I.2. The Riemann Boundary Value Problem

39

1.2.1.

1.2.

THE RIEMANN BOUNDARY VALUE PROBLEM Formulation of the problem

12.1.

L e t L be a smooth c o n t o u r , G ( * ) and g ( . ) f u n c t i o n s d e f i n e d on L , b o t h s a t i s f y i n g a H 6 l d e r c o n d i t i o n , a n d G(t)#O

(1.1)

for every t E L

The Riemann boundary v a l u e p r o b l e m f o r L i s : D e t e r m i n e a f u n c t i o n @ ( . ) such that

(1.2)

@ ( z ) is r e g u l a r f o r z E L+

i.

,

i s c o n t i n u o u s f o r z E L U L+ ;

.. 11.

~ ( z is ) regular f o r

z E L-

-

,

is c o n t i n u o u s for z E L U L- ;

... 111.

O(z) +A f o r I z I

iv.

O+(t)

.+

= G(t)O-(t)

t

with

g(t)

A a constant ; for t

E

L,

with O+(t)

lim

:=

z + t E+L z EL

Note t h a t f o r G ( t ) = l ,

O(Z)

;

o-(t)

:=

lim ~ ( z ) . z + t E_L z EL

t E L a n d A=O t h e s o l u t i o n h a s b e e n g i v e n

i n section 1.7. The homogeneous problem ( i . e . , g ( t ) discussed i n section section (1.2)

4.

0, t E

L) w i l l be

3 , a n d t h e inhomogeneous problem i n

A s i m p l e g e n e r a l i z a t i o n of t h e boundary v a l u e p r o b l e m

i s considered i n s e c t i o n

5.

1.2.2.

1.2.2.

Introduction to boundary value problems

40

The index of C(t), t E L I n t h e a n a l y s i s o f t h e problem f o r m u l a t e d i n t h e p r e c e d i n g

s e c t i o n t h e concept i n d e x

x

o f G(.)

o n L i s needed. T h i s i n d e x

i s t h e i n c r e m e n t of t h e argument of G ( t ) , when t t r a v e r s e s L Once i n t h e p o s i t i v e d i r e c t i o n , d i v i d e d by 2 ~ Because . G(')

i s c o n t i n u o u s on L i t i s s e e n t h a t

x

(2.1)

hence

x

1

= indG(t.1 = -

J

2TI t E L

d{argG(t)}

i s an i n t e g e r i f G ( t ) # 0 f o r e v e r y t

C o n s i d e r t h e case t h a t G(t), of a function G(z),

1

E

a f i n i t e number o f p o l e s i n L

d{lUgG(t)j,

L.

t E L i s t h e boundary v a l u e

z E Lt which i s r e g u l a r i n L t

J

t EL

. Then

t

except f o r

t h e i n d e x o f G ( ' ) on L i s

e q u a l t o t h e number o f z e r o s o f G(- ) i n Lt l e s s t h e number of poles i n L

t

, the

zeros and poles counted according t o t h e i r

m u l t i p l i c i t y , f o r a p r o o f see 111 p. 99.

1.2.3.

41

m e Riemann boun&ry value problem

12.3.

The homogeneous problem In this section the homogeneous problem (1.2) of section 1

is discussed, i.e. we consider here g(t).

(3.1)

0

,

It follows from ( 1 . 2 )

t E L iv and (3.1) that for t E L :

Because of (1.2) i,ii the relation (3.2) implies that

x

(3.3)

N++N-,

with Nt the number of zeros of a ( . )

L-, with

x

in L

+ , N-

t E L is the

the index of G(-), note that @'(t),

boundary value of a function regular in L

,

i

that of @ ( in- I

similarly

for

O-(t) with L+ replaced by L-. Consequently we should have,

which implies that if

x

indG(-)

<

0

then the homogeneous

problem has no solution, except for the trivial null solution. The cases x=O and

x>0

are discussed separately.

Case A. x = O . Hence N+n N- = 0, so that logO(z) has no z e r o s f o r z E L+, and also no zeros in L-.

Consequently, it follows from (1.2) i , ii, iii with A # O that (3.5)

i.

log O(z) should be regular for z

L

E

t

,

continuous €or z E L

t

U

L,

ii. log @ ( z ) should be regular for z E L-, continuous for z

E

bounded for I z I

m.

+

L- u L,

42

1.2.3.

Inttoduction to boundary value problem

Next n o t e t h a t G ( . )

s a t i s f i e s on L t h e H - c o n d i t i o n a n d

d o e s n o t v a n i s h on L s o t h a t log G ( t ) s a t i s f i e s on L t h e H-cond i t i o n ( s e e s e c t i o n 1 . 3 and [ 7 1 p. 16,E’).

Hence by w r i t i n g

( 3 . 2 ) as

it i s s e e n t h a t t h e problem of d e t e r m i n i n g a f u n c t i o n f y i n g ( 3 . 5 ) and ( 3 . 6 )

t h e homogeneous p r o b l e m

( w i t h ( 3 . 1 ) ) i n t h e c a s e x - 0 i s u n i q u e and g i v e n by

(1.2)

= Ae Note t h a t i n ( 3 . 7 )

is

satis-

i s i d e n t i c a l with t h a t formulated i n s e c t i o n

I t f o l l o w s t h a t t h e s o l u t i o n of

1.7.

@(-)

y

z E L-,

it i s i r r e l e v a n t which b r a n c h of log G(t)

chosen. Obviously, i f A=O, c f .

(1.2)

iii, t h e n t h e n u l l

s o l u t i o n i s t h e only solution.

If f o r t h e homogeneous p r o b l e m t h e c o n d i t i o n ( 1 . 2 )

Remark 3 . 1

iii i s r e p l a c e d by l O ( z ) l rn O ( l z l k )

(3.8)

for IzI-+m, k

>

0 , an integer, then

t h e general solution reads @(z)

(3.9)

where

P

k

(2)

r.o(z) = e P,(z),

z E

L + U L-,

i s a n a r b i t r a r y p o l y n o m i a l i n z of d e g r e e

k and

z E L+U L U L-.

43

The Riemann b o u n b y value problem

1.2.3.

O b v i o u s l y @ ( z ) as g i v e n by ( 3 . 9 )

Proof

PS f o r m u l a s , c f .

(1.6.4),

satisfies (3.8).

a p p l i e d t o ( 3 . 9 ) y i e l d for t E L,

@ + (=t )e i l o g G ( t ) + I ' O ( t ) P k ( t )

(3.11)

= ,-;log

@-(t)

and h e n c e ( 1 . 2 )

i,

...,i v

The

G(t)

+

3

r3(t)pk(t),

with g ( t ) = O and ( 1 . 2 )

by ( 3 . 8 ) a r e s a t i s f i e d . The

iii r e p l a c e d

uniqueness of t h e s o l u t i o n f o l l o w s

s i m i l a r l y as i n s e c t i o n 1 . 7 . Case B.

x>

0

Take t h e o r i g i n o f t h e c o o r d i n a t e s y s t e m i n L t and

0.

rewrite ( 1 . 2 )

i v with g(t)=O, c f .

(3.l),as

O b v i o u s l y on L (3.13)

indCt-XG(t)l

Put

r

(3.14)

X

(2)

:=

-

0

1 J 2 ll.l t E L

L + u L u L-

log I t - x G ( t ) l d t , t-z

n o t e t h a t log { t - X G ( t ) ] s a t i s f i e s t h e H - c o n d i t i o n on L.

,

Hence

from ( 1 . 6 . 5 )

C o n s e q u e n t l y from ( 1 . 2 )

i v with g ( t ) = O ,

Because

r

X

(2)

i s r e g u l a r f o r z E L+ u L-

,

i s continuous and f i n i t e f o r z

it i s seen t h a t ( 1 . 2 )

i , ii and ( 3 . 1 6 )

E

L

imply

+ U L-,

r

that @(z)/e

(z)

,

Introduction to boundary value problems

44

Z E

L'and

r

zXO(z)/e

( 2 ) 9

1.2.3.

zEL- areeach other'sanalyfic continuations.

Further ( 1 . 2 ) i i i implies Iz%(z)/e

(3.17)

r

X

(z)

1

= J A J J z J X for

JzI-tm.

Consequently L i o u v i l l e ' s theorem i m p l i e s t h a t

(3.18)

r

e X

O(Z)

r

e X

with

l i m Z-XP

(3.19)

I z I-

X

(2)

(2)

P

X

for z

(2)

z-XpX(z)

(z) =

E

L+,

f o r z E L-,

A

where P ( . ) i s a n a r b i t r a r y p o l y n o m i a l o f d e g r e e

X

x

satisfying

(3.19). T h e r e l a t i o n s (3.14), ( 3 . 1 8 ) and (3.19) p r e s e n t t h e g e n e r a l s o l u t i o n o f t h e homogeneous b o u n d a r y v a l u e p r o b l e m (1.2)

f o r t h e case

x

= ind G ( - )

>

coordinate system l i e s i n L

0,

+.

but note t h a t t h e o r i g i n of t h e

45

The Riemann bounday value problem

1.2.4.

1.2.4.

The nonhomogeneous problem In this section it will again be assumed that the origin of the

coordinate system lies in L i.

x>O.

Recalling ( 3 . 1 4 ) ,

t

. i.e.

so that

the relation ( 1 . 2 )

ivy t E L,

may be rewritten as,

Because g ( - ) satisfies the H-condition on L and so does e cf. lemma 1 . 6 . 1 , which is alwavs nonzero on L, we

may and do write

so that ( 4 . 4 ) becomes for

t E

L:

r+(.) x

Introduction to boundary value problems

46

Because eri(z) is never zero for z

-

z E

erx(’),

L

U

E

L

U

L+, and similarly for

L-, it follows that (note

e-r x ( 2 )

(4.8)

O(z)

(4.9)

zX~(z) e

h

r

z=O E

L

+

and

x

2 O),

+

is regular for z E L , is continuous for z E L u L + ,

Y(Z)

x (‘1 -

12.4.

~ ( z )

is regular for z E L-, is continuous for z E L

IJL-,

lzX~(z) e-rx(z) - ~ ( z ) Consequently ( 4 . 7 ) together with ( 4 . 8 )

and ( 4 . 9 )

imply that

the expressions in (4.8) and (4.9) a r e each other’s analytic continuations, and the asymptotic relation in ( 4 . 9 )

together with

Liouville’s theorem implies that

or

with P

X

(.)

(4.11)

an arbitrary polynomial of degree

x>

0 and such that

lim z-X P ( z ) = A. /zI+m

X

Consequently, f o r t h e c a s e

x

2 0

t h e r e l a t i o n s (4.10) and (4.11)

r e p r e s e n t t h e g e n e r a l s o l u t i o n o f t h e inhomogeneous boundary v a l u e p r o b l e m (1.21, it contains (apart from A ) X arbitrary

constants (coefficients of P ( - ) I . It is readily verified that

X

it is the unique solution (apart from thex arbitrary constants). ii.

x < 0.

In this case the relations ( 4 . 7 ) , . . . ,

(4.9) still hold;

The Riemann boundaty value problem

1.2.4.

47

together with Liouville's theorem they imply that

Because Y(z),

cf. (4.51, possesses in a neighbourhood of

a convergent series expansion in powers of z-h , h=0,1,... , i.e. Y ( -1) is regular in a neighbourhood of 2-0,

z

m

we may write

Y(z)

(4.13)

t-lchz-h for

m

[zI sufficiently large.

Hence if (4.14) then

c the

-X

=A,

ch=O for h = l,.... , -x-1,

x

-1,

inhomogeneous boundary value problem (1.2) has

a unique solution which is then given by (4.12); it is insoluble if (4.14) does not hold.

Introduction to boundary value problems

48

1.2.5.

A variant of the boundary value problem (1.2)

1.2.5.

The boundary value problem (1.2) may be generalized in several directions. We shall discuss here a simple generalization, see further

[6]

and [ 7 1 for more complicated cases.

Let again L be a smooth contour, g ( . ) and G(.) functions defined on L, both satisfying a H-condition on L and with G(-) non-vanishing on L, L

t

x

will be the index of G(-) on L, and

contains the origin or the coordinate system. It is required to construct a function @ ( * I such that i. ~ ( z )is regular for z

(5.1)

E L+U

L-;

is continuous from the left and right at L ; is bounded for t

ii. 0 (t) =

I zI +

-G(t) @-(t) (t-B)P

with a E L, B E L, a + B ,

m;

t

g(t),

t

E

L,

and m and p positive

integers. With for t

E

rX (z) as

defined:-n (4.1) we rewrite (5.1) ii as:

L,

with

As before we obtain by applying Liouville’s theorem that

The Riemann boundary value problem

1.2.5.

with P

X+m

(.)

49

a polynomial of degree X+m and satisfying the condi-

tions : (5.5)

+

i. t=B is a zero of multiplicity p of Y (t) + P

X +m

ii. t=a is a zero of multiplicity m of Y-(t) + P

X +m

(t) ; (t)

If the conditions (5.5) i,ii can be satisfied by a proper choice of the coefficients of P

X +m

(-),

then (5.4) represents the solu-

tion of (5.1) which is bounded for IzI + m . or p > l the relations of section 1.10

Note that if m

>

are needed, and g ( . )

should possess the relevant derivatives.

1

.