A R K I V F O R M A T E M A T I K B a n d 4 n r 44
Communicated 24 J a n u a r y 1962 by L. CA~LESON
An iterative method for conformal mappings of multiply-connected domains
By INGEMAR LIND
Preliminary remarks
Let D be a connected domain in the z-plane containing the points z = 0 and z = oo.
Suppose t h a t there exists a function w = F (z), analytic a n d univalent in D such t h a t F (co) = 0% F ' (oo) = 1 and F (0) = 0, eonformally m a p p i n g D onto a domain of prescribed canonical type. There are four fundamental, harmonic functions in D in certain meanings measuring a point's deviation from its original position under the mapping:
Re {F (z) - z}, I m {F (z) - z}, log F (z) and arg F (z)
Z Z
B y studying the behaviour of these very simple types of functions under iterative mapping processes, described in detail below, existence proofs can be obtained for the simplest types of canonical domains - - the parallel slit domain, the circular slit domain, the radial slit domain a n d the logarithmic spiral slit domain.
There are of course several simple proofs of these results. The iterative proof given here m a y have a certain value; it is in certain cases suitable for actual calculation of the mapping function, and it gives an explanation to the simplicity of these mappings.
The uniqueness proofs - - as exemplified in [1], p. 57 (the second proof) for a paral- lel slit domain - - are also based on simple properties of the n a m e d types of har- monic functions.
Preparations and denotations
The following paper demonstrates in detail an existence proof for conformal m a p - ping onto a parallel slit domain where the slits are parallel to the imaginary axis.
To be precise the following well-known theorem is reformulated:
Every domain D in the z-plane o/connectivity k can be con]ormally "mapped onto a parallel slit domain in the w-plane, where the k slits are parallel to the imaginary axis.
Two arbitrary points ~ and fl o / D can be carried into the origin and the point o] in- /inity, respectively. 1/ w = F (z) denotes the mapping/unction in question, F (z) may be normalized by the requirement that its residue at z = fi be equal to 1. This normali- zation determines the ]unction uniquely.
3 7 : 6 5 5 7
INGEMAR LIND,
An iterative method for conformal mappings of multiply-connected domains
D (n) :
k
L (~)= U F(~):
v = l
:For t h e uniqueness proof one is referred to [1]. The existence proof is based on induction. The above t h e o r e m is t r u e for k = 1 (Riemann) a n d hereafter it is assumed true for k - 1. W i t h this a s s u m p t i o n there is no loss of generality to assume t h a t t h e given k-connected d o m a i n is b o u n d e d b y k - 1 rectilinear slits parallel to t h e imagi- n a r y axis a n d a n o t h e r non-rectilinear, analytic slit, a n d t h a t a is chosen as t h e ori- gin a n d fl as the point of infinity. The following notations are used:
a d o m a i n of the a b o v e described t y p e in t h e zn-plane;
t h e b o u n d a r y of D(~); F(~ ~), v = 1, 2 ... k, denote t h e separate boun- d a r y c o m p o n e n t s one of which is non-rectilinear. Occasionally this c o m p o n e n t will be d e n o t e d b y F (n) (lower index omitted) a n d its corresponding rectilinear b o u n d a r y c o m p o n e n t in t h e z~.l-plane (see below) b y y(~-l);
D [F~ ~)] or D ~ ) : t h a t d o m a i n of connectivity k - 1 in the
zn-plane,
which b o u n d a r y can be written L <~) - 1~ n) (1 ~< ~ ~< k).The iterative process
Assume i. e. t h a t F (1) = F(21). Map eonformally D(1 ~) in the zl-plane onto D~ 2) in the z2- plane where all t h e b o u n d a r y slits F(~ ), v = 2, 3 ...k are rectilinear slits parallel to t h e i m a g i n a r y axis. The possibility of this m a p p i n g is assured b y t h e induction assumption. ~(1)= F(11) in D~ 1) is h e r e b y carried into a non-rectilinear, analytic slit F(12)= F (2) in D(12). N e x t m a p in t h e same w a y D(2 2) in t h e z2-plane o n t o D(2 ~) in t h e
l r ~ ( k + l )
z3-plane, t h e n D(3 a) onto D(a 4) a n d so on. T h e k : t h step is to m a p D(k ~) o n t o ~ k 9 Next, m a p D(1 k+l) o n t o D(1 k+2), t h e n D(2 k+2) onto D(2 ~+3) a n d so on. The m a p p i n g pro- cess is n o w iterated infinitely in t h e suggested cyclic manner. T h e analytic function performing t h e described m a p p i n g between t h e actual d o m a i n in the Z~_l-plane a n d t h a t in t h e z~-plane is d e n o t e d b y z~ = F~ (Z~-l), a n d it is required t h a t Fn (0) = 0, F~ (cr = ~ a n d 2'~ ( ~ ) = 1 (n = 2, 3 ...), t h e possibility of this again being assured b y t h e induction assumption. L e t y~(zl) = F~ { F ~ - I ( 9 9 (Zl) 9 .)} denote t h e composed function, analytic in D (~) a n d m a p p i n g D (D conformally onto D (~).
Induction statement and proof
yJ(zl) =~i=m ~n(zl) exists, is analytic in D <1), maps D ~> con/ormally onto a parallel slit domain with k rectilinear slits parallel to the imaginary axis, and v(O) = O, ~ ( ~ )
= ~ , ~ ' ( ~ ) = 1.
Proo/:
The functionsun(z~)=Re
{ F ~ + l ( z ~ ) - z ~ } , n = 1, 2 ... (1) are harmonic in D [y(n)], u~ ( 0 ) = 0, a n d since F . + l ( Z ~ ) - z~ is analytic a n d b o u n d e d in D [~,(n)], u~ (z~) attains its m a x i m u m a n d m i n i m u m values on F (~). W i t h the n o t a t i o nJ~ = sup ] R e {z~ - z~'}[ = sup [ u n ( z ~ ) - u~(z~')[
it thus follows for all z~ t h a t
l un (zn) I ~< J~. (2)
558
ARKIV F6R MATEMATIK. B d 4 nr 44 Since u,~(z,) is n o t constant, it follows from the m a x i m u m principle t h a t there exists a n u m b e r qn, 0 < q, < 1, so t h a t
sup lU.(Z~)--U.(Z~*)[<~Jnq,.
* * * c F ( n ) z n , z
Combining [4], l e m m a 1, p. 282 with [2], "Verzerrungssats V " , p. 235 it m o s t easily follows t h a t it is possible to fix a n u m b e r q, 0 < q < 1, so t h a t q, c a n be chosen
~< q for all n. T h u s
sup [u,~(z~)-u,~(z;*)[ < J ~ q . B u t this is equivalent to
J , + l = sup [Re (z~ +x - z~'+l} [ ~< J~ q.
~:,+r z~'+l~r(" +1) Thus
Jn ~ J l q n-i for all n. (3)
The functions Um (z.) = Re {Z.+m -- Z.} are harmonic in D (n) a n d
m 1
Um (Zn) = ~ U.§
(4)
is valid. F r o m (3) it follows for e v e r y point z~ t h a t
q n - 1
[Um(zn)[ ~<J1 1 _ q < e if n>n~. (5) T h u s R e {y)n (zl) - zl} converges u n i f o r m l y to a harmonic function in D (1), and, since the c o n j u g a t e functions are single-valued a n d well determined, the convergence is easily extended to the sequence of analytic functions {y)n (Zl) - zl}. F u r t h e r (3) states t h a t t h e limit d o m a i n is of t h e desired type. Obviously ~p(0)=0, 7)(oo)= oo a n d v 2' (oo) = 1 are valid. The i n d u c t i o n s t a t e m e n t is proved.
C o m p l e m e n t a r y r e m a r k s
I t has just been p r o v e d t h a t w = ~0 (zl) m a p s D (1) eonformally onto a parallel slit d o m a i n where u -- c., ~ = 1, 2 ... k, on the separate slits (w = u + iv = Re~~ c. are cer- t a i n real constants). Obvious modifications of t h e given proof yield the existence of t h e following m a p p i n g functions:
a) w=~p,~(Zl); o~u+flv=c~ on the slits, t g v a = -~---
8'
b) w = P (zl); R = c~ on the slits, (circular slit domain);
e) w = Q (zx); 0 = c, on t h e slits (radial slit domain);
d) w = R (Zl); ~ log R + 80 = c~ on the slits (logathmie spiral slit domain), ~, 8 ~= 0.
Various kinds of iterative processes were a. o. studied b y Koebe. A n interesting example giving the existence of a conformal m a p p i n g onto t h e circle d o m a i n is found in [3].
559
I~GEMAR LIND, An iterative method for conformaI mappings of multiply-connected domains
R E F E R E N C E S
1. I:~. COURANT, D i r i c h l e t ' s P r i n c i p l e , C o n f o r m a l M a p p i n g , a n d M i n i m a l S u r f a c e s , I n t e r s c i e n c e P u b l i s h e r s , I n c . , :New Y o r k (1950).
2. P . KOEBE, A b h a n d l u n g e n z u r T h e o r i e d e r k o n f o r m e n A b b i l d u n g V . , M a t h . Z. 2. B a n d (1918), p p . 198-236.
3. - - , A b h a n d l u n g e n z u r T h e o r i e d e r k o n f o r m e n A b b i l d u n g VI., M a t h . Z. 7. B a n d (1920), pp.
235-301.
4. L. SARIo, A l i n e a r o p e r a t o r m e t h o d o n a r b i t r a r y R i e m a n n s u r f a c e s , T r a n s . A m e r . M a t h . Sot., Vol. 72, :No 2 (1952), p p . 281-295.
560
Tryckt den 23 november 1962
Uppsala 1962. Almqvist & Wiksells Boktryckeri AB
