Generalizations of the bilateral laplace transform

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE

TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OF GEZAG VAN DE RECTOR MAGNIFICUS IR. H.R. VAN NAUTA LEMKE, HOOGLERAAR IN DE AFDELING DER

ELEKTROTECHNIEK, VOOR EEN COMMISSIE, AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN, TE VERDEDIGEN OP

WOENSDAG 20 DECEMBER 1972 TE lU.00 UUR

DOOR HA]^I LEMEI wiskundig ingenieur getoren te Bandoeng BIBLIOTHEEK TU Delft P 1895 2020 C 608276

Dit proefschrift is goedgekeurd door de promotor

Prof.dr. B.L.J. Braaksma.

Introduction

Chapter 1. Inversion formtilas for integral transforms related to

a class of second order linear differential equations 1

0. Introduction . 1

1. Special solutions 2 2. Inversion formulas for functions vanishing in a

neighborhood of i» 5 3. Proof of the inversion formulas 11

Chapter 2. Representation formulas for the integral transforms of chapter 1

k. Introduction 5. Preliminaries

6. Representation theorems for functions analytic in a strip

T. Applications to chapter 1 and 2

11+

m

23 31 Chapter 3. Inversion formulas for integral transforms related to

a class of even order linear differential equations 8. Introduction

9. Special solutions

10. Proof of the inversion formulas

UO

^1 53

References 57

0. Introduction.

This thesis deals with a couple of generalizations of the bilateral Laplace transform, (see Widder, The Laplace transform, ch. VI.)

The kernels of the bilateral Laplace transform exp(±Xx) are solutions of the differential equation —-^ - X^y = 0, -<» < x < «>, where exp(±Xx) are the "small" solutions when Re X > 0 and x->- +<» respectively.

In chapter 1 we study the pertiirbed equation -r-^ - {X^+q(x)}y=0, _oo < X < «>, where q(x) is a continuous function defined for all real values of x such that there exist constants a and b with q(x)-a e L(0,°°), q(x)-be L(-°°,0). In this case we are able to construct two solutions y (x,X) and yp(x,X) which behave as exp(-/x2+a x ) , as x->-+«>. Re /X^+a > 0; respectively as exp( /x^+b x ) , as x->-<». Re A ^ + b > 0, These solutions are then used as kernels instead of exp(+Xx), and we prove a couple of in-version formulas for these transforms linder L conditions.

In chapter 2 we continue the study of chapter 1 and state sufficient conditions that a function ^(X) be representable as an integral transform on -"^ < X < °°, with the above mentioned kernels.

(3^2n 2n

In chapter 3 the even order equation — r — y-{X + q(x)}y=0, _oo < X < a>, where q(x; is a continuous function defined for all real values of x and q(x) e L(-°°,»), is considered. We construct solutions y_^(x,X) of this equation which behave as exp(+Xx) respectively as x->+«> on the sector where larg Xl 5 •T/2n. We then prove a couple of expansion formulas, using the y+(x,X) and their rotations in the X-plane as kernels, under L conditions.

Inversion formulas for integral transforms related to a class of second order linear differential equations.

0. Introduction

We consider the differential equation: (0.1) ^ - ^^2, _{^ - {X2+q(x)}y = 0,}

where q(x) is a continuous function for all real values of x. Furthermore we assume that there exist constants a and b such that

(0.2) q(x) - a e L(0,OO), q(x) - b e L(-«>,0)

In section 1 we will construct solutions y (x,X) and y (x,X) of (O.l), which are characterized by the following asymptotic behaviour in x = +«> and x = -°o:

(0.3)

y (x,X) exp /X2+a x -v 1 , T- {y.(x,X) exp /x2+a x} ^ 0 as X ->• +00 and Re /X^+a > 0;

(o.M

y (x,X) exp - /x2+b x -> 1 , ^ {y2(x.^) exp - vV+b x} ->- 0 as X -> -00 and Re /P+b > 0,

The purpose of this paper is to prove that imder certain conditions on f(t) and the path of integration the following inversion formulas hold:

(0.5) and

(0.6)

X +ioo

dX _{W(X) •'2' o}

yAx^,'>^)

f(t)y^(t,X)dt = hi{f(x^+0)+f(x^-0)}, X -i"»

X +ioo dX

W(X) y.(x ,X) f(t)y,^(t,X)dt = ^TTi{f(x +0)+f(x -O)}, i o d o o X--100

2.

where W(X) denotes the Wronskian of y (x,X) and yp(x,X), and the integrals over X are defined as their principal values.

The exact formulation of (0.5) and (0.6) with the conditions will be given in theorem 3 in section 3. In section 2 we prove extensions for inversion formula (0.5) in the case f(t) = 0 for t < x , and for (0.6) in the case f(t) = 0 for t > x . These are formulated in the theorems 1 and 2.

o

These results are used in the proofs of (0.5) and (0.6) in section 3. A corollary of these theorems given in section 2 is related to a result of Titchmarsh on the differential equation (O.l) with q(x) defined and real for 0 < X < 00 [1^ chapter IX].

1. Special solutions of equation (O.l)

It is easily shown that the solution of the initial value problem (O.l) with (0.3) is equivalent to that of the integral equation:

1.1) u^(x,X) = 1 +

2/X2Ta for the function

[. -2/X2+a(t-x)i r i^\ \ /'x -v

\^4-{1-e }{q(t)-a} u.(t,X)dt

(1.2) u (x,X) = y (x,X) exp /X^+a x. Re /x^+a > 0, x2+a i 0,

occurring in (0.3)- This equation has a unique solution which may be found by the method of successive approximations:

u^ Q ( X , X ) = 1 , u. ^(x,X) = 1 + — ^ f {l-e-^'^^^^^*-''h{q(t)-a} u, ^ Jt,X)dt

^'"^ 2/x2n J ^'""-^

X (n = 1,2,...). Putting 00 (1.3) e^(x) = Iq(t)-a|dt, X

we obtain by mathematical induction:

^ {e^(x)}'' lu. (x,X) - u. ,(x,X)| < —r

(1.1+) I u , ( x , X ) - l | < - 1 + e x p and from (1 .2) and (1 .1+):

/F+a

0 ^ ( x )

(1 .5) | y , ( x , X ) - e x p - / X ^ + a xl <l exp-/X^+a x K - 1 + e x p • >

/)J^\

for Re/x^+a >0, x2+a7^0. In a s i m i l a r way t h e f u n c t i Lon

(1.6) u„(x,X)=y-(x,X)exp-/X^+b x is the solution of the integral equation:

X (1.7) u_(x,X) = 1 + — ^

2/X'^+b -00

{1_e-2*''^^+^(^-^)}{q(t)-b}u^(t,X)dt,

where ReA^+b>0, x2+b?^0. Putting X

02(x) = |q(t)-b|dt, (1.8)

we find in the sam,e way:

( 1 . 9 ) | / F T b | c 0 (x) ) ( 1 . 1 0 ) | y p ( x , X ) - e x p A ^ + b x | < | e x p /x2+b x H - l + e x p

e (x)

|up(x,X)-l|<-1+exp A^+b

5

for ReA2+b>0, x2+b^0. Furthermore we need in the proofs of the formulas another special solution y^(x,X) of (O.l). y (x,X) behaves like exp-A2+b x for x>-<», ReA2+b>0. Let c be an arbitrary real constant. Then we choose for the solution y-,(x,X) of (O.l) the function

(1.11) y_(x,X )=u^(x,X ) exp - A'^+b x,

where u_(x,X) is the solution of the integral equation

(1.12) U2(x,X)=1 +

L _ [ {l-e-2*^^^^(*-^)}{q(t)-b}u^(t,X)dt,

Putting

we f i n d i f x S c t h a t

( 1 . 1 ^ ) |y (x,X) - exp - A2+b x | < |exp - A2+b x | j -1 + exp

e3(x)

I

A^+b

for Re A2+b > 0, x2+b ^ 0. Now we approximate the Wronskian (1.15) W(X) = y^(x,X)y^(x,X)-y2(x,X)y'(x,X)

as X->oo in the region D of the X-plane where Re A^+a > 0 and Re A2+b > 0, From (1.2) we derive:

(1.16) y:|(x,X) = {u^(x,X) - A2+a u^(x,X)} exp - A2+a x, and from (1.1):

u'(x,X)= - [ e-^''^^^^^^-^) {q(t)-a} uJt,X)dt, J

X

In view of (1.3) and (l.l+) we have:

e/t) e^(x)

|u^(t,X)| < exp • < exp

| A ^ ^ | |/x2+al

for t > X, (1.17) |u'(x,X)|< exp i/x)

A?+a|

|q(t)-a|dt = e (x) exp

e/x)

A^

+a Hence u'(x,X) = 0( 1 ) as X->ooon D with x fixed.

For y-(x,X) we obtain from (1.5):

y^(x,X) = {1 + 0(^)} exp - A2+a x, and from (l.l6), (1.17) and (l. 1|):

y'(x,X) = - A ^ n {1 +0(^)} exp - A^TI x.

In a similar way

y - ( x , X ) = {1 + 0 ( | ) } exp A2+b x ,

Substituting in (l.15) we obtain

(1.18) W(X) = 2X (1 + Oij)), as X-x» on D.

In the same way we obtain for the Wronskian W (X) of y (x,X) and

(1.19) W__(X) = -2X (1 + 0(f)) as X-x» on Re A^+b > 0.

A property of the solutions of the differential equation (O.l) which will be used later reads as follows: the function K(x,t;X) satisfying (O.l) as a function of x with initial conditions

K (t, ti X) = 0, |- K ( X, t; X) I = 1 ,

''^ x=t

is equal to

(1.20) K(x,t;X) = /(x,X)y(t,X)-y(x,X)y*(t,X) W(y,y*;X)

for any pair y and y of linearly independent solutions of (0.1), where W(y,y ;X) is the Wronskian of y and y .

2. Inversion formulas for functions vanishing in a neighborhood of + oo or - 00

First we prove a generalization of the inversion formula (0.5) in the case that f(t) vanishes for t < x .

o

Theorem 1. Let x , X and X ^ be real numbers with X ^ > X , X, > 0.

As-o' o 1 l o l

sume q(x) is continuous for x > x and there exists a constant a with q(x) - a e L(x ,oo).

Let y (x,X) be the solution of (0.1) satisfying (0.3).

Let f(t) be defined for t > x , be of bounded variation in a right-hand neighborhood of t = x , and

6.

(2.1)

-X t

f(t) e ° e L(x ,00)

H is the region in the X-plane where Re A^+a" > O, and C is a contour in H consisting of the line Re X = X with contingently some finite detours in H.

Let cp(X) be a function analytic in the region G to the right of C and continuous on the closure G of G with

(2.2) cp(X) = e ° (1 + 0(1)) as X-^ on G. Then X +ioo (2.3) dX cp(X) y (t,X)f(t)dt = 7rif(x +0) X -i»

where the integral in X, taken over C, has its principal value.

Proof: Substituting v = A2+a we see that formula (2.3) is equivalent to

A x +iu)2+a

(2.1+) li _im dv ^•^ Ax^-iy)2+a x^ T T(X)y,(t,X)f(t)dt = TTif(x +0) A I O

Let G be the image of G iinder the transformation v = A^+a, Then the function

(2.5) g(t,v) = j'f{\)y^{t,X)

is analytic in G . From (1.3) we have 6,(t) < e j x ) for t > x . Hence from 1 1 1 o o (2.5), (2.2) and (I.5) we obtain:

(2.6)

g(t,v) = {exp v(x -t)}{l + i|^(t,v)}

where

(2.7) '('(t,v) = 0{±) as v->«o Qj^ Q uniformly in t for t ^ x .

Let l(t,y) be defined by

Ax^+iu)

-+a : 2 . 8 ) i ( t , u ) = g ( t , v ) d v , / ( X ^ - i y ) 2 + a From ( 2 . 6 ) , ( 2 . 7 ) and C a u c h y ' s t h e o r e m we f i n d f o r t > x^ ( 2 . 9 ) I ( t , y ) = o ° - i y o°+iy A X ^ - i y ) 2 + a / ( X ^ + i y ) 2 + a g ( t , v ) d v . Now we s p l i t I ( t , y ) i n t w o p a r t s I ( t , y ) a n d I ( t , y ) l / t , y ) = ° - i y 00+iy ( 2 . 1 0 ) ' ' A X ^ - i y ) 2 + a / ( X ^ + i y ) 2 + a exp v ( x - t ) d v = o ^^—^ { e x p C / ( X ^ + i y ) 2 + a ( x ^ - t ) ] - e x p [ / ( X ^ - i y ) 2 + a ( x ^ - t ) ] } , ( 2 . 1 1 ) _{l 2 ( t . y ) =} " - l y 00+iy {exp v ( x - t ) } i f j ( t , v ) d v . / ( X ^ - i y ) 2 + a / ( X ^ + i y ) 2 W F i r s t we p r o v e : ( 2 . 1 2 ) • l i m y-Ko --X o f ( t ) l f t , y ) d t = •rrif(x + 0 ) , I o -X t From ( 2 . 1 0 ) we h a v e I ( t , y ) = 0 ( e ° ) f o r t - ^ u n i f o r m l y i n y . Hence b y ( 2 . 1 ) : 2 . 1 3 ) l i m f ( t ) l ^ ( t , y ) d t = 0

uniformly in y. Furthermore, in view of (2.10) sin y(x^-t) X^(x_-t) (2.1^) l/t,y) = 2i 1 X -t o • r-0 "' . , ^ ^ V * ^ e + X(t,y) e where (2.15) X(t,y) = O(^)

8.

as y-x» uniformly on x < t < r. Since f(t) is of bounded variation in a right-hand neighborhood of t=x we obtain (2.12) from Dirichlet's theorem.

Next we prove:

(2.16) lim y-x»

f(t) l2(t,y)dt = 0,

which implies with (2.12)

( 2 . 1 7 ) l i m

X o

f ( t ) I ( t , y ) d t = TTif(x +0),

Again we s p l i t t h e i n t e g r a l i n ( 2 . l 6 ) i n t o two p a r t s ƒ and ƒ , where s

X s , s . . . . O

is chosen such that f(t) is of bounded variation on the interval x < t < s. o

The real and imaginary parts of f(t) therefore can be written as the difference of two monotonie functions on x < t ^ s. We will prove that

(2.18) l2(t,y)dt -^ 0

as y-^^ uniformly on x S s ^ s. Then applying Bonnet's mean value theorem it follows that

(2.19) lim y-x»

f(t) l2(t,y)dt = 0.

To prove (2.l8) we deduce from (2.1l) and (2.7):

l2(t,y)dt < K -ly / (X^-iy)2+a 00+iy /(X +iy)2+a |v| Re V ^^-^^P ^^ v(x^-s^)}|dv|

Since 0 ^ 1-exp Re v(x -s,) ^ 1, the last integrals tend to zero as y-»^ o 1

uniformly on x < s < s, which proves (2.l8). Furthermore from (2.7) and (2,11) we have:

-X t 1

l2(t,y) = e

° 0{p

oe

f

lim

y->oo J

f(t)l2(t,y)dt = O, which together with (2.19) implies (2.l6).

From (2.8), (2.5), (1-5) and condition (2.1) it follows that I(t,y)f(t)dt =

/(X +iy)2+a OS

(

dv \ 9(X) y^(t,X)f(t)dt. o /(X -iy)2+a o

Passing to the limit and using (2,17) we obtain (2.U).

From theorem 1 we may deduce a generalization of the inversion formula (0.6) in the case that f(t) vanishes for t > x .

Theorem 2. Let x , X and X, be real numbers with X > X , X^ > 0. As-o' o 1 1 As-o' 1

sume q(x) is continuous for x < x and there exists a constant b with o

q(x) - b e L(-oo,x^).

Let yp(x,X) be the solution of (O.l) satisfying (0.1+), Let f(t) be defined for t 5 X , be of bounded variation in a left-hand neighborhood of t=x , and

o ^ o (2.20)

X t

f(t)e ° e L(-o°,x )

K is the region in the X-plane where Re A2+b > 0, and C is a contour in K consisting of the line Re X = X with contingently some finite detours in K.

Let cp(X) be a function analytic in the region G to the right of C and con-tinuous on the closure G of G with

(2,21) (p(A) = e' ° (1 + 0(1)) as X-voo on G. Then (2.22) X +ioo dX 9(X) y^(t,X)f(t)dt = uif(x -0) X -ioo

where the integral in X, taken over C, has its principal value.

Proof: If y(x) is a solution of (O.l) then y (x) = y(-x) satisfies the differential equation:

2

10. Then a = b, y (x,X) = y (-x,X). If we replace x by -x , then it is easily seen that theorem 1 applied to the differential equation (2.23) gives formula (2.22).

Adding the results of theorems 1 and 2 we have:

Corollary 1. Let x , X and X be real numbers with X > X , X > 0. Assume q(x) is defined and continuous for all real values of x and there

exist constants a and b such that (0,2) holds,

Let y (x,X) and y (x,X) be the solutions of (O.l) satisfying (0,3), (0,1+) respectively.

Let f(t) be defined for all real values of t, be of bounded variation in a neighborhood of t=x , and

-X |t|

(2.21+) f(t) e ° € !(-«>.").

D is the region in the X-plane where Re A 2 + a > 0 and Re A^+b" > 0, and C is a contour in D consisting of the line Re X = X with contingently some finite detours in D.

Let 9.(X) and 9p(X) be functions analytic in the region G to the right of C and continuous on the closure G of G with

(2.25) 9,(M = e ° (1+0(1)), 9,(X) = e °(l+0(l)),

as X-x» on G. Then X,+ioo 1 ( (2.26) where $(X)dX = ïïi{f(x +0)+f(x -0)}, o o X^-io 00 o ( (2.27) $(X) = (p^(X) y^(t,X)f(t)dt + '^J^X) y2(t,X)f(t)dt, X -00 O

In (2.26) the integral is taken over C and has the principal value.

Corollary 2. Suppose q ( x ) , y (x,X) and y (x,X) are the functions of section 0. Then in theorem 1 we may choose 9(X) = y (x ,X) on account of (2.2) and (1.10). In theorem 2 we may choose 9(X) = y (x ,X) on account of (2.21) and (1.5). In the same way in corollary 1 we may take 9.(X) = yAy. ,X ) , 92^'^) = y^(x , X ) . Then we obtain (2.26) with

X

o $(X) = y-(x ,X)

d o y^(t,X) f(t)dt + y^(x^,X) y2(t,X)f(t)dt, X

o

A related result for the differential equation (O.l) with q(x) defined and real for 0 S x < oo can be found in Chapter IX, in particular 9.6, of [1].

Remark 1. It is easily seen that the preceding resiilts can be adapted to the case that q(x) also depends analytically on X. Then the assumption

(0.2) has to be replaced by the following conditions: |q(x,X)-a| < q^(x) with q^(x) e L(0,oo), |q(x,X)-b| < q2(x) with (i^{^) e L(-oo,0), for Re A2+a > 0 and Re A2+b > 0.

3, Proof of the inversion formulas (0.5) and (0.6) The exact formulation of (0.5) is given in

Theorem 3. The constants a and b and the functions y.(x,X), y (x,X) and W(X) are defined in sections 0 and 1.

Let e, X and x be real numbers with e > 0, X > 0. Let f(t) be defined for real t, be of bounded variation in a neighborhood of t=x , and

-(X -e)t -(X +e)t

(3.1) f(t) e e L(0,oo), f(t) e e L(-«',0). D is the region in the X-plane where Re A2+a i 0 and Re A2+b > 0, and C

is the contour in D consisting of the line Re X = X with contingently some finite detours in D such that the zeros of W(X) are to the left of C.

Then (0.5) holds.

Furthermore (0.6) holds if the condition (3.1) is replaced by (X +£)t (X -e)t

(3.2) f(t) e ' e L(0,«), f(t) e ' e L(-~,0). In (0.5) and (0.6) the integrals in X are taken over C and have their principal values.

12.

Proof: From (I.I8) it follows that W(X) / 0 for sufficiently large [xj in D. Hence there exists a contour C which satisfies the conditions of the theorem. It is sufficient to prove the theorem for the following two cases:

I. f(t) = 0 for t < X , o II. f(t) = 0 for t > X .

o

Case I is comprised in theorem 1 with 9(X) = 2Xyp(x ,X)/W(X), The con-dition (2,2) is satisfied because of (I,l8) and (l,10).

Now we restrict ourselves to case II, Introducing K(x,t;X) with y = y-|(x,X), y = yJx,X) in (l.20) we see that (0,5) for this case is equivalent to X +ioo

lia

y-x» dX _{y i ( x . ^ )} W(X) -^1' o y2(t,X) f(t)dt + X^-io X +io + lim y-H» XdX K(x ,t;X)f(t)dt = 5TTif(x - 0 ) , o o X -i^

By theorem 2 with 9(X) = 2Xy (x ,X)/W(X) the first limit in this equation is equal to 5irif(x - 0 ) ,

So we have to prove that the second limit vanishes, or putting a = A 2 + b that (3.3) l i m y-x» / ( » ,

^<S

+ i y J - i y )2+b )2+b ado K(x ,t;X) f(t)dt = 0. Now (3.1+) v'(X^+iy)2+b IP a K(x ,t;X)da = ''(X^-iy)2+b -ly -ly /(X^-iy)2+b •^(X^+iy)2+b ly = i^ ^ I2 - I3.

On the line Re o = 0, cr 9^ 0, the functions y„(x,X) and y (x,-X) are defined, and linearly independent. This last assertion follows from (l,10):

y2(x,X) = e^^ (l + o(l)), y2(x,-X) = e'^^ (1 + o(l)), as x-+ -oo. Therefore these functions may be substituted in formula (1.20) for

K(x,t;X). Then we see that K(x,t;X) is an even function of X, and conse-quently of a for Re a = 0. Hence I. in (3.^) vanishes.

For I and I we use:

y^(x,X)y (t,X) - y2(x,X)y (t,X)

(3.5) K(x,t;X) = -^ ^p-^-5 ^

(cf. (1.20), where in the definition of y we choose c > x ), From (I.1I+), (1.10) and (1,19) we deduce:

a K(x ,t;X) = i {1 + o{-)] exp o(x -t) - i {1 + 0(1)} exp a(t-x ),

as v-^ uniformly in a on the line segments joining -iy with /(X -iy)2+b, and iy with /(X +iy)2+b. Hence

12 + 1 3 = - ^ Ccos /(y+iX^)2b (tx^) -o

, 1 (X +e)(x -t) - cos /(y-iX )2_b (t-x )] + 0{-) e ^ ° .

1 o y

From this, the lemma of Riemann-Lebesgue and (3.1) we obtain: X

lim

/(X^+iy)2+b

dt f(t) aK(x ,t;X)dG = 0. /(X^-iy)2+b

From (3.5), (1.10), {1 .^h) and (3.1) the changing of the order of integration in the last formula can be justified.

So we obtain (3.3), and the proof of (0.5) is complete.

By using the same substitutions as in the proof of theorem 2, and condition (3.2) instead of (3.l), we may deduce (0.6) from (0.5).

Remark 2. In the theorems the condition that f(t) is of bounded variation in some neighborhood of t=x may be replaced by some other condition sufficient for the convergence of an ordinary Fourier series,

I l + .

CHAPTER 2 .

Representation formulas for the integral transforms of chapter 1.

k. Introduction

In this chapter we give sufficient conditions that a function $(X) be representable as an integral transform on -oo < x < o°:

eo

(U.1) ^ ^ ( ^ ) = f.(x)y (x,X)dx, with

(U.2) f , M - ^ $(X)y.(x,X)dX, (i=1, j=2 or i=2, j=1)

•p

where the kernels y.(x,X), (i=1,2), are solutions of the above differential equation characterized by the asymptotic behavior in x = +00 and x = -00 respectively. See section 1, where these kernels are constructed. The contour C in the complex X-plane will be specified below. The conditions for the validity of (l+,l) and (l+,2) are similar to those for the representa-tion of funcrepresenta-tions as a bilateral Laplace integral (cf, [3], ch. VI, §19, p. 265).

In section 5, which is an extension of section 1, further properties of the kernels y.(x,X), (i=1,2), are derived. In addition another solution, of the above differential equation, is derived along with an auxiliary function A(x,X,X ) which plays an important role in the proof of the main theorem.

The condition for the validity of (l+.l) and (1+.2) are given in theorems 1+ and 5 of section 6.

Section 7 is devoted to applications of the main theorems of the previous and this chapter.

5. Preliminaries.

First we define the region D of the X-plane which will be considered in the discussion of (l+.l) and (1+.2).

If neither of the complex numbers a and b lies on the negative real

axis or is zero, then D will be the X-plane cut along the curves where x2+a < 0 and those where x2+b < 0. These cuts are symmetric with respect to the origin and do not intersect the real X-axis. One such cut, A , is shown

o

in fig. 1. If a is on the negative real axis or is zero, the curve x2+a < 0 consists of the imaginary X-axis and the straight line-segment with

end-2

2

^A^A^ A .^"'AA^A,

X-plane Fig. 1.

Similarly the curve x2+b < 0 has the same shape when b lies on the negative real axis or is zero.

Consequently when a and/or b lies on the negative real axis or is zero, D will be the half plane Re X > 0 cut along the curves x2+a < 0 and x2+b < 0.

In the following the square roots A2+a and A^+b will denote their branches with positive real part in D .

Lemma 1. Let y.(x,X) and y (x,X) be the solutions of (O.l) as constructed in section 1.

Let 9.(x,X) and i p . ( x , X ) , ( j = l , 2 ) be d e f i n e d by: J J

( 5 . 1 ) y^(x,X) = {1 + 9^(x,X)} exp - A 2 + a x ,

16.

(5.3) y2(x,X) = {1 + ^^{x,X)} exp A2+b x,

(5.1+) y2'(x,X) = A2+b {1 + if; (x,X)} exp A2+b x, for X e D .

Let r be a positive constant and x a real constant. o

Then:

(i) cp.(x,X), \|j.(x,X) = 0(1/X) as X-»^ uniformly for (-1)"^ x s x and X e D .

(ii) 9-(x,X), 4/.(x,X) = o(l) as (-1)"^ x^ +oo, uniformly for X e D and |x2+a| > r if j=1 respectively for X e D and ix2+b| > r if j=2. Proof: We give the proof for the case j=1. The proof for the case j=2 is similar.

First we prove (i). From (1.2) and (1.1+) we have:

e.(x)

|cp,(x,X)| = lu,(x,X)-l| < exp _ - 1,

^ ^ | A 2 ^ | and from (l.l6) and (I.I7) we have:

u '(x.X) u '(x,X) |'i'(x,X)| = |u.(x,X) 1 1 1 < |u.(x,X)lI + ' ^ '

-' I /1 9 . I _{A ^ n ^ A 2}

e^(x) e^(x) e^(x)

i exp — — ^ 3 - - 1 + ——^;;;^ exp +a A27i:i | A 2 ^ i | A 2 T r i When X S X we see t h a t o 9.(x,X) < exp ^ ° - 1 = 0{j-) and e , ( x ) QAx ) e . ( x ) l ^ . ( x , X ) I < exp ^ - ^ - - 1 + ^ - ^ exp ^ - ^ - = 0 ( 1 ) ,

^ lA^TTI lA^^I \/x^\ ^

as X-x», X € D . o Hence ( i ) i s v a l i d .

Next we c o n s i d e r ( i i ) . Here we have:

e j x )

! 9 T ( X , X ) | < exp - ^ 1,

6^(x) e^(x) e^(x)

|9,,(x,X)| < exp -^— - 1 + -j^— exp /r

From (1.3) we deduce 6 (x) = o( 1 ) as x^«, therefore 9 (x,X), if» (x,X) = o( 1 ) as x-x» for X e D and |x2+a| > r.

o ' '

Lemma 2. Let y (x,X) be the solution of (O.l), defined by (1.11) and

(l.12), depending on the real constant c. Let e and r be positive constants. Let 9 (x,X) and i); (x,X) be defined by:

(5.5) y3(x,X) = {1 + 93(x,X)} exp -A2+b x, (5.6) y '(x,X) = -A2+b {1 + i|;3(x,X)} exp -A^+b x, for X 6 D .

o Then:

(i) There exists a constant c = c (e,r) such that for x < c < c |q5-(x,X)| < e and |t|^ (x,X)| < e, for X e D and |x2+b| > r.

(ii) 9„(x,X), if; (x,X) = 0(l/X) for fixed x 5 c as X-x» uniformly on D . Proof: First we prove (i ). From (1.11), ( 1 .11+) and (5.5) we have:

e^(x) 0 (-00) |9^(x,X)| = [u (x,X) - l| < exp ^ - 1 < exp ^j - 1,

^ ^ |A2TF| '^'^

and from formulas analogous to (l.l6) and (1.17) we have:

u_'(x,X) u '(x.X) k fx,X)| = lu^(x.X) - 1 - -^ i < I u (x,X) - 1| + 1-^ I <

^ ^ A^Tb ^ A2Tb

e (-C») e (-oo) 03(-") < exp -j^— - 1 + -^j^~ exp - ^ ^ — .

Since 0 (-0°) = ƒ ^| q(t)-b| dt -+ 0 as c-+ -oo, we may choose the real constant c such that |9^(x,X)| ^ e and |I|J (x,X)| < e for x ^ c S c .

The proof of (ii) is similar to the one given in lemma 1 for cp (x,X) and iJj.(x,X) and so is omitted.

V7e now construct another solution yi(x,X) of (O.l) which will be used below.

18.

(5.7) u^(x,X) = 1 + - ^ 2A2+a

{1 - exp - 2A2+a(x-t)}(q(t)-a)u^(t,X)dt, where Re A2+a > O, x2+a ^ O, x > d and d an arbitrary real constant, has a unique solution u, (x,X). The function

(5.7a) y^(x,X) = u^(x,X) exp A^+a x,

is a solution of the differential equation (O.l) with Re A^+a" > 0, x2+a i- 0, X s d.

Lemma 3. Let 9, (x,X) and i|^, (x,X) be defined by: (5.8) yi^(x,X) = {1 + 9i^(x,X)} exp A2+a x, (5.8a) y^'(x,X) = A^+a {l+i|^, (x,X)} exn /X^+a x, for X e D . Let e and r be positive constants.

o Then:

(i) There exists a real constant c> = c, (e,r) such that for x > d > c» : I9, (x,X)| < e and |ij;, (x,X)| < e for X e D and |x2+aj > r.

(ii) 9,(x,X), if;, (x,X) = 0( 1/X) for fixed x > d and X-x» uniformly on D . The proof is analogous to that of lemma 2 and therefore it will be omitted.

Let the Wronskian of y.(x,X) and y.(x,X), (i ,j = 1 ,2,3,1+), be denoted by W^.(X). We write as in ch.1 W(X) for W (X).

ed Lemma 1+. Let r and e be positive constants with e < — . Consider the

functions y (x,X) and yr(x,X) in the cases that the constants c and d us in their definition (cf. (I.II), (1.12), (5.7) and (5.7a) satisfy c < c (e,r) and d > c. (e,r). Here c (e,r) and Ci (e,r) are the quantities occurring in lemmas 2 and 3•

Then:

(i) W„_(X) and W,, (X) have no zeros for X e D and |x2+b| > r and |x2+a| > r 23 14 o ' ' ' ' respectively.

(ii) W^_(X) = 0(1) as X-x» uniformly on any subset of D on which Re X is

13 o bounded.

Proof: First we prove (i). Using (5.3), (5.1+), (5.5), (5-6) and the definition of W (X) we get

(5.9)

y^^{X)= -2A2+b[1 + H93(x,X)+i(;3(x,X)+92(x,X)+iJ;2(x,X) + + ^^{x,X)\l,^{x,X)+<f^{x,X)^^{x,X)}l.

From this, lemma 1 (ii) and lemma 2 (i), the result follows for V7 (X). The proof for W , is analogous.

Secondly we prove (ii). Using (5.1), (5.2), (5-5), (5.6) and the definition for W (X) for some fixed x < c S c_ we get:

W (X) = [(A2+a-A2+b)-A2+b{9^(x,X)+;|; (x,X) +

+ 9^(x,X)ii;3(x,X)}+A2+a{93(x,X)+i(^^(x,X)+i|;^(x,X)93(x,X)}3

. exp -(A2+a+A2+b)x.

From lemma 1 (i), lemma 2 (ii) and the boundedness of Re X the result follows, Let W{y.(x,X), y.(x,X )} denote the Wronskian of the functions y.(x,X) and y.(x,X^), ( i,j = 1,2,3,1+).

We now define an auxiliary function A(x,X,X ) by:

(5.10) A(x,X,X^) = p ^ X T W{y^(x,X),y2(x,X^)}. o

Lemma 5. Let X and X be point of D , X ^ ± X . Let r be a positive constant. o o o

Then A(x,X,X ) has the following properties: (i) (d/dx)A(x,X,X ) = y.(x,X)y^(x,X ).

o I 2 o

(ii) A(x,X,X ) = 0(l/X) as X-x» uniformly on any subset of D on which Re X

o o is bounded, for any fixed x.

(iii) (X2-A2)A(X,X,X ) = {(v+v )/2v } exp {(v -v)x}{W(X )+o(l)}, as x-x» o o o o o o

uniformly for X e D and |x2+a( > r.

(iv) (x2-x2)A(x,X,X ) = W(X){(a+o )/2a} exp {(a -a)x}{1+o(1)}+o(1), as x->- -00, uniformly for X on any subset of D on which Re X is bounded and |x2+a| > r, |x2+b| > r,

20.

Proof: It is sufficient to consider values of r with r < |x2+a|. ' o '

First we prove (i). Differentiation of A(x,X,X ) with respect to x and substitution from the differential equation (O.l) with X and X yields the result (i).

Assertion (ii) follows from lemma 1 (i) and the boundedness of the real part of X.

Next we prove (iii). Choose the constant e such that 0 < e < — and choose the constant d in the definition of y, (x,X) (cf. (5.7) and (5.7a)) such that d = c, (e,r) (cf. lemma 3). From lemma 1 (ii) and lemma 3 (i), it follows that there exists a constant d > c,(e,r) such that |9-(x,X)| < e,

|iJ;.(x,X)| < e for j = 1,l+ and x > d , X e D , |x2+a| > r. By lemma 1+ (i) W.i (X ) / 0. We may therefore write yp(x,X ) as the linear combination of y/x,X^) and yj^(x,X^):

^2U^^o^ W(X^)

^5-"^ yg^^'^o^ = w - T r T ^ / ^ ' ' o ^

^rTöTT^u^^'^o^-14 o 1i+ o S u b s t i t u t i o n i n t o ( 5 . 1 0 ) y i e l d s : Wp,(X ) (x2-x2)A(x,X,X^) = / (;^°) W{y^(x,X),y^(x,X^)} + ll+ o

^

wT(fT'^^^i^^'^^'yi+^^'^^^-14 O We w r i t e t h i s a s ; (X2-X2)A(X,X,X^) - W(X^) ^ exp ( v ^ - v ) x = ^ ^ o 11+ o ( 5 . 1 3 ) . W { y ^ ( x , X ) , y ^ ( x , X ^ ) } + W(X^) W { y / x , X ) , y ^ ( x , X ^ ) } ^ l + ( ^ o ) v+v 2 ^ exp ( v ^ - v ) x o

For the first term on the right hand side we have by lemma 1 (ii), W , (X )

(5.11+) -—__- w{y^(x,X),y^(x,X^)} = vo(l) exp(v -v)x as x^-oo, ll+ o

Using (5.1), (5.2), (5.8), (5.8a) and the definition of W{y^(x,X),y^(x,X^)}

we obtain:

W{y,(x,X),y, (x,X )} = [(v +v) + v {9,(x,X)+ii;, (x,X ) +

1 4 0 O 0 l 4 0

+ 9^(x,X)i(;^(x,X^)} + v{i|;^(x,X)+9j^(x,X^)+<i;^(x,X)9)^(x,X^)}3 exp (v^-v)x.

If we choose X=X in this formula we obtain an expression for Vr, (X ).

o

'

14 o

Hence:

W{y^(x,X),y^(x,Xj} v+v^ 3^ lvl + |v^,

—- - gxp (v -v)x "

^ l + ( ^ ) ^\ < 1-3e Iv^!

(5.15)

exp Re (v -v)x, for x > d , X e D and |x2+a| > r,

o o o ' '

From (5.13), (5.II+) and (5.15) we see that there exists a constant d, > d

such that

v+v _

| ( X 2 - A 2 ) A ( X , X , X J

- W(X^) ^ exp (v^-v)xl < {e + : ^ .

o

(c

T^\ |V|+|V I

^^•'^'

. w(X )} — i

r-

exp Re (v -v)x,for x > d, , X e D and

o | v | o 1 o

o

|x2+a| > r.

Now (iii) follows since e is arbitrary.

Next we prove (iv).

Let D be a subset of D on which Re X is bounded, |x2+a| > r, |x2+b| > r,

Let

t

and r be positive constants with

e < rr .

We choose the constant c in the definition of y (x,X) (cf. (1.11 ) and

(1.12)) such that c = c(e,r) (cf. lemma 2 ) . From lemma 1 (ii), and lemma

2 (i) it follows that there exists a constant c < c (e,r) such that for

X < c , X e D and |x2+b| > r we have |9.(x,X)l < e, |4).(x,X)| < e, j=2,3.

The Wronskian Wp_(X) does not vanish because of lemma 1+ (i). We may

there-fore write y.(x,X) as the linear combination of y^(x,X) and y (x,X):

^n^^^ wfx)

(5.17) y ^ ( x , X ) = ^ y ^ ( x , X ) - ^ y3(x,X).

22. (x2-X2)A(x,X,X ) O ' ' o W (X) d ü T W{y2(x,X),y2(x,Xj} 23V (5.18) ' ^ W{y2(x,X^),y3(x,X)},

We write this as:

0+0 W (X) (X2-X2)A(X,X,X^) - W(X) - ^ exp {a^-o)x = ^ ^ ^

(5.19) . W{y2(x,X),y2(x,X^)} + W(X)

rW{y2(x,X^),y3(x,X)

a+0

2a exp [a

o-'^ J

The quotient W (X)/W (X) is analytic on D and 0(l/X) as X-x» on D Fiorthermore,

W{y2(x,X),y2(x,X^)} = {(o^-a)+(a^+a)o(1)} exp(a^+a)x, as x->- -00 uniformly for X on D by lemma 1 (ii).

Hence:

W (X)

(5.20) W ( X T W{y2(x,X),y2(x,X^)} = o( 1) as x^ -00,

uniformly for X on D

Using ( 5 . 3 ) , (5.I+), ( 5 . 5 ) , ( 5 . 6 ) and t h e d e f i n i t i o n of W{y2(x,X^) ,y3(x,X)} we o b t a i n

W{y2(x,X^),y3(x,X)} = -[(a+a^)+a{92(x,X)+iJj (x,X)+92(x,X^)il;3(x,X)} +

+ o^{(p^{x,X)+ilj^{x,X^)+<pAx,X)^^{x,X^)}2 e x p ( a ^ - a ) x ,

From this and (5.9) we deduce:

W{y (x,X ),y (x,X)} a+a

- - ^ ^ exp(a^-a)x W23(V ,^ \a\+\o \ 3e o (5.21) " 1-3e lol exp Re(a -a)x for x < c , X on D

From (5.19), (5.20) and (5.21) we see that there exists a constant c' < c such that

a+o

|(x2-x2)A(x,X,X ) - W(X) ^r-^ exp(a -a)xl < e + ' o o 2a o '

^'•'^^

, khla I

+ - r ^ {| W(X)| — exp Re(a -a)x}, f or x < c' and X on D, .

1-3e II - ^ o ' o 1 Since e is arbitrary the result (iv) follows.

6. Representation theorems for functions analytic in a strip

Let C be a curve in the domain D (defined in section 5) of the complex X-plane defined by Re A2+b = c , where c > 0 and c is such that C does not intersect the cuts of D . Then Re X is bounded on C^. If b is not on the

o 1 negative real axis or zero and 0 < c < Re /b then C, consists of two contours A and A (cf. fig. I); one. A., in the upper half plane around the upper part of the cut A : x2+b < 0, and the other, A^, in the lower half plane around the

o 2

lower part of the cut A . If b is not on the negative real axis or zero, and

o

c . 2 Re / b , t h e n C, c o n s i s t s of two contoiirs A^ and A^ from ImX = -00 t o

1 1 p D

Im X = +00 one t o t h e r i g h t of A and t h e o t h e r t o t h e l e f t of A ( c f . f i g . I ) .

o o If c = Re /b, these contours intersect at the origin (cf. A and A, ). If b

is not on the negative real axis or zero, C^ consists of one contour A^ from

& ' 1 5 Im X = -00 to Im X = +00.

A similar discussion applies to the curve C^ in D defined by Re A2+a = = Cp, where Cp > 0 and C does not intersect the cuts of D .

Now we suppose that c. and c are chosen in such a manner that C and C„ are in D and do not intersect each other and the cuts. Let G be the open

2 o

subset of D where Cp > Re A^+a, Re A^+b < c or c < Re A2+a, Re A2+b < c. Then the boundary of G consists of C and C and G consists of one or two subdomains. The real part of X is bounded on G, There are three types of sets G.

Type I. c 2 Re vb and c 2; Re /a. Then G consists of one or two strips extending from Im X = -00 to Im X = +0°, Two strips occur in the case that none of the numbers a and b is negative or zero. Then there is a strip to the right and one to the left of the cuts, with boundaries of type A or A^ (cf. fig. I). In the case that one of the numbers a and b is negative or zero there is only a single strip to the right of the cuts.

2U.

Type II. Neither of the numbers a and b is on the negative real axis or zero and 0 < c < Re /b, 0 < c < Re /a. (In this case c = Re /b and c = Re /a cannot occur simultaneously since then C and Cp would inter-sect at the origin.) There are now two strips, one in the upper half plane around the corresponding parts of the cuts, and a similar piece in the lower half plane.

Type III. Suppose neither of the numbers a and b is on the negative real axis or zero. Suppose c < Re /b, c > Re /a or c > Re /b, c < Re /a. Then G is a H-shaped subset of D (cf. fig. 1, the boundaries of G are of

o the form A , A , A and A^).

The numbers a, b, c and c have to satisfy certain conditions in order that C and C do not intersect the cuts and each other. This is also why regions of type II and III do not always occur, there may not exist a c or c , with 0 < c < Re /b, 0 < c < Re /a, such that C or C does not intersect with the cuts.

For strips G of type I and II with c > Re A^+a, Re A2+b > c on G, we may formulate the following theorem:

Theorem h. Let y (x,X) and y2(x,X) be the solutions of (O.l) satisfying (0.3) and (0,1+) respectively. Let W(X) be the Wronskian of y (x,X) and

y2(x,X), Let c.,c ; C , C and G be defined as in the beginning of this section Suppose G is of type I or II and c > Re A2+a, Re A^+b > c on G, Let C

be a contour in G with the following properties:

If G is of type I, then C lies in the half plane Re X > 0 or in the half plane Re X < 0 and the orientation of C is such that the cuts are to the left of C,

If G is of type II, then C surrounds the upper cuts or the lower cuts of D and the orientation of C is such that the surrounded parts of the cuts are to the left of C.

Let 'I'(X) be analytic in the component G of G containing C and let ^(X) be continuous on G., 1 Suppose: (6.1) [ |$(X)| |dX| < 00, C (6.2) 'I'(X) = X o( 1 ) as X^^ uniformly on G^

Then:

(6,3) f(x) =

2Tri $(X)y (x,X)dX, exists for -oo < x < oo,

and

(6.1+) W(X) $(X) = f(x)y2(x,X)dx,

for X e G , Furthermore, (6.1+) also holds for X on the boundary of G (i,e, on C or C ) , X ?i 0 if:

(6,5) $(X) satisfies a Holder-condition with exponent y, 0 < y S 1, in the restriction to G of a neighbourhood of X.

Remark 3. The uniformity of (6.2) on G need not be assumed since it is a simple consequence of the Phragmen-Lindelöf theorem.

Proof of Theorem 1+. If G is of type I we need only to consider the portion of G in the right half plane, since G is symmetric with respect to the origin. Similarly if G is of type II we need only to consider the portion of G in the upper half plane. Therefore in what follows the symbols C , Cp and G will designate their parts in the right half plane or upper half plane. So we may assiome the contour C to be in the right half plane (Type I) or upper half plane (Type I I ) ,

Consider a contour in G, If X=0 is not on the contour, we deduce from (6.6), (6.1), (6.2) and Cauchy's theorem that along the contour $(X)/X is absolutely integrable.

Let X be a point of G. Then we write (6.3) and (6.1+) as o

(6.7) TTl

W(X )

- ^ $ ( X ) =

X o y2(x,X^)dx y^(x,X)$(X)dX,

From (6.1) and (l.5) the integral in X is uniformly convergent for -oo < a < < X 2 6 < 00, Hence

6 e

(6.8) y (x,X )dx

26.

Using lemma 5 (i) we obtain for the right-hand side of (6.8) $(X) {A(B,X,X ) - A(a,X,X )}dX,

o o

The integrand is analytic in G and continuous on G. Applying Cauchy's theorem, lemma 5 (ii), and (6.2) we obtain:

(6.9)

y2(x,X^)dx y^(x,X)0(X)dX = $(X) {A(6,X,X ) - A(a,X,X )}dX,

° o

where C is a contour in G from a -ioo to ap+ioo for G of type I and from -a +ioo to a +ioo for G of type II (a > 0, a > 0 ) .

The proof is completed by consideration of the following cases: 1) X e C,, 2) X e G, 3) X e C^.

o I o o 2 Case 1. X e C,.

o 1

Let 6 be a positive number such that the contour defined by |a-a | = 6 is disjoint from Cp and the cuts.

For C in (6.9) take the contour consisting of C excluding those points where la-a I ^ 6 and the contour la-a I = 6, Re {a~a ) > 0. Let r be the

' o _' o o

minimiom of jx2-Ha| and |x2+b| on G and on | a-o | < 6, clearly r is positive. First we prove:

(6.10) i-im 3-xo

0(X) A(B,X,X )dX = 0.

Now the function $(X)A(6,X,X ) is analytic in G and continuous on G for X?^ X .

o

For every fixed B we have, using Cauchy's theorem, lemma 5 (ii) and (6.2) (6.11) $(X)A(B,X,X )dX = $(X)A(B,X,X )dX,

Now since C^ is to the right of C, and X is on C, we have Re v < Re v 2 '^ 1 o 1 o on Cp. So from lemma 5 (iii) and (6.6) we obtain

lim 6-x»

<I'(X)A(B,X,X )dX = 0.

Secondly we shall show that (6.12) l i m -a-v -oo W(X ) <f(X)A(a,X,X )dX = TTi X o

Let us denote the contour |a-0 | = 6, Re {a-a ) > 0 by L., the contoior 10-0 I < 6, Re {a-a ) < 0 by L and the contour \a-a \ < 6, Re (0-0 ) = 0 by L . The orientation is chosen such that Im 0 increases along the respec-tive contours.

Using lemma 5 (iv), (6.5), (I.I8) and the Riemann-Lebesgue lemma we see that the contribution of the integral along the portion of C where Re 0 = Re a , tends to zero as a-*- -«>.

o

The remaining integral may be written as:

$(X)A(a,X,X^)dX = -$(X^) A(a,X,X )dX

- I {$

(X)-$(X )}A(a,X,X )dX.

1

With the condition (6.5) and Cauchy's theorem we deduce:

{$(X) - $(X )}A(a,X,X )dX = {$(X) - $(X )}A(a,X,X )dX, 1 3 Now from (6.5), lemma 5 (iv) and the Riemann-Lebesgue lemma we deduce;

lim

a->- -00

{$(X) - $(X )}A(a,X,X )dX = 0. o o

F i n a l l y from t h e theorem of r e s i d u e s and ( 5 . 1 0 ) we o b t a i n W(X )

A(a,X,X )dX = A(a ,X ,X )dX - -X" _{' o X} T T l .

But the integral along L tends to zero as y tends to -00 by lemma 5 (iv), So (6.12) holds.

From (6.9), (6.10) and (6.12) we deduce (6,1+). Case 2: X e G.

o

if

Now we choose C = Cp in (6.9).

We can obtain (6.10) by the same argument used in case 1. By the theorem of residues, lemma 5 (ii), and (6.2) we get

28.

(6.13)

e w(x )

Kx)A(a,X,X )dX = - $(X)A(a,X,X )dX + Tii , ° $(X

O o A

^2

S

Now since Re(a -o) = Re A^+F-c, > O we deduce from lemma 5 (iv) and o o 1

(6.6) lim ƒ $(X)A(a,X,X )dX = 0.

^.1 o

a^- -00 1

Thus we have (6.12) again.

From (6,9), (6.10) and (6.12) we deduce (6.1+). Case 3: X e C^.

o 2

The proof is similar to that given in case 1; the difference lies in the fact that the roles of A(a,X,X ) and A(B,X,X ) are interchanged.

o o This completes the proof of theorem 1+.

Now we consider the representation (6.1+) and (6.3) for X and C in a domain G of type III.' In this case we assume that ^(X) is an odd function because the functions y. (x,X) .y (x,X) and W(X) occurring in (6.1+) are even in G.

For the contour C we now have three possibilities: 1) C runs from ±(-a +ioo to a +ioo), around two cuts.

2) C runs from ±(a -io° to -a +ioo), i.e. from right bottom to top left or from left bottom to topright in G.

3) C runs from ±(a -ioo, to ap+ioo), i.e. C is to the right or to the left of the cuts.

'"from ±(-a +io° to ap+ioo)" means from -a +ioo to a +ioo in the case where the upper sign is taken and from a -ioo to -ap-ioo in the case where the lower sign is taken.

Theoremi 5. Let y (x,X) ,y (x,X) and W(X) have the same meaning as in

theorem 1+. Let G be a domain of type III such that c < Re /b, c > Re /a. Let C be one of the contours in G discussed above.

Suppose $(X) is an odd continuous function on G, which is analytic in G and satisfies (6.1) and (6.2), the latter uniformly on G. Then f(x) de-fined by (6.3) exists for -oo < x < oo.

Furthermore (6.1+) holds for X e G and C of type 1 or 3, see above. If C is of type 2 and X e G then

(6.1M

yp(x,X)f(x)dx = 0,

Formula (6.1+) remains valid at points X on the boundary of G which satisfy (6.5) with G replaced by G,

Remark 5. As in theorem

h,

we need not assiome that (6.2) holds uniformly

on G (cf. remark 3 ) .

Remark 6. If $( X) = o( 1 ) as X-x» on G and C is of type 2, then f(x) = 0.

* . . . .

For let C be a contour similar to C but symmetric with respect to the

origin. Then we see from (1.5), the condition on $(x) and Cauchy's

theorem that we may replace C by C . It then follows that f(x) = 0 for

every x since the integrand $(X)y (x,X) is an odd analytic fxinction and

C

is symmetric with respect to the origin.

Proof of theorem 5. As in the proof of theorem

U

we may deduce (6.6) and

(6.9) where the contours are of the same type as C. The integrand in

(6.9) is an odd analytic function of X in G (cf. (5.10)).

If C is of type 2 we may replace C by a similar contoxir C which is

symmetric with respect to the origin. Hence the left-hand side of (6.9)

vanishes, and so (6.11+) holds for X e G.

"k it

Now suppose C in (6.9) is of type 1. We may assume that C passes

through the origin. The lower and upper parts of this contour will be

•jf if

denoted by L and U . Then because the integrand in (6.9) is odd, it

"k ic "k

follows that we may replace C by the contour consisting of L. and -U.

which is of type 3. In the same way contours of type 3 may be replaced

by contours of type 1. The proof of (6,1+) is now similar to that of

theorem 1+,

Let C' and C" be the upper and lower contours which together form

C.. Their orientation is chosen such that the upper cuts and the lower

cuts of D are on their left-hand sides.

o

Let Cp be the part of Cp where Re X > 0, traversed from below to above,

*

If X € C' we choose for C the contour chosen in case 1 of the proof

o 1

^

of theorem 1+, with C replaced by C'. Then (6,10) may be proved again

by the same arguments. From the equivalence of contoiors of type 1 and

•type 3, Cauchy's theorem, lemma 5 (ii) and (6.2) we deduce

r

$ X)A(B,X,X )dX = I 3'(X)A(e,X,X )dX.

o j o

* C'

C "^2

30,

Remark 7. If W(X) has no zeros on G then we can make the following

substitution 4'(X) = {W(X)/2X}4i(X) which brings (6.3) and (6.1+) to the form: (6.16)

_'^^^-Éï

^5^H'(X)y^(x,X)dX, ?X

(6.17) H'(X) = f(x)y2(x,X)dx.

Here the conditions on H'(X) are the same as those stated for <I>(X). This follows since 2X/W(X) is analytic in G and tends to 1 as X^oo on G.

We have proved theorems 1+ and 5 for the set of formulas (U.1) and (1+.2) with the second indices. Now we formulate analogous theorems, 1+a and 5a, for the first indices in (l+,l) and (l+,2).

Theorem 1+a, Let the assumptions of theorem h be satisfied with a and b interchanged. Then:

(6.18) _{^(^) = 2 ^} «'(X)y2(x,X)dX, exists for -oo < x < oo, and

(6.19) ^ $ ( X ) = f(x)y (x,X)dx, for X e G.

Furthermore (6.19) also holds for points XT^O on the boundary of G (i.e. on C^ and C^) if (6,5) holds.

Theorem 5a, Let the assumptions of theorem 5 be satisfied with a and b interchanged. Then f(x) defined by (6.l8) exists for -oo < x < oo. Further-more (6.19) holds for X e G, and C of type 1 or 3 (see above theorem 5). If C is of type 2 and X e G then (6.1I+) is replaced by:

(6.20)

J

—00

y^(x,X)f(x)dx = 0.

Formula (6.19) remains valid at points X on the boundary of G where (6.5) with G replaced by G is valid. In (6.19) we interpret the left-hand side as

Proof of theorems 1+a and 5a. If y(x) is a solution of (O.l) then y (x) = y(-x) satisfies the differential equation:

^^•^""^ ^ - tx2+q*(x)}y* = 0,

where q (x) = q(-x), and the condition (0.2) for q (x) is satisfied by

•k "k

constants a = b and b = a. Furthermore this equation has solutions satisfying (0.3) and (0.1+) given by y^ (x,X) = y2(-x,X) and y^ (x,X) = = y.,(-x,X). Now applying theorem 1+ and 5 to the equation (6.21) we get:

^ ( ^ ) = ^ *(X)y^ (x,X)dX,

4 ^ ^(X) =

f(x)y2*(x,X)dx.

* J

Substitution in the above expressions for y. (x,X) and y (x,X), changing 1 2 X to -x yields (6.l8) and (6.19).

7. Applications.

Remark 8, If a=0, b^O then we may replace the conditions (3.1) and (3.2) of theorem 3 by the less restrictive conditions:

-X t -(X +e)t

(7.1) f(t)e e L(0,oo), f(t)e e L(-«',0),

X^t (X -e)t

(7.2) f(t)e e L(0,OO), f(t)e e L(-<»,0), If a?^0, b=0 they may be replaced by

-(X -e)t -X t

(7.1a) f(t)e e L(0,<»), f(t)e e L(-«>,0), (X +e)t X t

(7.2a) f(t)e e L(0,OO), f(t)e e L(-~,0),

When a=b=0 we may take e=0 in (3.1) and (3.2), This can be seen from the proof of Theorem 1 where we do not need the substitution v = A^+a when a=0, and I^(t,y) = 0(e '' ). (See the line following (2,12)),

32.

A.

As a trivial application of the theory consider the case q(x) E 0. The differential equation (0.1) reduces to d2y/dx2 - x2y = 0; the regions D and D are the half'planes Re X > 0 and Re X > 0 respectively.

^ -Xx Xx The special solutions are: y,(x,X) = e and y (x,x) = e ; their

Wronskian is W(x) = 2X.

Applying theorem 3, taking into account remarks 3 and 8 we may formulate the following theorems.

Theorem 6. Let X. and x be real numbers with X > 0. Let f(t) be defined for real t, be of bounded variation in a neighbourhood of t=x and

-X t (7.3) f(t)e € L(-oo,oo). Then: (7.1+) X +ioo e dX f(t)e"^*dt = Tri{f(x+0)+f(x-0)} X -100

When condition (7.3) is replaced by X t (7.3a) f(t)e e L(-~,<») Then: (7.1+a) X +i«' -Xx ,, e dX f(t)e^* dt = ïïi{f(x+0)+f(x-0)}, X -ioo

The integrals in X in (7.1+) and (7.1+a) have their principal values.

Theorem 6 is similar to theorem 5a in [33, Ch. V I , §5, the condition on X in the latter is however less restrictive.

In the application of theorem k we only consider the case where G is a strip of type I.

Theorem 7. Let G be the strip defined by 0 < c < Re X < c . Let C be a contour in G, traversed from below to above. Let 't'(X) be analytic in G and continuous on G. Suppose (6.1) and (6.2) hold.

( 7 . 5 ) f ( x ) = •" 2TTI C e x i s t s f o r -oo < x < oo, a n d (7.6) $(X) = $(x)e-^'' dX, f(x)e dx,

for X e G. In (7.5) and (7.6) the upper and lower signs belong together. Furthermore suppose X is on the boundary of G. If $(X) satisfies a Holder-condition, with exponent y, 0 < y < 1, in the restriction to G of a

neighbourhood of X, then (7.6) also holds for those points.

Remark 9- As in theorem 1+, we need not assiime that (6.2) holds uniformly on G (cf. remark 3 ) .

Remark 10. A theorem similar to theorem 7 can be found in [3], Ch. VI, §19, theorem 19a, with somewhat different conditions.

B.

We now consider integral transforms with a hypergeometric function as kernel. The theory of this type of transform acting on Lp(-oo,oo) functions has been given by Titchmarsh Ccf.Cl], section I+.I9 and 1+.30]. We will treat this transform under L, conditions. Therefore we consider.

1

following Titchmarsh, the hypergeometric equation:

(7.8) X(1+X) Ip- + {c+(a+b+l)X} ^ + ab Y = 0, X > 0,

which has solutions;

(7.9) Y.J = X"^ F (a, 1-c+a; 1-b+a; - 1 ) ,

and

Putting: e^ = X,

y(x) =X^(-^\l+X)^(^^^"^-=Mx),

(7.11 a = a+X+A2+A, b = a+X-A2+A, c = 2X+1, the equation (7.8) becomes;

d2y

(7.12) ^ - {x2+q(x)}y = O,

where:

x X

(7.13) q(x) = A - — — + a(l-a) , _oo < x < oo, e +1 (e +1) From (7.13) we infer: q(x) -+ A as x-x» q(x) -+ O as X-»- -00 and " X |q(t) - A|dt < 00, |q(t)|dt < <».

Hence q(x) satisfies the conditions (0.2) The special solutions are:

(7.11+) y/x,x) = X ^ ^ = - ^ ^ ( I + X ) ^ ( ^ ^ ^ ^ ^ - ^ ) Y ^ ( X ) ,

which t a k e s t h e form:

(7.15)

, , ^ -A^+Ax , l+e^,*^

y/x,x) = e (~ir^

F(a+X+A2+A, a-X+A2+A; 1+2A2+A; - e ~ ^ ) and

which takes the form:

Xx,, XvOi^

(7.17) y2(x,X) = e^(l+e ) F(a+X+A2+A, a+X-A^+A; 1+2X; -e"").

For the Wronskian of y.(x,X) and y (x,X) we find using [6], section 2.10, formula (2):

W^{y^(x,X), y2(x,X)} = {X^^^'^ ^ l+X)^^^^^^^-^^^. . W {X"^ F(a, 1-c+a; 1-b+a; - 1 ) , F(a, b; c; -X)} =

= {...}^ .X.W {X"^ F(a, 1-c+a; 1-b+a; - 1) , F(a, b; c; -X)} = - ( \2 Y r(c) r(a-b)

- 1 . . . ; .A. j , ^ ^ _ ^ ^ p ^ ^ ^

W {X F(a, 1+a-c; 1+a-b; - - ) , X F(b, 1+b-c; 1+b-a; - -)}

A A A

„ ,1+X a+b+1-c r(c) r(a-b) , ^^ „ r(c) r(a-b)

, ^. ^^

" ^ ~ ) r(c-b) r(a) (^-^) " r(c-b) r(a) (^-^)'

^'

^""-Since V/ {y (x,X), yp(x,X)} is independent of x we obtain:

(^.18) w(x) = ^ f 2 i I % H (-M = r(i^2X)r(i+2A2^) _

r(c-b)r(a) r(i-a+x+A^TI)r(a+x+A2^)

Applying theorem 3 to this case we can formulate the following theorem. Theorem 8. Let y (x,X), y (x,X) and W(X) be given by (7.15), (7.17)

and (7.l8) respectively. Let the conditions of theorem 3 be satisfied with (3.1) replaced by (7.1a). Then (0.5) holds. If (3.2) is replaced by (7.2a) then (0.6) holds.

In the application of theorem h and 1+a to this case we need only con-sider strips G of type I, since b=0.

Theorem 9" Let the conditions of theorem 1+ be satisfied, with c. > 0, C2 ^ R e A , then (6.3) and (6.1+) take the form:

f(x) = -1^

27ri

,(x) e'^^^^ ( : ^ ) "

^ X '

e (7.19)

36. and ( 7 . 2 0 ) r ( 2 x ) r ( 1+2A2+A) $ ( x ) = r ( 1 - a + X + A 2 + A ) r ( a + X + A 2 + A ) F ( a + X + A 2 T A , a + X - A 2 + A ; 1+2X; - e ^ ) d x . „/ V Xx, ^ , x.a. f(x)e (1+e ) .

Theorem 9a. Let the conditions of theorem 1+a be satisfied with c > RerA, C2 > 0, then (6.18) and (6.19) take the form:

^^^) = iï

(7.21) and , , X X x , , . x.o. )(X)e ( 1 + e ) . F ( a + X + A 2 + A , a + X - A 2 + A ; 1+2X; - e ^ ) d x . r ( 2 X ) r ( l + 2 A 2 + A ) r ( 1 - a + X + A 2 + A ) r ( a + X + A 2 + A ) >(x) = ^, . - A ^ T A X ,1+6""" f(x) e (——) (7.22)

F(a+X+A2-(-A, a-X+A2+A; 1+2A2+A; -e~^)dx.

Using [1+] and formula (3) from C5], we may rewrite the special solutions y (x,X) and y (x,X) in terms of generalized Legendre's associated functions, defined by Kuipers and Meulenbeld in [1+]. For y (x,X) and y (x,X) we obtain:

(7.23) y/x,X) = 2^-^'^^e--*'^^r(l+2A^)

PI^^^^^'

"^^ (tgh f

(7.21+) y^(x,X) = 2 ' ' ^ ^ - ^ e-^i^r(l+2X) p - 2 ^ - 2 A ^ ^_ ^^^ | )

Hence we may write (0.5), (0.6), (7.19), (7.20), (7-21) and (7.22) in the form: X.j+io X^-10 dX . xr(i-a+x+A^;A)r(a+x+A^;^) e-^i^^-*-'^^^^) .

(7.25) p ^ ,-2/X^ (_ ,g, :^) I p-2A^,-2X (^^^ 1^ ^(^^^^ ^

iri ^ {f(x^+0)+f(x^-0)},

( 7 . 2 6 ) X +io X ^ - i

f dX X r(i-a+x+A?7I)r(a+x+.^r^:^) ^-^A^^>/x2:i^)

oo 2 A 2 + A , - 2 X - a

(tgh

-f)

-a f ' - 2 ^ ' ^ ( - t g h | ) f ( t ) d t T i l {f(x +0)+f(x - 0 ) } , (7.27) f ( x ) = 2Tri

U^-'^^^e-"i''^^r(i+2A^) .

_ 2 A 2 + A , - 2 X . ^ x^ . , . , , , P_o( ( t g h - ) $ ( X ) d X , (7.28) TTiX ^ X - A 2 + A - 1 e 2 r ( l + 2 A 2 - h A ) Xr( 1 - a + X + A 2 + A ) r ( a + X + A 2 + A ) 00 - 2 X , - 2 A 2 + A $ ( X ) = ( - t g h - ) f ( x ) d x , ( 7 . 2 9 ) f ( x ) = 27Ti i ^^X^-X ^ - . i X ^ ( ^ ^ 2 X ) . 2X , - 2 A2+A , . , Xs . , , > , , P ' ( - t g h - ) $ ( X ) d X , —a ^ (7.30) TTiA2+A ^A2+A-X e 2 r ( 2 X ) r ( 1 - a + X + A 2 + A ) r ( a + X + A 2 + A ) 2 A 2 ' Ï A , - 2 X ^(X) = -a (tgh |) f(x) dx respectively.

If we take A=0 then (7-13) reduces to:

(7.31) q(x) a(1 - g )

I 1.2 X

h cosh —

and the special solutions 7.(x,X) and yp(x,X) can be expressed in terms of associated Legendre functions the formulas easily follow from (7.23), (7.2I+) since the P ' (x) reduce to P, (x) when m=n.

38.

(7.32) y/x,X) = e'^^^ r(l+2X) P"^^ (tgh f ) ,

(7.33) y2(x,X) = e-^^^ r(l+2X) P ^ (- tgh |) The Wronskian (7.l8) takes the form:

(7.31+) W(X) = r2(l+2X) r(l-a+2X)r(a+2X)

This case is also treated by Titchmarsh (cf. Cl], section 1+.19). Application of theorems 3, 1+ and 1+a to this case gives the following results.

Theorem 10. Let the conditions of theorem 3 be satisfied, with e = 0, while a = b = 0.

If (3.1) i s s a t i s f i e d then:

X.j+ioo dX. Xr(l-a+2X)r(a+2X) e"^""^^ P"^^ ( - t g h - ^ )

(7.35) S-^"

p - f (tgh | ) f ( t ) d t = ^ {f(x^+0)+f(x^-0)}.

If (3.2) is satisfied then: A +i«> dX X r(l-a+2X)r(a+2X) e"^""^^ V'^^ (tgh ^ ) (7.36) ^-i"' P"f (- tgh |) f(t) dt = ^ {f(x^+0)+f(x^-0)}.

-a

Theorem 11. Let the conditions of theorem 1+ be satisfied, with c and Cp positive constants; then (6.3) and (6.1+) take the form:

(7.37) f(x) =

and

— 0 0

respectively.

Theorem 12. Let the conditions of theorem Ha be satisfied, with c and c

positive constants. Then (6.l8) and (6.19) take the form:

(7.39)

r M ' ^

and

e-""^^ r(l+2X) P-^^ (- tgh |) $(X)dX,

/„ ,.x TTiX r(2X)

f .

^^•'^°^ ^ r(l-a+2X)r(a+2X) ^^^^ =

P i f (tgh |) f(x)dx.

respectively.

If we take a = 0 or 1 then we get our example A, as can be inferred

from (7.32), (7.33), (7.31+) and [63, 3.2, (3).

1+0.

CHAPTER 3.

Inversion formulas for integral transforms related to a class of even order linear differential equations.

8. Introduct ion.

W e consider the differential equation: ,2n

(8.1) ^ - 2 ^ - {X^" + q(x)}y=0, - o o < x < « . , n=2,3,..., dx

where q(x) is a continuous function defined for all real values of x, and ƒ 1 q(x) Idx < oo.

In section 9 we construct solutions y (x,X) of (8.1) which are charac-terized by their asymptotic behaviour in ±oo respectively:

(8.2) y^(x,X) exp(Xx) ^ 1, . .

{y^(x,X) exp(Xx)}^'^^ -> 0, j = 1 ,, , , ,(2n-1 ) , (the upper index (j ) denotes differentiation with respect to x ) , as x-+ +oo and | arg x| < TT/2n,

(8.3) y_(x,X)exp(-Xx) ^ 1,

{y_(x,X) exp(-Xx)}^'^ ' -> 0, j = 1 ,, ,. ,(2n-1), as x-+ -oo and | arg x| < •n-/2n . , , - , - , (2n-l), (2n-2),(l) ,(2n-l) ^ ^ ^^ ^.,.

Let [91;;] = 9 ijj - 9 ijj + . . . -91JJ denote the bilinear concomitant of 9(x) and i|;(x).

Let the transpose of a matrix be indicated by the upper index t. In section 10 we prove the following theorem:

Theorem. Let x be real. Let q(x) be a continuous function for all real CO . I

values of x and S^ |q(x)|dx < °°. Let y^(x,X) b e the solutions of (8,1) characterized b y (8.2) and ( 8 . 3 ) . Let f ( x ) , defined for all real x , b e of bounded variation in a neighborhood of x = x and

o CO

(8.1+) |f(x)|dx < 0°.

n— 1 Let Y_^(x,X) b e t h e n - v e c t o r (y_|_(x,X ) ,y_^(x,(jjX ) , , . . ,y_^(x,a) ~ X)) \{X) = ƒ " f ( t ) Y ^ ( t , X ) d t , w h e r e u = e x p ( - T r i / n ) .

Let A b e the n x n matrix with (g,h) element [y (x,u X)y (x,w X ) 3 . Let G b e t h e part o f the X-plane where (— - — ) < arg X < — .

TT TT Let C b e a contour m G consisting of the ray with arg X = — - — from oo exp i (— - — ) t o a exp i {—-—), a > 0 , then a. curve in G from

•^ 2 n ^ 2 n

a exp i(-^ - —) to pi, B > 0, such that the non-removable singularities of t — 1

Y (x ,X) A~ (X) $ (X) are to the left of it, and the positive imaginary axis from Bi to ioo,

Let p/. , y > max(a,B), denote integration along the contour C from iwy to '^ iwy

iy.

Then the following t w o f o r m u l a s , in matrix n o t a t i o n , hold:

(8.5) li"^ c y-xo iwy iy (8.6) lim y-x» ^-l luy

2n X^" ^ Y*(x ,X)A~\x )$_!_( X)dX = TTi{f(x -0)+f(x +0)},

2n X^'^"^ Y^(x ,X){A"Vx)}*$ (X)dX = ui{f (x^-0)+f (x^+O)}.

9. Special solutions of the equation (8.1 ).

The general solution of (8.1), obtained by using the method of variation of constants is:

2n 2n I

(9.1) y ( x , X ) = I c exp(cj-^Xx) + ^ - — r T, w^ exp{u)lx(x-t)}q(t)y(t,X)dt

1=1 2nX^'' 1 = 1

I

/ TTi \

ü)= exp(.- — J , where the x are constants which m a y b e chosen suitably, From (9.1) it is easily seen that the function u (x,X) defined b y

(9.2) u (x,X) = y (x,X)exp(Xx) is the unique solution of t h e integral equation: ^ 2n 00 (9.3) u (x,X) = 1 - —- E w^ exp{(üJ^+l)X(x-t)}q(t)u (t,X)dt, 2nX 1 = 1 -' X when largXl < -r—, X_{I D I 2n} ^ 0,

1+2. u_^(x,X) = 1 o + ^u^(x,X) = 1 When we put 2nx2-^ 1=1 j-^ exp{(cü-^+l)X(x-t)}q(t) _.|U_|_(t ,X )dt, r=1,2,3,. (9.1+) 0(x) = q(t)|dt,

we obtain by mathematical induction:

,u^(x,X) -^_^u^(x,X)| , ^ ]yrl2n-^) > - ^ ' ^ , 3 , r.' XI

Hence: lim u (x,X) = u (x,X) exists and: r + +

r+oo

(9.5) |u^(x,X) - 1| < exp ( ^^ll_^) - 1, |u^(x,X)| < exp (.^^^1 J , 2n-1 and

(9.6) |y^(x,X) - e x p ( - X x ) | < { e x p ( ^ ^ g n - l ^ " ^^ | e x p ( - X x ) | ,

U

for |arg X| < - ^ , X ?« 0.

We also need approximations to t h e derivatives o f y (x,X) with respect to x, denoted b y y^ ( x , X ) , j = 1 ,... ,(2n-1 ) . F r o m (9-2) and (9.3) w e d e d u c e :

00

(9.7) y|.'^^x,X) = (-X)J e x p ( - X x ) - ^2n-1 f ^ ('^^)^'*"^exp{(^^X(x-t )}q(t) y_^(t,X )dt,

2nX

1=1

and from this w e derive with ( 9 . 2 ) , (9.1+) and ( 9 . 5 )

oo

(9.8) |y|.^^x,X) - (-X)J e x p ( - X x ) | < - ^ — ^ l e x p ( - X x ) | | | q ( t ) | | u^(t ,X ) | dt ' ' X

^ |X|J | e x p ( - X x ) | { e x p - ^ ^ ^ - 1} .

N

2n-1

Approximations to t h e solution y (x,X) o f ( 8 . 1 ) defined b y ( 8 . 3 ) , and its derivatives u p t o t h e order (2n-l) are easily obtained from t h e solution y (x,X) i n the following w a y .

"k

If y(x) is a solution of (3.1) then y (x) = y(-x) satisfies the differential equation: ,2n (9.9) 2 _ - y * - {X^" + q*(x)} y* = 0, dx where q (x) = q(-x). Hence: (9.10) y_(x,X) = y*(-x,X).

From {9.6), (9.8) and (9.10) we obtain:

(9.11) |y (x,X) - exp(Xx)l < {exp( ^ A"' ) - 1} |exp(Xx)|,

|x|

(9.12) |y^J^x,X) - xJ exp(Xx)| < |x|J|exp(Xx)| {exp - ^ ^ - 1} , x|2-^ j=1,...,(2n-l), if I arg X| < -r^ , X y^ 0, where 2n (9.13) 0 (x) = X q(t)|dt.

From the uniform convergence of the successive approximations it is clear that the solutions y_^(x,X) are analytic functions of X in the sector

I arg x| < — , X 5^ 0 and continuous on | arg X| < — , X?^0.

For the proof of the theorem in the next section we need a number of auxiliary solutions of (8.1) which will be derived below,

Let S , m=1,,.,,2n, be the sector: -r— TT < arg X < - — TT .

m 2n 2n

In each sector S we may construct solutions of (8,1), y, (x,X), k=1,,,,,2n,

^ k-1

which behave as exp{ü) Xx} as x-*- ±0». Putting:

(9-11+) \ ( x , X ) = \ ( x , x ) exp(a)^-''xx), we may infer from (9.1):

1 2n ^ (9.15) \ ( x , X ) = 1 + 1 — - I

^ 2nX'^""' 1=1 '

1 k-1 1 k-1

Now choose x,= +00 when Re (w -a) )X > 0 and x-, = -00 when Re (w -w )X < 0. 1 k-1

On the path of integration we then have Re (u -u )X(x-t) < 0, If a = m-k then the choice is as follows:

1+1+.

X = +00 v h e n :

1) 1 < k - 1 and a < 1 < 2 n - 1 + a

2) 1 > k-1 and 2n+a < 1 < Un+a-1 or -2n+2+a < 1 < a-1

3) 1 = k-1, k=2,...,2n; k=»1, l=2n

and X.. = -o» when:

1+) 1 < k-1 and 2n+a < 1 < l+n+a-1 or -2n-».2+a :£ 1 < o-l

5) 1 > k-1 and a < 1 < 2n-l+a.

Now (9-15) has a unique solution which may be found by the method of

successive approximations.

^,(x,X) = 1

^v,(x,X) = 1+

^

r f E' 0.^ exp{(ü)^-u)^-'^)X(x-t)} q(t) . >-(t,X)dt +

oa — c s

- I E "

J-

exp{(ü)-'--ü>^"'')X(x-t)} q(t) ._^Vj^(t ,X)dt1,

X

j=1,2,3,..., where E' means summation over those indices 1 for which 1+) and

5) hold, and E" means summation over those 1 for which l ) , 2) and 3) hold.

Using (9.1+) and (9.13) we obtain by mathematical induction:

|-Vx.x)-^.,v,(x,x)| . ( y ^ ) ^

hence: . ,.

l i m V ( x , X ) exists when

\x\ >

{e(-«<.)}^""'' ={ƒ" | q(t) |dt}^""''> = M.

j-K» J ^ -"

For X on S and Ixl > M we have:

m ' '

(9.16) |\(x,X) - 11 5

—

^

, l\(x.X)| < i

|x|^-Ue(..) ' ' V - " - , . _ 6 ( ^

Hence:

xl^"-^

(9.17) I y^(x,X) - exp(ü,^-^x)|<

^

^

|exp(co^"\x)|, X e S and

^ |x|2"-i_e(-co)

^

\X\> M . + Vi

Proceeding similarly as before in (9-7) we derive an approximation of the j

derivative of y,(x,X), j=l,...,(2n-1), we have:

,(j)^^ . ^ , ^ > ^ ^ j „ ^ ^ ^ . k - 1 w ^ ^ -A j 2n "" (9.18) "y;^'(x,X) = (io""'X)'Jexp(w""'Xx)+ x—r E ^ 2nX 1=1 • q(t)"'yj^(t,X)dt. ü^^'^*^^exp{(ü'^X(x-t)} ^1 I Ti 12n-1 , ., F r o m ( 9 . 1 6 ) a n d (9.1I+) it f o l l o w s t h a t T y , (x,X)I < - ^ — |exp(ü) " X x ) |

I. I <in—1 „ / \ and t h i s y i e l d s w i t h (9-18) ' ' - « ^ - " ^

(9.19) ry['^^x,X) -

{J'-h)Kxp{J"\x)\ <

|x|J|exp(/-^x)!

— ^ } ^

,

^ |x|'=^"-^-0(-oo) for X e S and 1x1 > M . m A s b e f o r e it is clear t h a t t h e s o l u t i o n s y, ( x , X ) , k = l , . . . , 2 n , a r e a n a l y t i c f u n c t i o n s o f X o n S and |x| > M . m ' ' S i n c e :

y_{x,J-h)

= ^Jy.(x,X) = 2J-V.(x,X),

a . ^9-^5 ^ y^{x,J-\) = 2Jy.^^(x,X) = ^^-\.^Jx,X), j = 1,..,,n

we have better approximations for the right sides in (9.6), (9.8), (9.11) and(9.12). From Green's formula we may deduce for any pair of solutions 9(x,X), i|)(x,X) of

(6.1), belonging to the same value of X and any pair x ,x e (-00,00): ^2 cL^" d^"

ƒ {^{—^ 9 - q(x)9) - 9 ( — 2 ^ ^ - q(x)i|))} dx = L(f\l>l{x^) - [9ijj](x.|) = 0 •'^X d x d x

Hence l(fiil = 9 ^ ^ ^ " ^ ^ - i^^^^~'^\^'^'+ . . . _9,|;(2"~''\ t h e s o c a l l e d " b i l i n e a r c o n c o m i t a n t " i s i n d e p e n d e n t o f x a n d t h u s o n l y a f u n c t i o n o f X ( e v i d e n t l y [91I,] = -[9,1,]). R e m a r k 1, F r o m ( 9 . 1 7 ) and ( 9 . 1 9 ) f o r fixed X e S ( s u f f i c i e n t l y l a r g e ) a n d t h e m i n d e p e n d e n c e o f t h e b i l i n e a r c o n c o m i t a n t o f a n y t w o s o l u t i o n s o f ( 8 , 1 ) o n x w e i n f e r :

(9.20) [ y y.] = 0, j,k=l,,,,,2n except the concomitant where the

^ J

indices differ by n, C y y ] ?^ 0, k=1,,,,,n.

Moreover from (9-17) and (9-19) for fixed n and X-x» on S we infer: m

(9.21) C V V . , ] = 2n J-^ X^"-"" {1 + 0 ( — 1 — ) } as X ^ on S , k=1,..,,n, K K+n 2n-l m