r, we can genera.te new solutions

by forming linea.r combinations of known solutions. In this wa.y we

can construct a.n infinite set

{tp

₂n+1}, n =

o,

.±

1, ••• , of

solu-tions by mea.ns of the relasolu-tions

(2.17)

If n "'

o,

the sum I: is vaouous, and its valu.e bas

to

ba in• 1 <k<n

terpreted as

o.

The same convention holds for I: , if

n+1 <k<-1 n • -1. These functions, which are analytic solutions of (1.1) for Re (x) <

o,

play an important part in the oase ex <

o.

By means of lemma 2.1 and formula (2.1)) one ea.sily verifies that

(2.19) co +(2n+1 ) 1ti z 1

J

1~

""' 1'

Jas

-1 cp (x) .. 2 -1 z ex exp{- xz +-., - ds }dz. 2D+1 1t ex' S .co +(2n+1) 1ti 0

Dealing similarly wi tb the adjoint equation, we define a set

{H ,

ÎÎ} ,

n ..

0,

.:!:

1,

.:!:

2, ••• , of solutions by the relations n

.

(2. 21)

B:n (x) "'

11!' +(x) + I: 11!' 2k+1 (x) O<k<n-1 (n > 1 ), (2.22)

(n

< -1).

These funotions are analytic in the balf-plane R (x) > -1.

e I f n

f

0, we have (2.23) co+2n1t i Z 8 H (x)=-1

I

z-'/ex-1exp {(x+1)z..1

I~

ds}dz. ·n ex ex s -co+2nn i

o

In the following chapters we shall prove tbat the solutions F(x),

Fn (x) (n •

o,

.:!:

1 , .t 2, ••• ), eens ti tute a fundamental system of

the equation (1.1), that is to say they are linea~ly ~ndependent,

and

any

.function f (x) tbat is a solution for x > y >

o,

can be ex•

pressed, in a uniqua way, in terms of the functions of the system

by means of finite or infinite linea.r combinations. We have (see

GO

f(x) • {f,Îi} F(x) + l: {f,H } F (x)

. .oo n n (x>

y)J

the ooeffioients in the above series are defined by the expression

(1.

7)

for the inner produots.

It should be noted that the funotion N(x) does not appea.r in

(2.24)J it is needeel only if we shoUld develop solutions of (1.1)

for x <

o.

Although N(x) is originally defined for R (x) < 0 only, e

it oan be oontinued analytioally to the right half-plane R (x)>O

e out along the positive real axis from the origin to infinity. We shall disouss this problem here as it is a very interesting one

in saveral respeots. Our metbod for establishing the continuatien

of N(x) depende on Cauohy 's theerem whioh enables us to deform the path of integration in (2.8) in suoh a way that we obtain formu-lae whioh hold in a wider region.

Theorem 2.1

The analytio continuatien of N(x) (which is also denoted by N(x))

throughout the upper half-plane ~(x)

>

0 is given by

0 N(x) •

2!i

J

(-z)

1/a.

-s

z s { 1 J e - 1 } exp -xz

+a

- a - ds dz' •ioo 0

and this relation still holds for

any

rea.l x <

o.

Again, (- z) 1

/a: -

1 is defined by its prinoipal value.

Proof

Let v be a po si tive number > 1. From lemma 2.1 we infer tha t the

pathof integration (.oo,O) in (2.8) may be replaced by a curve

consisting of the parts

(i) the half-line z • u - iv

(ii) the vertioal line segment

(-oo<u~O)

"

z - - iu (v> u>

o).

It is easily seen that the integral along the horizontal line (i)

30

(vR (x) ) .. 1 e -vim,(x). '!'hen, mak:ing v...,. co , i t follows that the equal-e

ity (2.25) holds whenever I (x);;.. 0, R (x)<

o.

Obviously, the

m

e

right-hand side of (2.25) is an analytic function in the

half-plane ~(x)> 0, and this completes the proof.

'l'heorem 2. 2

'!'he analytic oontinuation of N(x) to the lower half-plane

~(x) < 0 is given by 0 z N(x) .. -1-

J

(-z) 1

/a..,

exp {-xz

+1

J

es - 1 ds }dz , (2. 26) 2ni

a

s ioo 0

and this relation also holds for any real x <

o.

This theorem can be proved in the same way as theorem 2.1. Both

theorema together establish the continuatien of N(x) over the whole complex x-plane cut along the positive real axis from 0 to "" • Of course, N(x) satisfies the linear differential- differenoe

equation (1.1 ) in that region. The behaviour on the upper and the

lower sides of the out is very remarkable. In order to see this

we substitute, in (2.25), z = .. iu. Aftarsome trivial

oalcula-tions we obtain (2. 27) z s (-z)1/a.., e:xp {-xz

+~

J e 6• 1 ds} = 0 c {. 1 =-exp 1Xu--u

a

""

J

7 d s } -is u

where c is a constant (depending on a only).

Fartial integration yields

00

(2 .. 28)

--

1

ex

_J

r

e-is ds s = cxu e-iu + 0

(-1-)

2

u

u

(u ;;..

1 ),

Hence, z (2. 29) ( )1

J:

-1 { 1

I

e 8 - 1 -z a exp -xz +- -a s ds}"' 0

=

~

eixu

{1

+.2:_e-iu +

o(..1...)}

u au 2 (u ~ 1 ).

·u

Then it can easily be verified that the integral on the

right-hand side of (2.25) is a continuous function in the region

I (x)~ 0 (x

r

0), satisfying (1.1) on the positive real axis m

for x > 1. In the same way i t can be shown that

the right of (2.26) is continuous for I (x).;;; 0

m

isfies (1.1) for x> 1.

the integral on (x

I

0 ), and

sat-The jump of N(x) at the cut - which is defined by its value on the upper side minus its value on the lower side - equals

(2. 30) ioo 2:i

I

(-z)1/a. -1 z s { 1 I e - 1 } exp -xz

+ä

-

₈- ds dz (x> 0 ). -i"" 0

I f we now combine this result with theorem 1.4, we find that we have established the following theorem.

T.heorem 2.3

The~ump of N(x) at the positive real axis is equal to

eJOf Gcx(O,x), wherey is Euler's constant.

In the half-plane R (x)> 0 the function N(x) can be expressed in

e

termsof the solutions F+' F_, Fn (\n\ ~ 1 ).

'fheorem 2.4

I f R (x)> 0, I (x)> 0, then

e m

(2.31) N(x) = enifa F (x)- eni/a I: F (x),

32

and this relation also holde for x

>

0 with the value of N(x)

oor-responding to its value in the upper half-plans. Ir R (x) > O, I (x)

<

O, then

e m

N(x) • e-rd./a. F (x)+ e-rd./a. I: F (x),

+ n>1 n

and this equality is still true for x > 0 with the value of N(x)

taken from the lower half-plane.

P.roof

We first take x in the region R (x) > O, I (x) >

o.

For a.ny

inte-s m

ger m > 1 we have

- F (x ) + I: F (x ) ..

- _{-m..;;n..,.;;-1} n

0 -(2m+1 )ni+co

• 2:i

r

exp{~J~(z)}

dzz -

2~i

[, exp{IP(z)}

~z

-&m+1 )ni - 2m+1 )ni

where, wi th the prinoipal value of the logari thm,

liS s ( ) 1 1

Ie

-1 lP z "' -

_a.

log z - xz + -

_a.

- -_s ds • 0 z 1

/a.

1

I

es -1

Sinoe the funotion z exp {- - - ds} is uniformly bounded

a.

s

0

wi th respect to m on the half-linea z • -(2m + 1 )n i+ u (u > O,

m > 1), the second term on the right-hand side of (2.33) is, in absolute value, lesa than a constant multiplied by m-1 (the

con-stant depending on x and a only ). Then, keeping x fixed and making

m - co , we obtain

0

(2.34) - F (x) + I:

- n..o;-1 Fn(x) •

2~i

I

exp{cp(z)}~z

•

Flnally, (2.31) follows. from (2.13), (2.25) and (2.34). '!he seoond part of the theorem oa.n be proved in the same wa.y.

2.

3

'!he case «

<

0

In order to obtain an infinite set of solutions of (1.1), we

start from the system

{V

,

"f , "f } defined in the previous

sec-2n+1 o +

tiono Replacing x by - x- 1 and « b:y - ex , we find as solutions of

(1.1) the functions (n •

o,

.:!:

1,

.:!:

2, ••• )

(2.35) f2n+1(x) • 2n:i 1

I

z ex

1~

-1 exp

{-xz

+ ; Jz e s-1 ds}dz s

W+ (2n+1 )d 0

z

fo(x) •

2~1

I<-z)1/ex-1 e:x:p{-xz

+~I

es;1 ds}dz,

r

o

z s

n(x) •

2

~

1

j(-z)

1

/ex-1

exp{-xz

+~Je

; 1 ds}dz.

1

The multipliaa.tive factor 1/2ni bas been chosen in accordance with

formula (2.3

h

W, rand P

1 are defined in sec 2.2. It should be

observed tha.t f an+

1 , f 0 are entire funotions, while n (x) is

a.na.-lytic for R (x)

<

o.

e

Dea.ling simila.rly with the adjoint equa.tion, we obta.in, sta.rting

from the set

{<p

₂n+1 (x), N(x)} defined in the previous section, a

system {h

0, h2n+1 }, n ...

o,

.:!:

1, ••• , of analytic solutions for Re(x)

>

-1. We have, n ..

o,

±

1, •••,

(2.~) h2n+t (x) ..

00 +(2n+1 )d z

1

I

-1

~

-1 {( ) 1

I

e8 -1 }

• - ëi

z ex exp x + 1 z -

ëi

- s - de dz ,

-~(2n~)~ 0 0

z

h₀(x)

"'-~

I

(-z)-

1

/«-

1

e:x:p{(x+1)z-~Je

8

;

1 _ds}dz. -oo 0 33

34

It will be shown later on that the functions f

0, f2n+1 (n = O, .;t1,

+ 2, ••• ) constitute a fundamental system of

(1.1).

Any function f(x) that is a solution of

(1.1)

for x> y >

o,

can be expreseed in terms of these solutions by means of finite or infinite linear combinations. We have (see theerem 8.4)

co

f(x) .. {f,h } f (x) + .E {f,h } f (x)

0 0 2n+1 2n+1

-co

(x>

y).

The function n(x) does not appear in the above formula; it is need-ed only if we develop solutions of

(1.1)

for negative values of x. It should be noted that n(x), which is originally defined for R (x)< 0 only, can be continued analytically over the whole

x-e

plane cut along the positive real axis from the crigin to infini-ty, the jump at the positive real axis being equa] to (q'cxGa(O,x)

(see also theerem 2.3). Finally, we note that n(x) oan be expand-ed in termsof the solutions fo' f2n+1 (Jnl >

o),

if x lies in the half-plane R (x) _e > 0 (see also theerem 2.4).

CHAPTER 111

THE

CASE a>O

)o 1 Introduetion

So far we have only aarried out some preliminary investiga-tions needed for a further detailed discussion of the linear

func-tional equation. In the previous chapter we had to distinguish

be-tween a positive and a negative in order to obtain sets of special solutions. For this reasen it is convenient to deal from now on with the two cases separately. Throughout the next chapters we as• sume a to be a fixed positive number. We already mentioned that the most important theorem to be derived statea that any function

f(x) constituting a solution of (1.1) for x~ y ~ O, can be writ•

ten as an infinite linear combination of the form

00

{r,H}F(x)+

l:

{r,H }F

(x)

-oo n n (x > y).

From this representation the asymptotic properties of f(x) for positive values of x can be deduced. This gives rise to the following problems. First, in order to establish the converganee of the above series, we must know the behaviour of F (x) and H (x)

n n

for large values of lnl when x is any positive real number. Sec-ondly, we have to determine the asymptotic behaviour of the spe-cial solutions as x approaches infinity. Both problems can be solv-ed by means of saddle point analysis. Satisfaotory results can

be obtained by using the saddle points of the funotion ~(z), where

z

(3.1)

_~ ( ) z = 1

ä

log z - xz + 1

;x

J

ess-1 ds

0

3.2 The saddle points of ,(z)

The saddle points of ,(z) are the roots of the equation ''(z)

= o.

We have ''(z)

=

•X + ez/(az), and therefore ''(z) = 0 i f ez -=axz. Let S(e,ö) be the sector: e ~!x! <oo,

iarg xl ~ n/2- b, where e and bare small positive numbers (& < n/2). Fbr each integer k, denote by

Tk

the half-strip

(2k -1 )n ~ I (z) ~ (2k + 1 )n, R (z) ~ 1 •

m e

Now we shall prove the following theorem concerning the distribu-tion of the saddle points of ,(z) over the half-plane R (z) ~ 1.

. e

Theorem 3.1

There exists a large positive number p

0, depending on

a, e

and b

only, such that the equation ''(z) ~ 0 has for each x € S (e,ö) one and only one salution in every horizontal strip

Tk

with 12kni + log cxxi ~ p

0• :FU:rther, this root in the k-th strip,

de-noted by ~k' is the sum of an absolutely convergent double power series 1 m ~k = p + log p + -. I: I: clm c:1 -. 1=0 m=O 00 00 where

(3-3) p = 2k ni +log ax, c:1

=

1

jp, -.

= log

pjp,

all the logarithms having their principal values; the olm are con-stante, and we note that o

=

c

=

1, c

=-

!.

00 10 01

Pro of

Let P be the set conaisting of the complex numbers

p .. 2kn i+ log cxx, where k runs through all integer values and x € S (e,ö). I f p € Pand [p.j is su.fficiently large, say,

!PI

~c

1

, 36 then larg pj~n/2 + &/2. Now choose x in the sector S(e,ö) and,

keeping x fixed, choose k such that

jpj

~ c

1,

IPI

~ e

2_{/(a~). The}

linear transformation

(3.4) u

=

z - p - log p

maps the part of Tk where Re (z) ~- 1 + logl axl + log lP

I,

into

the infinite strip

(3-5) R (u)~ -1.

e

Under this transformation the equation ~1_(z)

=

_{0 takes the form}

e u - 1 - au - -r = 0

where o and -r are defined by (3.3).

Fbr the time being, we ignore the relation existing between o and -r , and we shall consider them as small independent complex.

param-eters.

N.G.

de Bruijn proved (see

[9], §

2.4)

that there exists

a small positive number 1'lt such that, if

j

cl

< 1'},

I

-rj < TJ, the

equation (3;6) has just one salution in the infinite strip (3.5),

and that this solution lies in the domain I uj <

b/2.

FUrther, this

solution, denoted by u, is the sum of an absolutely convergent double power series

u "" 't

r:

E olm a -r 1 m 1=0 maO co (3. 7) (jol < 1'lt 1-rl < TJ), where the c 1m are oonstants.

We now return to the special values of a and -r given in formula

(3.3), viz. o = p -1, -r "' log pfp. For

I

pI sufficiently large, say, jpl ~ c

2, we have joj

<

1'}, 1-rl < TJ• Keeping x fixed, choose k large enough in order to guarantee that

I

pI • 12k ni + log a

xl ;;..

max: ( c , o , e2 /(a~)). For those values of

1 2

k, each half-strip

I

Im (z) - 2knj .,;;; n, Re (z) ;;,. -1 + logj axl +log

jpl

oontains just one saddle point of the funotion ~(z), and this

It still remains to be proved that

''(z)

does not vanish through-out the part of Tk where 1 .;:; Re (z) .;:; -1 + logj axl + log I

PI•

In this region, for any fixed value of I (

z),

the modulus of ez/z

m

is an increasing funotion of R (z). On the vertical line segment

e

z = -1 + logjax! +log

IPI

+ 2k1ti + vi (- 1t.;:; v.;:; n) we have lzl ~~ IPI, provided !PI is large enough. Hence

(3.8)

I

1

<

Re(z).;:; -1 + logiaxl +log IPI ), and this completes the proof of the theorem.

Since p = 2k ni +log ctx, we have

IPI

~

p

0, if at l!east one of

ikl, lxl is a large positive nurnber. This means that, if x lies

far to the right inside the sector S(e,ó), say, lxl ~x , eaoh

hor-a

i:;ontal strip Tk Cl kj ~ 0) contains just one sa.ddle point, and

this is given by the series (3. 2 ). We will localise these saddle

points more precisely~ In order to do this we start from (3.2) and

take a few terms only Clxl ~x

0

, jkj ~ 0)

(3. 9) l;,k ""p +

2

p

+~

+O((lo; p ) ) .

Now write

arg x =

e,

{2k 1t +

e

)/log

I

a xl q.

After some simple oaloulations we find

(3.1 0) (l;, ) .. k

arctan q - q

log~

1 + q2 - q loglog

I

ax

I

2k7t+

e + arotan

q + - _

{1 + q2 )log

I

ax!

with an error term which, in absolute value, is leas than a

~

log

I

cal )-1

log(~

•log

I

cal ). Considering the above expression for large a.nd sma.ll vs.lues of lql respectively, it fel-lows tha t (

I

xl large )

(k

>

0 ),

(k < 0 ).

Next we deal with the oase where x is real and positive but not neceasarily large. I f x ;;;. E a.nd lkl is large enoUBh, aay, jkj ;;;. k

0 we have I pI

=

12k ni+ log a x I :;;;. p • Fbr those vs.lues of k each

0

half-strip

Tk

containa just one root of the aquatien ~'(z)

=

0, and this root is given by (3.2). If a.;;; x.;;; b (e.;;; a< b), we may write

(3., 12) tk = 2k n _{i + log 2knl.aX + · 2kni + 0} . log 2k n:i (

k

1 )

the constant implied in the 0-symbol does not depend on x. Hence, if also a .;;;x .;;;b,

I(!;.)= 2kn+1!.agn[k] _logJ 2knl +0(1)

mk 2 · 2kn k

where sgn [k] denotes the sign of the integer k.

The saddle points l;.k are not the only roots of the equation

~ '(z)

=

o.

From the graphs of the functions ez and a x z (z real) we see that ~1_{(z) = 0 has a solution lying close to the origin,}

provided x is a large positive number. Denoting this solution by z, we have z

=

(ax)-1 + (ax)-2 + O(x-3 ).

~.3 The behaviour of Fn(x) for large vs.lues of

I

xl

In this section we shall apply saddle point analysis to the integral (2.3) in order to obtain information concerning the be-haviour of F (x) for x far to the right in the sector S(e,ö): _n

39

40

e

<\:x\

< oo, \arg

:x\ .;;;

n/2 - b (e >

o,

0 < b < n/2). All formu-lae labelled

":x ..

oon will hold uniformly wi th respect to both n (n ..

o,

.±

1,

.±

2, ••• ) a.nd arg

:x

as

:x

~oo in

s(e,b).

Formule. (2. 3) rea.ds

(3.14) Fn(x) .. 2:i

I

exp{cp(dd:,

W+2nni

where, referring to our previous notation,

z S

( ) 1 1 J e - 1

cp z .. - log z -_a :xz + -_ex - -_s ds , 0

a.nd Wis described in seo. 2.1.

In the previous seotion we pointed out tha.t every ~orizbntal strip

Tk(\k\ ~ 0) contains just one.sa.ddle point of the function cp(z),

provided that

\:x\

is large enough. As the pathof integration

W + 2nni ca.n be kept entirely inside the strip Tn we may expect

to be able to restriet ourselves to the saddle point tn• We there-fore have to lookfora path tha.t crosses tn' while tn itself is the "highest point" on it. Clearly, tha.t path will depend on n,

arg

x

a.nd

I

x\ ,

a.nd the major trouble is caused by the dependenee on

arg x. This diffioulty may be overcome by application of

conform-al mapping. Substituting z • t + t, the sa.ddle point is shifted

n to the origin a.nd we obta.in

c~.16)

where W + 2n ni • t is defined as the contour described by

n

t + 2n ni- tn' i f t describes

w.

From lemma 2.1 a.nd fom.Ula (3.11)

we infer that we ma.y repla.ce this contour by the pa.th W-i arg

x,

whioh does not depend on n. :Ebr convenianee we put a.rg x •

e.

The integra.nd ia an analytio funotion in the complex t-plane cut

along the line joining the points •t and - t - oo. We notioe

n n

the.t thia he.lf-line lies far to the left in the he.lf-plane Re (t)

<

o.

We are free

to

modify the path auoh the.t it passes

through the se.ddle point t ..

o.

In order to know how i t must cross

the point t == 0 we must have a olear idee. about the behe.viour of

the integrand in the neighbourhood of the origin. By Taylor series

expansion we have, for all I tI <

I

tn I '

(3.18) cp (tn + t) .. cp (tn) + cp" (tn)

TI

t2

+ epi" (tn)

.:t.:.

3

l + • • • •• • • The derivatives of q(z) at the point tn oan be evalue.ted by dif· ferentiation of the right-he.nd side of.the expresaion (3.15).

(3.1'9)

IJl''

(t ) • x(1 - j_ ) ,

n tn

and, i f k OIO :;, k fixed,

(3.20) cp(k) (t ) • x(1 +

o(J-))

n '"n (x - oo ).

The e.xis of the se.ddle point (for this oonoept see

[9],

P• 84)

is

the straight line through the origin defined by

IJl''

(t )t2 rea.l and <

o.

n

The argument of the a.xis is ~-i arg(,"(t

11)) and this tends to

1t 9 (

2 - 2

i f x .. 00 _{the argument is uniqualy determined apart from}

an additional multiple of

n).

Along this straight line the value

ta

of exp { cp11 _(tn)

2 }

_{is in all points exponentially eme.ll , apart}

from a sma.ll segment e.round t •

o.

Now i t is vecy important to know

whether in the Te.ylor expansion (3.18) the sum of the terms

t2

is small compared to the term f" (tn)

2 ,

if t lies close to the origin. If it is small, the second-order term determines the con-tribution of the saddle point.

In fact, it oan bl! p:roved that for any integer N;;.. 2 and for eaoh point t inside the circle with centre 0 and with radius R (R is a fixed, but arbitrarily large, positive number)

(3 .. 21 )

The constant in the 0-symbol depende on both N and R. In order to verify this expression we start from formula (3.15). Putting z = F;.n + t, we obtain

t

(3. 22)

q~(F;.n

_{+ t)}

₌

q~(F;.n)

_{- xt +}

x~;.n

J

_{s ::n}

d~

_' where the integral is taken along a straight line.

The function (s +F;.n)-1 can be written as (N;;.. 2,

lal

ç R)

Then we easily find (N~ 2, ltl ç R)

N-1

q~(F;.n + t) = q~(F;.n) - xt + x I:

k=O

(x - co).

Expanding the functions on the right into powers of t and

neglect-. N+t tN+2 h mak . 0 (X· tN+1 ) and

~ng terms t , , ••• , t e error we e 1s this completes the proof.

We now choose the path of integration such that it crosses the saddle point while the angle it makes in t

=

0 with the axis of the saddle point, is leas than ~/4. Then it is easy to evaluate