SCHWARZ-CHRISTOFFEL MAPPING OF MULTIPLY CONNECTED … · SCHWARZ-CHRISTOFFEL MAPPING OF MULTIPLY CONNECTED DOMAINS T.K. DELILLO, A.R. ELCRAT, AND J.A. PFALTZGRAFF Abstract. A Schwarz-Christoﬀel

SCHWARZ-CHRISTOFFEL MAPPING OF MULTIPLYCONNECTED DOMAINS

T.K. DELILLO, A.R. ELCRAT, AND J.A. PFALTZGRAFF

Abstract. A Schwarz-Christoffel mapping formula is established for polyg-onal domains of finite connectivity m ≥ 2 thereby extending the results ofChristoffel (1867) and Schwarz (1869) for m = 1 and Komatu (1945), m =2. A formula for f, the conformal map of the exterior of m bounded disksto the exterior of m bounded, disjoint polygons, is derived. The derivationcharacterizes the global preSchwarzian f ′′ (z) /f ′ (z) on the Riemann spherein terms of its singularities on the sphere and its values on the m boundarycircles via the reflection principle, and then identifies a singularity functionwith the same boundary behavior. The singularity function is constructedby a ′′method of images′′ infinite sequence of iterations of reflecting prevertexsingularities from the m boundary circles to the whole sphere.

1. Introduction

Recently, it was remarked that “It is a longstanding dream to generalize [Schwarz-Christoffel] to multiply connected domains...”, [5], p.7. In this paper we carry outthat generalization for domains of connectivity m ≥ 2 by deriving a formula forthe conformal map of an m-connected, unbounded circular domain Ω onto an un-bounded polygonal domain, P. The boundary of P consists of m disjoint polygonalcurves. The point at infinity is in the interior of both domains and is held fixedby the mapping f : Ω → P, f (∞) = ∞. The boundary components of P may bepolygonal Jordan curves, simple polygonal ”two-sided” slits, or polygonal Jordancurves with some attached simple polygonal ”two-sided” slits protruding into P.Our construction of the mapping function is based on infinite sequences of iteratedreflections that generate infinite products in our mapping formula

(1.1) f (z) =∫ z m∏

i=1

Ki∏

k=1

∞∏j=0

ν∈σj(i)

(ζ − zk,νi

ζ − sνi

)

βk,i

dζ.

The zk,νj are generated by reflections of the prevertex points on ∂Ω. The sνj

are generated by the point at infinity and its reflections and the βk,jπ are theturning angles at the vertices of ∂P. The ν′s are multiindices. We explain all ofthese matters in the paper. Our proof of the convergence of the infinite productsin (1.1), and hence the validity of the formula, for connectivity m ≥ 3 is validfor domains whose boundary components are disjoint and satisfy an additional

Date: October 1, 2002.1991 Mathematics Subject Classification. Primary 30C30.Key words and phrases. Conformal map, analytic continuation, reflection principle, conformal

invariant.

1

2 T.K. DELILLO, A.R. ELCRAT, AND J.A. PFALTZGRAFF

separation condition described in the proof and discussed further in the paper. Itshould be possible to prove that disjointness is sufficient for (1.1). Our techniqueof using an infinite sequence of iterated reflections and analytic continuation byreflection to construct the mapping formula is quite similar to the ”method ofimages” technique that appears in electrostatics, c.f., [1].

The problem of finding a conformal map from a simple type of canonical domainto a general domain has a long history and has led to much elegant mathematicaltheory. If the domains are simply connected the canonical domain is usually takento be the unit disk and the theory is that associated with the Riemann MappingTheorem. For multiply connected domains various choices of “canonical” domainhave been identified and studied, [9], [8], [17]. The one that we have chosen is theexterior of a finite number of nonoverlapping disks, which seems especially appro-priate with the target domain being an exterior domain. Furthermore, there arevarious classical explicit formulas relating pairs of some of the canonical classes ofslit domains, but the circular domains seem to stand apart by themselves in this re-spect. However, since the parallel slit canonical domains are unbounded polygonaldomains, our S-C mapping formula provides an explicit formula connecting map-pings in the circular class with those in the parallel slit class of canonical domains.Also, the circular boundary components are convenient for the use of Fourier anal-ysis of boundary data in applied problems. For a general target domain there is noexplicit mapping formula. On the other hand, in the simply connected case, if thetarget is bounded by a polygon then the Schwarz-Christoffel (S-C) formula (3.1)gives an explicit representation of the map in terms of quadratures and a finitenumber of parameters (which must be determined numerically [6]).

The (S-C) formula for the conformal map of an annulus onto a doubly connectedpolygonal domain (3.2) has been known for more than fifty years, [12]; c.f., [3].However, with the exception of special configurations involving much symmetryallowing reduction to a simply connected mapping problem, [7], [15], no explicitgeneralization to connectivity m ≥ 3 seems to be known; see also [13], [14].

In their recent paper [3] the authors gave a new derivation of the annulus for-mula based on the characterization of the global preSchwarzian, f ′′(z)/f ′(z), ofthe mapping function in terms of its singularities and boundary behavior, a con-struction of a global singularity function S(z) for the analytic continuation of thepreSchwarzian, and then a proof that f ′′ (z) /f ′ (z) = S (z). The construction con-sists of an infinite sequence of repeated reflections that generate infinite productsand the theta functions in the final formula for the mapping function (3.2). Ourpresent work for higher connectivity uses the same general strategy, but the stepto connectivity greater than two introduces new and substantial difficulties. Theprincipal new feature that distinguishes the case m ≥ 3 from m = 1 or 2 is anexponential increase of terms by a factor of m − 1 at each of the infinitely manylevels of reflections in the construction of the singularity function and the attendantdifficulties in describing and establishing the properties of S(z).

The remainder of the paper is organized as follows: In section 2, we set notationand present some supporting results on reflections, moduli, separation and estimatesof radii. The global preSchwarzian, its singularity function, the main result (Theo-rem 1) and its corollaries for connectivity m = 2 are in section 3. Convergence ofthe limit defining S(z) for a region whose separation modulus, ∆, satisfies the bound

MAPPING MULTIPLY CONNECTED POLYGONAL DOMAINS 3

∆ < 1/ (m− 1)1/4 is established in section 4. In section 5 we show that S (z) satis-fies the boundary condition necessary for proving that S(z) = f ′′(z)/f ′(z). Someelementary graphical illustrations are given in section 6.

2. Preliminaries

Throughout this work C is the complex plane and C∞ = C∪∞ is the Riemannsphere. We let Ω denote an m-connected, unbounded, circular domain containingthe point at infinity that is conformally equivalent to the unbounded polygonaldomain P. That is ∂Ω consists of m disjoint circles Cj = z : |z − cj | = rj , j =

1, . . . , m, and Ω = C∞\m⋃

j=1

clDj where Dj = z : |z − cj | < rj, Dj = cl (Dj), are

the m disks with mutually disjoint closures, Dj∩Di = φ, i 6= j. The correspondingpolygonal boundary components, f (Cj), of P will be denoted by Γj , j = 1, . . . ,mand given the counterclockwise orientation. The Kj vertices of Γj will be denotedby wk,j , k = 1, . . . ,Kj numbered counterclockwise around Γj . The correspondingvertex angles of the Γj at the vertices wk,j , measured from the interior of P, aredenoted by παk,j , where 0 < αk,j ≤ 2, and βk,jπ is the turning of the tangentat wj,k where βk,j = αk,j − 1. The prevertices are denoted by zk,j ∈ Cj withf(zk,j) = wk,j .

2.1. Reflections. Throughout this work the reflection of z through a circle C withcenter c and radius r is given by

z∗ = ρC(z) := c +r2

z − c,

i.e., z and z∗ are symmetric points with respect to the circle C. If C = Cτ whereτ is an index of a circle, we will denote ρCτ by ρτ .

In order to explain the reflection process and the indexing of reflections of bound-ary circles, prevertices and the centers sj in our work we first describe the initialsteps in the triply connected case which anticipate the general situation when theconnectivity exceeds three. We begin with the unbounded, triply connected cir-cular domain Ω, ∂Ω = C1 ∪ C2 ∪ C3 that f maps conformally onto an unboundedtriply connected polygonal domain P. On each circle Cj there are Kj preverticeszk,j , k = 1, ...,Kj , j = 1, 2, 3. The reflection process begins by reflecting Ω throughC1 to form Ω1 := ρ1 (Ω), reflecting Ω through C2 to obtain Ω2 := ρ2 (Ω) and Ωthrough C3 to obtain Ω3 := ρ3 (Ω). Each of the three domains Ωj is a boundedtriply connected circular domain with circular boundary components Cj , Cji, i 6= j,1 ≤ i ≤ 3:

∂Ω1 = C1 ∪ C12 ∪ C13, ∂Ω2 = C2 ∪ C21 ∪ C23, ∂Ω3 = C3 ∪ C31 ∪ C32.

At the next level there are six reflections: Ω1 is reflected through its interior bound-ary circles C12 and C13 and similarly for Ω2 and Ω3 producing the six bounded,triply connected circular domains

Ω12:=ρ12 (Ω1) , Ω13:=ρ13 (Ω1) , Ω21:=ρ21 (Ω2) , Ω23:=ρ23 (Ω2) ,

Ω31:=ρ31 (Ω3) , Ω32:=ρ32 (Ω3) .


The new boundary circles are

C121 = ρ12 (C1) , C123 = ρ12 (C13) , ∂Ω12 = C12 ∪ C121 ∪ C123,

C131 = ρ13 (C1) , C132 = ρ13 (C12) , ∂Ω13 = C13 ∪ C131 ∪ C132

with similar results and notation for the boundaries of Ω21, Ω23, Ω31,and Ω32. Onesees that by continuing in this fashion the number of new regions and new boundarycomponents created by the reflections at a given level is twice that at the precedinglevel, see Figure 1. Furthermore, the general m-connected case is similar with m−1replacing the factor 2 in the exponential rate of increase of regions and boundarycircles.

−1 0 1 2 3 4 5−1

0

1

2

3

4

5

C1

C2

C3

C12

C13

C21

C23

C32

C31

z1,1

z2,1

z3,1

Fig.1 Circle reflection notation

We now describe the iterated reflection process for the general m-connectedunbounded circular domain Ω with boundary components C1, ..., Cm. Fixing anindex ν1 ∈ 1, ..., m we reflect Ω through the circle Cν1to form an m-connectedbounded circular region Ων1with outer boundary Cν1 and m− 1 holes bounded bycircles Cν1j ,

Ων1 := ρν1(Ω) , Cν1j = ρν1 (Cj) , j 6= ν1, j = 1, ...,m.

Similarly, reflecting Ω through each of the other m−1 boundary circles, Ck, k 6= ν1,produces circular domains with outer boundary Ck, m−1 holes and inner boundarycircles

Ωk := ρk (Ω) , Ckj := ρk (Cj) , j 6= k, j = 1, ...,m.

This set of m reflections and corresponding circular subdomains, Ωk, of the comple-ment is the first step in the sequence of iterations of reflections of domains through


boundary circles that leads to an exhaustion of the complement of Ω except for aset of limit points.

At the next level we produce m(m−1) domains Ωkj by reflecting each Ωk domainthrough each of its m − 1 interior boundary circles Ckj . For each k this producesthe m − 1 circular domains Ωkj := ρkj (Ωk) (j 6= k, j = 1, ...,m) of connectivity meach with outer boundary Ckj and the m− 1 inner boundary circles

Ckjk := ρkj (Ck) , Ckji = ρkj (Cki) , i ∈ 1, ..., m \ j, k .

We note that for each fixed k and j 6= k the first circle Ckjk is the reflection of theouter boundary of Ωk, while each of the other m− 2 circles is the reflection of oneof the inner boundary components of Ωk.

In general the reflected regions and circles are labeled with multi-indices ν =ν1ν2 · · · νn with νj ∈ 1, ..., m, νk 6= νk+1, k = 1, ..., n − 1, and we write |ν| todenote the length of ν, i.e., |ν| = n.

Definition 1. The set of multi-indices of length n will be denoted

σn = ν1ν2 · · · νn : 1 ≤ νj ≤ m, νk 6= νk+1, k = 1, ..., n− 1 , n > 0,

and σ0 = φ, in which case νi = i. Also

σn (i) = ν ∈ σn : νn 6= i ,

denotes sequences in σn whose last factor never equals i.

Thus, if ν ∈ σn, n > 1, Ων = ρν

(Ων1...νn−1

)is a circular domain with outer

boundary Cν and m− 1 interior boundary circles(2.1)

Cννn−1 = ρν

(Cν1...νn−1

), Cνj = ρν

(Cν1...νn−1j

), j ∈ 1, ..., m \ νn−1, νn .

Note that if ν ∈ σn then for j 6= νn, Ωνj = ρνj (Ων). Clearly σn containsm (m− 1)n−1 elements which is consistent with our earlier comment that the num-ber of circular domains Ων at a particular level of reflections, say ν ∈ σn, is m− 1times the number of domains Ωeν , ν ∈ σn−1, at the preceding level.

It will be necessary to follow the successive reflections of c1, ..., cm, the centersof the boundary circles, C1,..., Cm. Each ck is the center of the circular domainΩk because it is the reflection of ∞ through Ck. Clearly, none of the successiveiterated reflections of ck will be the centers of the corresponding reflected domains,Ων ν ∈ σn. However, we can index each of these reflected points with the indexof the reflected domain in which it lies, i.e., sν = ρν

(sν1...νn−1

), ν ∈ σn, and

sk := ck, k = 1, .., m, (to avoid confusion with the ”center” cν of the reflectedcircular domain). We also note that if ν ∈ σn then for j 6= νn, sνj = ρνj (sν).

Clearly, one should index the reflections of the prevertices, zk,j , with the indicesof the reflected circles on which they lie. Thus, if ν ∈ σn, then zk,νn is the (iterated)reflection of zk,ν1 that lies on the circle Cν and from (2.1)

zk,ννn−1 = ρν

(zk,ν1...νn−1

), zk,νj = ρν

(zk,ν1...νn−1j

), j ∈ 1, ...,m \ νn−1, νn .

Similarly, we let rν denote the radius of a circle Cν , ν ∈ σn.The following elementary fact about successive reflections will be useful.

Proposition 1. Reflection of a set of points U through circle Cλ followed by re-flection through circle Cτ is the same as its reflection through circle Cτ followed by


reflection through Cτλ = ρτ (Cλ), the reflection of Cλ through Cτ . Symbolically, weexpress this as

ρτ (ρλ(u)) = ρτλ(ρτ (u)).

Proof. For simplicity in visualizing the geometry one may think of the case when thetwo circles are nonintersecting. Since Moebius transformations preserve symmetry(reflection) in circles it is enough to assume that Cτ is the real axis. Thus ρτ (u) =u, the circles Cλ and Cτλ are reflections of each other in the real axis, cτλ = cλ,rτλ = rλ and

ρτ (ρλ (u)) = cλ +r2λ

u− cλ= cτλ +

r2τλ

ρτ (u)− cτλ

= ρτλ (ρτ (u)) .

¤

The next result is used in section 5.

Lemma 1. Let ν = ν1 · · · νn ∈ σn and j ∈ 1, · · · ,m \ ν1 then

Cjν = ρj(Cν), sjν = ρj (sν) and zk,jν = ρj (zk,ν)

Proof. We first show that Cjν = ρj(Cν). This follows from Proposition 1 and isneeded to apply Proposition 1 when τ = j and λ = ν. Our proof is by induction on|ν|:

For |ν| = 1, ν = ν1 6= j, we have Cjν1 = ρj(Cν1) by the definition of Cjν1 in(2.1).

For |ν| = 2, ν = ν1ν2, ν1 6= j there are two cases. First, for ν2 6= j, we have

Cjν1ν2 = ρjν1(Cjν2) by (2.1)

= ρjν1(ρj(Cν2)) by definition of Cjν2

= ρj(ρν1(Cν2)) by Prop. 1 with τ = j, λ = ν1

= ρj(Cν1ν2) by definition of Cν1ν2 .

Next, for ν2 = j, we have

Cjν1j = ρjν1(Cj) by (2.1)

= ρjν1(ρj(Cj)) since ρj(Cj) = Cj

= ρj(ρν1(Cj)) by Prop. 1

= ρj(Cν1j) by definition of Cν1j .

We now use our induction hypothesis, Cjν = ρj(Cν) for |ν| < n and show thatthe result is true for |ν| = n > 2. Thus, let ν = ν1ν2 · · · νn, ν1 6= j. First, ifνn 6= νn−2 we have

Cjν = ρjν1···νn−1(Cjν1···νn−2νn) by (2.1)

= ρjν1···νn−1(ρj(Cν1···νn−2νn)) by the induction hypothesis

= ρj(ρν1···νn−1(Cν1···νn−2νn)) by Prop. 1 and the induction hypothesis

= ρj(Cν) by (2.1).


When νn = νn−2 we have

Cjν = ρjν1···νn−1(Cjν1···νn−2) by (2.1)

= ρjν1···νn−1(ρj(Cν1···νn−2)) by the induction hypothesis

= ρj(ρν1···νn−1(Cν1···νn−2)) by Prop. 1 and the induction hypothesis

= ρj(Cjν1···νn−1νn−2) by (2.1)

= ρj(Cν) since νn−2 = νn.

Similar reasoning yields the result zk,jν = ρj (zk,ν) since zk,jν ∈ Cjν .We now note that sνq = ρνq (sν) q 6= νn(by Defn. of sνq) and then argue

inductively. Recalling that sj = ρj (∞), sν1 = ρν1 (∞) and applying Proposition 1we find that

sjν1 = ρjν1 (sj) = ρjν1 (ρj (∞)) = ρj (ρν1 (∞)) = ρj (sν1)

and similarly sjν1ν2 = ρjν1ν2 (sjν1) = ρjν1ν2 (ρj (sν1)) = ρj (ρν1ν2 (sν1))=ρj (sν1ν2).Using the induction assumption, sjν1...νn−1 = ρj

(sν1...νn−1

), the first part of the

lemma, and Proposition 1 we argue that

sjν = ρjν

(sjν1...νn−1

)= ρjν

(ρj

(sν1...νn−1

))= ρj

(ρν

(sν1...νn−1

))= ρj (sν) .

¤

2.2. Moduli and separation. We let µ−1jk denote the conformal modulus of the

doubly connected region bounded by curves Γi and Γj . That is, the region isconformally equivalent to the annulus µjk < |z| < 1. For the region exterior to twomutually exterior disks with centers cj , ck, radii rj , rk and distance between centersdj,k = |cj − ck| one has the elementary formula,

(2.2) µjk =d2

j,k − r2j − r2

k −√[

(dj,k − rj)2 − r2

k

] [(dj,k + rj)

2 − r2k

]

2rjrk,

for the conformal invariant, µjk, obtained with the Moebius mapping of the regiononto the annulus µjk < |z| < 1.

The following lemmas will be used in our convergence proofs. The symbol α(Γ)denotes the area enclosed by a curve Γ.

Lemma 2. Let B be a bounded doubly connected region with finite modulus µ−101 > 1,

bounded on the outside by a Jordan curve Γ0 and on the inside by a Jordan curveΓ1. Then

α(Γ1) ≤ µ201α(Γ0).

Proof. See [10], Lemma 17.7c (a), p. 503. ¤

Note, if Γ0 and Γ1 are two circles of radii r0 and r1, then the lemma is r1 ≤ µ01r0.It is essential to have the disjointness (separation) of the m boundary circles Cj

expressed analytically. We do this with the nonoverlap inequalities

(2.3) µsepjk :=

rj + rk

|cj − ck| =rj + rk

dj,k< 1, j 6= k, 1 ≤ j, k ≤ m.

We now define the separation modulus of the region

(2.4) ∆ := maxi,j;i6=j

µsepij


for the m boundary circles Cj , c.f., [10], p.501. Let Cj denote the circle withcenter cj and radius rj/∆, then geometrically, 1/∆ is the smallest magnification ofthe m radii such that at least two Cj ’s just touch.

The quantities µsepjk in (2.3) are not conformally invariant, but one can express

the nonoverlapping property in terms of the conformally invariant quantities αj,k

defined by

(2.5) αj,k :=12

(µj,k +

1µj,k

)=

d2j,k −

(r2j + r2

k

)

2rjrk.

The second equality follows from (2.2). Two mutually exterior circles, Cj and Ck,are nonoverlapping iff d2

j,k > (rj + rk)2 iff αj,k > 1.

Proposition 2. µjk <(µsep

jk

)2

≤ ∆2.

Proof. To simplify notation we consider a typical pair of circles Cj , Ck and writeµs = µsep

j,k , α = αj,k, d = dj,k. Thus (µsd)2 = (rj + rk)2 and from (2.5) µ2s

(r2j + r2

k + 2αrjrk

)=

r2j +r2

k+2rjrk. Further manipulation and the geometric-arithmetic mean inequalitygive

αµ2s − 1 =

(1− µ2

s

) r2j + r2

k

2rjrk≥ 1− µ2

s

with equality iff rj = rk. Continuing, we find that

1µ2

s

≤ α + 12

=[12

(√µj,k +

1√µj,k

)]2

and hence √µj,k < µs since √µj,k < 2√µj,k/ (µj,k + 1) ≤ µs. The inequality isstrict since µj,k < 1. Clearly, for many nonoverlapping circles, the result followssince ∆ = max

j,k;j 6=kµsep

jk . ¤

Lemma 3. ([10], p.505, )

∑ν∈σn+1

α(Cν) ≤ ∆4nm∑

i=1

α(Ci)

∑ν∈σn+1

r2ν ≤ ∆4n

m∑

i=1

r2i .

Proof. (Idea of proof for m = 3). Here (Cj)ν denotes the reflection of Cj throughCν . (This is not the magnification of Cν by 1/∆ unless the circles are concentric.)By Proposition 2, Lemma 2 and the geometry of the areas α(Cν) ≤ ∆2α(Cν).Thus, for instance,we have that

α(C13) + α(C12) ≤ ∆2α(C1) ≤ ∆4α(C1)

α(C23) + α(C21) ≤ ∆2α(C2) ≤ ∆4α(C2)

α(C32) + α(C31) ≤ ∆2α(C3) ≤ ∆4α(C3).

This gives the first step and α(Cν) = πr2ν gives the second inequality. ¤


3. Analytic continuation and the mapping formula

In this section we describe the analytic continuation by reflection of the mappingfunction, the singularities of the preSchwarzian, f ′′ (z) /f ′ (z), and its singularityfunction, S (z). We give a precise statement of our mapping formula and its proofin Theorem 1. The convergence and boundary property of S (z) are used here, butthe somewhat cumbersome details of these proofs are postponed to the followingtwo sections of the paper.

We begin with some general observations about Schwarz-Christoffel type map-ping formulas of the unit disk and the annulus that motivate our present work forthe m-connected circular domain. In each case the formula is an integral of aproduct function,

f (z) =∫ z

P (ζ) dζ

or equivalently f ′ (z) = P (z) . For mapping the unit disk to a polygonal domainthe geometry of the polygon determines that arg(f ′

(eit

)ieit) is constant on arcs of

|z| = 1 with jumps βkπ at the prevertices, zk, and hence the product has the form

(3.1) f ′ (z) = P (z) =K∏

k=1

(z − zk)−βk .

Mapping the annulus, µ < |z| < 1 onto a conformally equivalent doubly connectedpolygonal domain leads to infinite products

(3.2) f ′ (z) = P (z) =K∏

k=1

[Θ

(z

µz0,k

)]−β0,k J∏

j=1

[Θ

(µz

z1,j

)]−β1,j

involving the theta function, Θ (w) =∏∞

ν=0

(1− µ2ν+1w

) (1− µ2ν+1/w

), [12], [3].

Although the local analysis of the boundary mapping of the annulus is drivenby the same geometric idea as for (3.1), it is not obvious why there are infiniteproducts and what they should be. One approach to this problem is to use thegeometric properties of the mapping function, f , under reflections and the affineinvariance of the preSchwarzian to generate a formula for the analytic continuationof a globally defined, single-valued preSchwarzian, [3]. Note that in terms of theproduct function one has f ′′ (z) /f ′ (z) = P ′ (z) /P (z). We now use this generalstrategy for the m-connected domains in the present work.

Our derivation of the mapping formula (1.1) begins by considering the analyticcontinuation of f from Ω to a domain Ωj by reflection across an arc γk,j betweensucessive prevertices zk,j , zk+1,j on Cj . Such an extension, fk,j , has the form

fk,j(z) = f(z), z ∈ Ω ∪ γk, fk,j(z) = ak,jf(cj + r2j /(z − cj) + bk,j , z ∈ Ωj ,

with ak,j , bk,j determined by the line containing the edge f(γk,j) joining wk,j andwk+1,j in the boundary of P. This reflection maps Ωj conformally onto Pk,j ,the unbounded, m-connected polygonal domain obtained by reflecting P acrossthe line containing the segment f(γk). In general, if ν ∈ σn, one can obtain acontinuation of f to the reflected m-connected circular domain Ων = ρν

(Ων1...νn−1

)by a finite number of successive reflections. By repeated application of the reflectionprocess one obtains from the initial function element (f, Ω) a global (many-valued)analytic function f defined on C∞\clzk,ν , sν where zk,ν , and sν denote the originalprevertices and centers and all of their images under sequences of iterated reflections


and cl denotes closure of the set. Any two values, fr (z) and fs (z) of f at a pointz ∈ C \ clzk,ν , sν are related by the composition of an even number of reflectionsin lines and hence fs(z) = αfr (z) + β for some α, β ∈ C. The preSchwarzian of f ,f ′′(z)/f ′(z), is invariant under affine maps w 7−→ aw+b, i.e., (af(z)+b)′′/(af(z)+b)′ = f ′′(z)/f ′(z). Thus, if one begins with the preSchwarzian of the mapping inΩ, the reflection process yielding the many-valued f also defines a global analyticpreSchwarzian, f ′′(z)/f ′(z), that is defined and single-valued on C\clzk,ν , sν.

As we shall see, the preSchwarzian is determined by its singularities, i.e., itsbehaviour on the singularity set zk,ν , sν. Since f extends analytically by reflec-tion across each boundary circle except at the prevertices it maps a circular arccontaining a prevertex zk,i onto a pair of line segments meeting at the vertex wk,i

with angle αk,iπ. Hence the function (f (z)− wk,i)1/αk,i maps the circular arc onto

a straight segment and by the reflection principle is analytic in a neighborhood ofzk,i with a local expansion

(3.3) (f(z)− f(zk,i))1/αk,i = (z − zk,i) hk,i (z)

where hk,i (z) is analytic and nonvanishing near zk. This gives the local expansion

f ′′ (z) /f ′ (z) = βk,i/ (z − zk,i) + Hk,i (z) , βk,i = αk,i − 1

where Hk,i (z) is analytic in a neighborhood of zk,i, and βk,iπ is the jump in thetangent angle at the vertex wk,i. Recall that elementary geometry of the turningtangent shows that −1 < βk,i ≤ 1 and

∑mk=1 βk,i = 2.

The point at ∞ and its iterated reflections are singularities of the preSchwarzian.This is a departure from the behavior of S-C maps of the disk and annulus tobounded polygonal domains. This feature is common to exterior maps and intro-duces double poles 1/ (z − p)2 in the integral formula for the S-C mapping formula,[16], p. 329, Thm. 9.9. Since f(∞) = ∞ there is a simple pole of f(z) at infinityand an expansion

f(z) = az + a0 +a1

z+

a2

z2+ . . . , a 6= 0

at ∞. This givesf ′′(z)f ′(z)

=2a1z3 + . . .

a− 2a1z2 + . . .

=2a1

az3+

[1z4

]

at ∞. Here and elsewhere in the context of a series expansion we use the standardnotation

[(z − ζ)k

]to denote a series in powers of (z − ζ) that has a factor of

(z − ζ)k.If f(z) is an analytic continuation by reflection of f across an arc of a circle with

center c and radius r then ∞ reflects to c and

f(z) = Af(r2

z − c+ c) + B = A

(ar2

z − c+ ac + a0 + a1

z − c

r2+ . . .

)+ B

for z near c. This gives

f ′′(z)f ′(z)

=A

(2ar2

(z−c)3 + . . .)

A(− ar2

(z−c)2 + . . .) = − 2

z − c+ b0 + [(z − c)].


A similar calculation shows that the extension of f by reflection through interiorcircles leads to the same behavior near the reflections, sν , of the centers of thecircles Cj ,

f(z) =A

z − sν+ B + [(z − sν)] ,

f ′′(z)f ′(z)

= − 2z − sν

+ b + [(z − sν)] .

This can be seen by considering reflections through a circle of radius r and centerc, replacing ρc(z) − ρc(sν) by its conjugate in the expansion of f near ρc(sν) andusing

r2

z − c− r2

sν − c=

r2(z − sν)(c− sν)(z − c)

= [(z − sν)]

for z near sν . This gives the desired expansion around the simple pole ρc(sν).

Definition 2. The singularity function, S (z), of the global preSchwarzian is theinfinite sum of the singular parts of the local expansions of f ′′(z)/f ′(z) at all of itssingularities, zk,ν and sν .

Thus one should think that

S (z) =∞∑

j=0

m∑

i=1

∑

ν∈σj(i)

(Ki∑

k=1

βk,i

z − zk,νi− 2

z − sνi

)

and with more care formulate the

Definition 3. Analytically S (z) is the following limit

S (z) = limN→∞

SN (z)

where

SN (z) =N∑

j=0

m∑

i=1

∑

ν∈σj(i)

Ki∑

k=1

βk,i(zk,νi − sνi)(z − zk,νi)(z − sνi)

=m∑

i=1

Ki∑

k=1

βk,i

N∑j=0

ν∈σj(i)

(zk,νi − sνi)(z − zk,νi)(z − sνi)

Note the equivalent forms of the summandKi∑

k=1


=Ki∑

k=1

(βk,i

z − zk,νi− βk,i

z − sνi

)=

(Ki∑

k=1

βk,i

z − zk,νi

)− 2

z − sνi

since the βk,i sum to 2. We defer to a later section the convergence proof of thesequence SN (z).

The principal idea in our work is that the singularity structure and the follow-ing boundary behavior of the preSchwarzian which is shared by S (z) (see sec. 5below) enable one to deduce that f ′′ (z) /f ′ (z) = S (z) and hence its completecharacterization.

Lemma 4. Re (z − cj) f ′′ (z) /f ′ (z)|z−cj |=rj= −1, j = 1, ..., m.

Proof. The tangent angle ψ(t) = argirjeitf ′(cj+rje

it) of the boundary Cj is con-stant on each of the arcs between prevertices. Hence ψ′(t) = Re (z − cj)f ′′(z)/f ′(z) + 1 =0, |z − ci| = ri, z 6= zk,j . ¤

We now present our main result.


Theorem 1. Let P be an unbounded m-connected polygonal region, ∞ ∈ P and Ω aconformally equivalent circular domain. Further, suppose P satisfies the separationproperty ∆ < (m− 1)−1/4 for m > 1. Then Ω is mapped conformally onto P by afunction of the form Af (z) + B where

(3.4) f (z) =∫ z m∏

i=1

Ki∏

k=1

∞∏j=0

ν∈σj(i)

(ζ − zk,νi

ζ − sνi

)

βk,i

dζ.

Where the turning parameters satisfy −1 < βk,i ≤ 1 and∑m

k=1 βk,i = 2. Theseparation parameter, ∆, is given explicitly in terms of the radii and centers of thecircular boundary components of Ω, (2.4).

Proof. The central idea is to prove that f ′′ (z) /f ′ (z) = S (z) by means of the argu-ment principle. We shall use the following two results whose proofs are postponedto the following two sections of the paper in order to keep the essence of the presentproof from being obscured by calculation details.

(i) Convergence: S (z) = limN→∞

SN (z) uniformly on closed subsets of cl (Ω) \ PV ,where PV denotes the set of prevertices on the m circular boundary componentsof Ω and cl denotes closure of a set.

(ii) B.C.: Re (z − sj) S (z)z∈Cj= −1, j = 1, ..., m.

For z ∈ (clΩ)\ PV we define the functions

H(z) :=∫ z

S(ζ)dζ, HN (z) :=∫ z

SN (ζ)dζ, P (z) := eH(z).

We first note that

HN (z) =∫ z

SN (ζ)dζ =N∑

j=0

m∑

i=1

∑

ν∈σj(i)

Ki∑

k=1

βk,i

∫ z 1ζ − zk,νi

− 1ζ − sνi

dζ

is defined and analytic in Ω since its periods are zero. Indeed∫

Cτ+SN (z) dz =

0, τ = 1, ..., m, where Cτ+ is a circle concentric with the boundary circle Cτ withradius slightly larger than that of Cτ , since the residues add out in pairs, and for the”point at infinity” lim

N→∞z2SN (z) = 0 which eliminates any period over large circles

enclosing all of the boundary components of Ω. Furthermore, H (z) is analytic inΩ since

H (z) = limN→∞

HN (z) = limN→∞

∫ z

SN (ζ)dζ =∫ z

S(ζ)dζ, z ∈ cl (Ω) \ PV

with SN (z) → S(z) uniformly on closed subsets of cl (Ω) \ PV .


The next step is to develop a formula for the antiderivative (up to an additiveconstant)

HN (z) =∫ z

SN (ζ)dζ =N∑

j=0

∫ z m∑

i=1

∑

ν∈σj(i)

Ki∑

k=1

βk,i

(1

ζ − zk,νi− 1

ζ − sνi

)dζ

=N∑

j=0

m∑

i=1

∑

ν∈σj(i)

Ki∑

k=1

βk,i

∫ z 1ζ − zk,νi

− 1ζ − sνi

dζ

=N∑

j=0

m∑

i=1

∑

ν∈σj(i)

Ki∑

k=1

βk,i log(

z − zk,νi

z − sνi

)=

m∑

i=1

Ki∑

k=1

βk,i

N∑j=0

ν∈σj(i)

log(

z − zk,νi

z − sνi

)

where each logarithm is the branch that vanishes at z = ∞, i.e. log 1 = 0. Fromthe preceding formula one has

P (z) = limN→∞

exp HN (z) = limN→∞

m∏

i=1

Ki∏

k=1

N∏j=0

ν∈σj(i)

(z − zk,νi

z − sνi

)

βk,i

and hence the product formula for P (z),

(3.5) P (z) = eH(z) =m∏

i=1

Ki∏

k=1

∞∏j=0

ν∈σj(i)

(z − zk,νi

z − sνi

)

βk,i

.

Our theorem, f (z) = A∫ z

P (ζ) dζ+B, is equivalent to showing that the quotient

Q(z) :=f ′(z)P (z)

≡ constant.

To accomplish this, we will apply the argument principle to Q(z). First, observethat P ′(z) = H ′(z)eH(z) = S(z)P (z), that is, P ′(z)/P (z) = S (z), and

Q′ =f ′

P

(f ′′

f ′− P ′

P

)= Q

(f ′′

f ′− S

).

Then, for z = cj + rjeiθ ∈ Cj , the boundary conditions of Lemma 4 and Theorem

5 on f ′′/f ′ and S respectively give

∂

∂θarg Q(z) =

∂

∂θImlog Q(z) = Re(z − cj)

Q′(z)Q(z)

= Re(z − cj)(

f ′′(z)f ′(z)

− S(z)) = 0.

By our construction of S(z), f ′′(z)/f ′(z)−S(z) is continuous on all of Cj includingat the prevertices. Therefore, arg Q is constant on each of the m boundary circles,Cj . Equivalently, Q(Cj), the image of Cj , lies on a half-ray emanating from theorigin. It is clear by the local behavior (3.3) and formula (3.5), that Q = f ′/Pis continuous on each Cj and not equal to 0 or ∞ there, since f ′, P 6= 0. Thusfor any w0 ∈ C\Q (Cj), j = 1, . . . , m, the winding number of Q(Cj) around w0,n(Q(Cj), w0) = 0 for all j. Let CR be a large circle of radius R centered at the originand containing w0 and all the Cj ’s in its interior, and write C = C1∪· · ·∪Cm∪CR


with the curves oriented so that the region interior to CR and exterior to the Cj ’sis on the left. Since Q has no poles in the region, by the argument principle (forbounded regions), the number of times Q(z) assumes the value w0 is

n(Q(C), w0) = n(Q(C1), w0)+ · · ·+n(Q(Cm), w0)+n(Q(CR), w0) = n(Q(CR), w0).

We now will show that n(Q(CR, w0) = 0. First,

n(Q(CR), w0) =1

2πi

∫

|z|=R

Q′(z)Q(z)− w0

dz =1

2πi

∫

|z|=R

Q′(z)/Q(z)1− w0/Q(z)

dz.

Recall that Q′(z)/Q(z) = f ′′(z)/f ′(z) − S(z) =[1/z3

] − [1/z2

]= O(1/z2) for z

near ∞, and that, Q(∞) = f ′(∞)/P (∞) is a finite constant. It suffices to assumew0 6= Q(∞). Then w0 6= Q(z) for R sufficiently large and there are constantsA,B > 0 such that

∣∣∣∣∣∣∣

∫

|z|=R

Q′(z)/Q(z)1− w0/Q(z)

dz

∣∣∣∣∣∣∣≤ A

∫

|z|=R

|Q′(z)/Q(z)||dz| ≤ B

2π∫

0

1R2

Rdθ → 0

as R → ∞. Therefore n(Q(C), w0) = 0 and Q(z) 6= w0 for w0 /∈ Q(Cj) andw0 6= Q(∞). Thus, Q assumes values only on the radial segments Q(Cj) (orQ(∞)) and hence, by the open mapping property of analytic functions, Q must beconstant on Ω. ¤

In the special case when m = 2 there is no restrictive separation hypothesissince then ∆ < (m− 1)−1/4 = 1 is equivalent to merely having the two boundarycomponents be nonoverlapping; an obvious necessity for double connectivity ofΩ. Clearly, the doubly connected case is easier because there is no exponentialdoubling of singularities in the reflection process. We conjecture that the result istrue for the general case m > 2 when ∆ < 1, i.e., the boundary components aredisjoint with no additional separation resriction. Note also the remark followingthe proof of Theorem 3.

The following corollaries show the relative simplicity of the mapping formulafor doubly connected unbounded polygonal regions. The second corollary showsthat with appropriate normalization of the circular domain the theta function inKomatu’s result (3.2) appears in the formula, and hence, removes some of themystery about the nature of the infinite products.

Corollary 1. Let P be an unbounded doubly connected polygonal region, ∞ ∈ Pand Ω a conformally equivalent circular domain. Then Ω is mapped conformallyonto P by a function of the form Af + B where

(3.6) f (z) =∫ z K1∏

k=1

∞∏j=0

ν∈σj(1)

(ζ − zk,ν1

ζ − sν1

)

βk,1

K2∏

k=1

∞∏j=0

ν∈σj(2)

(ζ − zk,ν2

ζ − sν2

)

βk,2

dζ.

The multi-indices contain only the integers 1 and 2 hence in the first productthere is only one ν in each σj(1), i.e., as j runs through the integers 1, 2, 3, 4, ...the corresponding sequence of subscripts ν1 is 1, 21, 121, 2121, ... respectively.Similarly in the second product the subscripts ν2 run 2, 12, 212, 1212, etc. With


some inspired rearranging and intricate manipulations it is possible to get an inter-esting formula with theta functions that is equivalent to (3.6) and related to (3.2).There is no loss of generality in letting the two boundary circles of Ω be C1, theunit circle and a circle C2 with center c > 1 and radius r.

Corollary 2. Let P be an unbounded doubly connected polygonal region, ∞ ∈ Pand Ω a conformally equivalent circular domain with boundary components C1 beingthe unit circle and C2 a circle of radius r and center c > 1. Then Ω is mappedconformally onto P by a function of the form Af + B where

f (z) =

z∫ K1∏

k=1

[Θ

(T (ζ)

µT (z1,k)

)]βk,1 K2∏

k=1

[Θ

(µT (ζ)T (z2,k)

)]βk,2

×

Θ(−pT (ζ)

µ

)Θ

(−T (ζ)µp

)−2 1− p2

(1− pζ)2dζ(3.7)

µ is the conformal modulus (2.2) of Ω (and P),

T (ζ) =ζ − p

1− pζand p =

c

1 + rµ.

4. Convergence of S(z)

In this section we prove that

S (z) = limN→∞

SN (z)

uniformly on closed subsets of (clΩ)\ PV when the regions satisfy the separationcondition ∆ < (m− 1)−1/4.

4.1. Convergence for m=2. To illustrate our technique, we will first constructthe singularity function by a reflection argument for the case where m = 2. Herei = 1, 2 and ν = 121 · · · or ν = 212 · · · . For the map w = f(z) from the exterior oftwo disks to the exterior of two polygons the singularity function is

S(z) =∑

ν

(K1∑

k=1

βk,1

z − zk,ν1− 2

z − sν1

)+

∑ν

(K2∑

k=1

βk,2

z − zk,ν2− 2

z − sν2

).

Our next task is to establish convergence of this expression.

Theorem 2. For the exterior 2-disk case SN (z) converges to S(z) uniformly onclosed subsets of (clΩ)\ PV by the following estimate

|S(z)− SN (z)| = O(µN12).

Proof. Using∑Ki

k=1 βk,i = 2, we can write the partial sums SN (z) of the series forS(z) in a form suitable for establishing convergence of

S(z) =∞∑

j=0

Aj(z)


where

Aj(z) =∑

ν∈σj(1)

(K1∑

k=1

βk,1

z − zk,ν1− 2

z − sν1

)+

∑

ν∈σj(2)

(K2∑

k=1

βk,2

z − zk,ν2− 2

z − sν2

)

=∑

ν∈σj(1)

K1∑

k=1

βk,1(zk,ν1 − sν1)(z − zk,ν1)(z − sν1)

+∑

ν∈σj(2)

K2∑

k=1

βk,2(zk,ν2 − sν2)(z − zk,ν2)(z − sν2)

.

Letting G be a closed subset of (clΩ)\ PV , z ∈ G and applying Lemma 2 givesthat

|Aj(z)| ≤ C(|zk,ν1 − sν1|+ |zk,ν2 − sν2|) ≤ Cµj12(r1 + r2).

This estimate establishes the convergence S(z) = limN→∞ SN (z) as claimed. ¤

4.2. Convergence for General m. We let H = cl(Ω)\zk,i : k = 1, . . . ,m, i = 1, . . . , Ki.For j = 0, 1, 2, . . . , we write

Aj (z) =m∑

i=1

∑

ν∈σj(i)

(Ki∑

k=1

βk,i

z − zk,νi− 2

z − sνi

)=

m∑

i=1

∑

ν∈σj(i)

Ki∑

k=1


and hence, in brief notation,

SN (z) =N∑

j=0

Aj(z), S(z) = limN→∞

SN (z).

If G is a closed subset of H, and

δ = δG = infz∈G

|z − zk,ν |, |z − sν | : k = 1, . . . , m, ν ∈ σ= inf

z∈G|z − zk,i| : k = 1, . . . ,m, i = 1, . . . , Ki ,

then, clearly δ > 0 and the second expression for δ holds since the zk,ν ’s and thesν ’s lie inside the circles.

We have the following

Theorem 3. For connectivity m ≥ 2, SN (z) converges to S(z) uniformly on closedsets G ⊂ H by the following estimate

|S(z)− SN (z)| = O((µ2√

m− 1)N+1).

for regions satisfying the separation condition

∆ <1

(m− 1)1/4.

Proof. Note that the number of terms in the Aj(z) sum is O((m − 1)j). This ex-ponential increase in the number of terms is the principal difficulty in establishingconvergence. Recall that rνi is the radius of circle Cνi. We bound Aj(z) for z ∈ H,by using the facts −1 < βk,i ≤ 1, |zk,νi − sνi| < 2rνi, Kmax := max

iKi, and the


Cauchy-Schwarz inequality, as follows:

|Aj(z)| = |∑

ν∈σj(i)

m∑

i=1

Ki∑

k=1


|

≤∑

ν∈σj(i)

m∑

i=1

Ki∑

k=1

|βk,i||zk,νi − sνi||z − zk,νi||z − sνi|

≤ 2δ2

∑

ν∈σj(i)

m∑

i=1

Ki∑

k=1

rνi

≤ 2Kmax

δ2

∑

ν∈σj(i)

m∑

i=1

rνi

≤ 2Kmax

δ2

∑

ν∈σj(i)

m∑

i=1

r2νi

1/2 ∑

ν∈σj(i)

m∑

i=1

1

1/2

=2Kmax

δ2

∑

ν∈σj(i)

m∑

i=1

r2νi

1/2

√m(m− 1)j/2

≤ 2Kmax

δ2∆2j

(m∑

i=1

r2i

)1/2√m(m− 1)j/2

≤ C∆2j(m− 1)j/2

by Lemma 3 where δ = δG. Therefore the series converges if ∆2√

m− 1 < 1. ¤

Remark. For m = 2 by Proposition 2, we have µ12 < ∆2, so the results forTheorem 2 are sharper than those of Theorem 3. This suggests the convergenceresults can be improved for the general case. Our current convergence estimatesbased on ∆ indicate that convergence is fast if the circles are well-separated andslow if some of them are very close to each other.

5. S(z) Satisfies the Boundary Condition

Here we prove that S (z) satisifies the boundary condition

Re (z − sj)S (z)z∈Cj= −1

as claimed in the proof of the main theorem. We will use the formulas

(5.1) Re

w

w − 1

= 1/2 for |w| = 1

and

(5.2) Re

w

w − 1+

w∗

w∗ − 1

= 1

where w and w∗ = 1/w are symmetric points with respect to the unit circle.


5.1. The Boundary Condition for m=2.

Theorem 4. For the unbounded, doubly connected case

Re (z − ci)SN (z) = −1 + O(µN12)

for z ∈ Ci, i.e., |z − ci| = ri.

Proof. To illustrate our proof, we write out SN (z) for N = 2:

S2(z) =K1∑

k=1

βk,1

z − zk,1− 2

z − s1+

K2∑

k=1

βk,2

z − zk,2− 2

z − s2+

K1∑

k=1

βk,1

z − zk,21− 2

z − s21

+K2∑

k=1

βk,2

z − zk,12− 2

z − s12+

K1∑

k=1

βk,1

z − zk,121− 2

z − s121+

K2∑

k=1

βk,2

z − zk,212− 2

z − s212

Next we rearrange S2(z) in a form convenient for the reflection calculation on thecircle |z − s1| = r1,

S2(z) =K1∑

k=1

βk,1

z − zk,1− 2

z − s1+

K2∑

k=1

βk,2

z − zk,2− 2

z − s2+

K2∑

k=1

βk,2

z − zk,12− 2

z − s12

+K1∑

k=1

βk,1

z − zk,21− 2

z − s21+

K1∑

k=1

βk,1

z − zk,121− 2

z − s121+

K2∑

k=1

βk,2

z − zk,212− 2

z − s212

=K1∑

k=1

βk,1

z − zk,1− 2

z − s1+

K2∑

k=1

βk,2

z − zk,2+

K2∑

k=1

βk,2

z − zk,12− 2

z − s2− 2

z − s12

+K1∑

k=1

βk,1

z − zk,21+

K1∑

k=1

βk,1

z − zk,121− 2

z − s21− 2

z − s121+

K2∑

k=1

βk,2

z − zk,212− 2

z − s212,

and therefore,

(z − s1)S2(z)

=K1∑

k=1

βk,1(z − s1)z − zk,1

− 2 +K2∑

k=1

βk,2(z − s1)z − zk,2

+K2∑

k=1

βk,2(z − s1)z − zk,12

− 2(

z − s1

z − s2+

z − s1

z − s12

)

+K1∑

k=1

βk,1(z − s1)z − zk,21

+K1∑

k=1

βk,1(z − s1)z − zk,121

− 2(

z − s1

z − s21+

z − s1

z − s121

)

+ (z − s1)

(K2∑

k=1

βk,2

z − zk,212− 2

z − s212

)

=K1∑

k=1

βk,1(z − s1)/(zk,1 − s1)(z − s1)/(zk,1 − s1)− 1

− 2


+K2∑

k=1

βk,2

((z − s1)/(zk,2 − s1)

(z − s1)/(zk,2 − s1)− 1+

(z − s1)/(zk,12 − s1)(z − s1)/(zk,12 − s1)− 1

)

− 2(

(z − s1)/(s2 − s1)(z − s1)/(s2 − s1)− 1

+(z − s1)/(s12 − s1)

(z − s1)/(s12 − s1)− 1

)

+K1∑

k=1

βk,1

((z − s1)/(zk,21 − s1)

(z − s1)/(zk,21 − s1)− 1+

(z − s1)/(zk,121 − s1)(z − s1)/(zk,121 − s1)− 1

)

−2(

(z − s1)/(s21 − s1)(z − s1)/(s21 − s1)− 1

+(z − s1)/(s121 − s1)

(z − s1)/(s121 − s1)− 1

)+(z−s1)

K2∑

k=1

βk,2(zk,212 − s212)(z − zk,212)(z − s212)

.

In the first term of the expression above, |(z−s1)/(zk,1−s1)| = 1 for |z−s1| = r1.Also, we have paired each zk,ν and sν with their reflections in C1. Thus, for instance,with |z − s1| = r1 we have zk,12 − s1 = r2

1/(zk,2 − s1), and so (z − s1)/(zk,2 − s1)is the reflection of (z − s1)/(zk,12 − s1) in the unit circle. Noting also that, withδ < |z − zk,212|, |z − s212|, we have for the last term that |zk,212 − s212| ≤ 2r212 =O(µ2

12), since r212 ≤ µ12r21 ≤ µ212r2 by Lemma 2. The remaining terms are

treated similarly. Therefore, using (5.1) and (5.2), the expression above gives for|z − s1| = r1

Re (z − s1)S2(z) = 1− 2 + 2− 2 + 2− 2 + O(µ212) = −1 + O(µ2

12).

For the reflection calculation on the circle |z − s2| = r2, we use the form

S2(z) =K2∑

k=1

βk,2

z − zk,2− 2

z − s2+ · · ·+

K1∑

k=1

βk,1

z − zk,121− 2

z − s121

It is clear that the general case for SN (z) follows by arranging terms as above. ¤5.2. The Boundary Condition for General m. The following Theorem shows,for general m, that S(z) satisfies the boundary condition for f ′′(z)/f ′(z).

Theorem 5. If ∆ < (m− 1)−1/4 then for z ∈ Ci, z 6= zk,i

Re (z − si)SN (z) = −1 + O((∆2√

m− 1)N )

andRe (z − si)S(z) = −1

Proof. First, we illustrate the idea of the proof for connectivity m = 3. The mapw = f(z) from the exterior of three disks to the exterior of three polygons has thesingularity function

S(z) =∑

ν

(K1∑

k=1

βk,1

z − zk,ν1− 2

z − sν1

)+

∑ν

(K2∑

k=1

βk,2

z − zk,ν2− 2

z − sν2

)

+∑

ν

(K3∑

k=1

βk,3

z − zk,ν3− 2

z − sν3

),

where the sνi’s are the iterated reflections of the centers ci of the m boundarycircles. We now write out S1(z) to illustrate our derivation:

S1(z) =K1∑

k=1

βk,1

z − zk,1− 2

z − s1+

K2∑

k=1

βk,2

z − zk,2− 2

z − s2+

K3∑

k=1

βk,3

z − zk,3− 2

z − s3


+K2∑

k=1

βk,2

z − zk,12− 2

z − s12+

K3∑

k=1

βk,3

z − zk,13− 2

z − s13+

K1∑

k=1

βk,1

z − zk,21− 2

z − s21

+K3∑

k=1

βk,3

z − zk,23− 2

z − s23+

K1∑

k=1

βk,1

z − zk,31− 2

z − s31+

K2∑

k=1

βk,2

z − zk,32− 2

z − s32.

Arranging the expressions in a manner similar to the two disk case, we have

(z − s1)S1(z) =K1∑

k=1

βk,1(z − s1)/(zk,1 − s1)(z − s1)/(zk,1 − s1)− 1

− 2

+K2∑

k=1

βk,2

((z − s1)/(zk,2 − s1)

(z − s1)/(zk,2 − s1)− 1+

(z − s1)/(zk,12 − s1)(z − s1)/(zk,12 − s1)− 1

)

−2(

(z − s1)/(s2 − s1)(z − s1)/(s2 − s1)− 1

+(z − s1)/(s12 − s1)

(z − s1)/(s12 − s1)− 1

)

+K3∑

k=1

βk,3

((z − s1)/(zk,3 − s1)

(z − s1)/(zk,3 − s1)− 1+

(z − s1)/(zk,13 − s1)(z − s1)/(zk,13 − s1)− 1

)

−2(

(z − s1)/(s3 − s1)(z − s1)/(s3 − s1)− 1

+(z − s1)/(s13 − s1)

(z − s1)/(s13 − s1)− 1

)

+(z − s1)

(K1∑

k=1

βk,1(zk,21 − s21)(z − zk,21)(z − s21)

+K3∑

k=1

βk,3(zk,23 − s23)(z − zk,23)(z − s23)

)

+(z − s1)

(K1∑

k=1

βk,1(zk,31 − s31)(z − zk,31)(z − s31)

+K2∑

k=1

βk,2(zk,32 − s32)(z − zk,32)(z − s32)

)

=K1∑

k=1

βk,1(z − s1)/(zk,1 − s1)(z − s1)/(zk,1 − s1)− 1

− 2

+K2∑

k=1

βk,2

((z − s1)/(zk,2 − s1)

(z − s1)/(zk,2 − s1)− 1+

(z − s1)/(ρ1(zk,2)− s1)(z − s1)/(ρ1(zk,2)− s1)− 1

)

−2(

(z − s1)/(s2 − s1)(z − s1)/(s2 − s1)− 1

+(z − s1)/(ρ1(s2)− s1)

(z − s1)/(ρ1(s2)− s1)− 1

)

+K3∑

k=1

βk,3

((z − s1)/(zk,3 − s1)

(z − s1)/(zk,3 − s1)− 1+

(z − s1)/(ρ1(zk,3)− s1)(z − s1)/(ρ1(zk,3)− s1)− 1

)

−2(

(z − s1)/(s3 − s1)(z − s1)/(s3 − s1)− 1

+(z − s1)/(ρ1(s3)− s1)

(z − s1)/(ρ1(s3)− s1)− 1

)

+(z − s1)

(K1∑

k=1

βk,1(zk,21 − s21)(z − zk,21)(z − s21)

+K3∑

k=1

βk,3(zk,23 − s23)(z − zk,23)(z − s23)

)

+(z − s1)

(K1∑

k=1

βk,1(zk,31 − s31)(z − zk,31)(z − s31)

+K2∑

k=1

βk,2(zk,32 − s32)(z − zk,32)(z − s32)

).

The truncation error will be given by the terms in the last two lines. The real partof the initial terms can be calculated with a reflection argument as in the doubly


connected case. For the first term, let w = (z − s1)/(zk,1 − s1). Note that |w| = 1for z ∈ C1. Then by (5.1) we have

Re

(z − s1)/(zk,1 − s1)(z − s1)/(zk,1 − s1)− 1

= Re

w

w − 1

=

12

For the second group of terms, we have, for instance, w = (z − s1)/(zk,2 − s1) andw∗ = (z − s1)/(ρ1(zk,2)− s1). Using (5.2), this gives

Re

(z − s1)/(zk,2 − s1)(z − s1)/(zk,2 − s1)− 1

+(z − s1)/(ρ1(zk,2)− s1)

(z − s1)/(ρ1(zk,2)− s1)− 1

= Re

w

w − 1+

w∗

w∗ − 1

= 1.

To bound the error in the final terms, note that |zk,ij − sij | ≤ 2rij . Then usingLemma 3 for |z − s1| = r1 the error bound is

2r1

δ2z

(r21 + r23 + r31 + r32) ≤ C√

r221 + r2

23 + r231 + r2

32

√4

≤ C√

r22 + r2

3

√4∆2 = O(

√3− 1(3− 1)1/2∆2),

and therefore,

Re (z − s1)S1(z) = 1− 2 + 2(2− 2) + O(√

3− 1(3− 1)1/2∆2)

= −1 + O(√

m− 1(∆2N (m− 1)N/2)

when m = 3 and N = 1. The general case is similar using Proposition 1 to groupterms related by reflection ρp through Cp with z ∈ Cp as follows:

(z − sp)SN (z) =Kp∑

k=1

βk,p(z − sp)/(zk,p − sp)(z − sp)/(zk,p − sp)− 1

− 2

+N−1∑

j=0

m∑

i=1

∑

ν∈σj(i),νi,ν1 6=p

Ki∑

k=1

βk,i

((z − sp)/(zk,νi − sp)

(z − sp)/(zk,νi − sp)− 1+

(z − sp)/(ρp(zk,νi)− sp)(z − sp)/(ρp(zk,νi)− sp)− 1

)

−2N−1∑

j=0

m∑

i=1

∑

ν∈σj(i)νi,ν1 6=p

((z − sp)/(sνi − sp)

(z − sp)/(sνi − sp)− 1+

(z − sp)/(ρp(sνi)− sp)(z − sp)/(ρp(sνi)− sp)− 1

)

+(z − sp)m∑

j=1,j 6=p

m∑

i=1

∑

jν∈σN (i)

(Ki∑

k=1

βk,i(zk,jνi − sjνi)(z − zk,jνi)(z − sjνi)

).

The first term and the terms grouped by reflection through Cp are handled asbefore. The final m − 1 terms, all lying inside circles Ci, i 6= p, approximate thetruncation error and are estimated by

∑ν∈σn+1

r2ν ≤ ∆4N

m∑

i=1

r2i .

This gives our final resultRe (z − sp)SN (z)


= 1− 2 + (m− 1)(2− 2) + (m− 1)2(2− 2) + · · ·+ (m− 1)N−1(2− 2)

+ O(√

m− 1(m− 1)N/2∆2N )

= −1 + O(√

m− 1(∆2N (m− 1)N/2).

¤

6. Examples

In this section we give some graphical examples of specific mappings computedwith our formula. These examples are done with a primitive computational proce-dure and are intended only to give a preliminary indication of what can be obtainedby computing with our mapping formula. In particular, our computatonal workhere does not yield the conformal invariants of a given polygonal region. Thiswill be an automatic byproduct of a complete numerical implementation that is inprogress. In this work the polygonal domain is specified and the side lengths ofthe polygons are used as constraints that force the numerical implementation toproduce the conformally equivalent circle domain along with the mapping.

For the present paper, we have developed an initial MATLAB code to demon-strate computationally the feasibility and correctness of our formula for some simpleexamples. One such example is given (for m = 3) of the mapping of the exteriorof three circles, Figure 3, to the exterior of two oblique slits and a rectangle (thesolid lines) in Figure 2. For this example we have put the parameters into theproduct formula and performed two levels of reflections, i.e., truncating the prod-ucts after N = 2. Since the computational circles are well-separated, the sum ofthe squares of the radii decrease very rapidly, as seen in Table 1 and illustrated inFigure 3. The trapezoidal rule is used for the numerical integration. This givesrough accuracy for the turning angles we have chosen. The map is evaluated on theboundary circles, 3 concentric circles and on Cartesian grid lines (dashed lines inFigure 2). Note that integrating around closed curves in the computational regionresults in (approximately) closed curves in the image plane, confirming the (near)single-valuedness of our truncated mapping formula. A quick ”reality check” ofthe accuracy of our approximation can be made by looking at a typical factor inthe infinite product

ζ − zk,νi

ζ − sνi= 1 +

sνi − zk,νi

ζ − sνi

and the extremely rapid shrinking of the circles in the first reflection as shown inFigure 3, and realizing that |sνi − zk,νi| is bounded by the diameter of the νi circle.This supports the belief that the factors for N > 2 must be near 1 to very highprecision, and also why the sides of the approximate image polygons appear (to theeye) to be straight rather than somewhat ”wavy”.

The parameters for the example in Figure 2 are

c1 = 1 + 2i, r1 = 0.5, β1,1 = β2,1 = 1, z 1,1 = c1 + r1ei3π/4, z2,1 = c1 + r1e

i7π/4,

c2 = 3 + i, r2 = 0.4, β1,2 = β2,2 = β3,2 = β4,2 = 0.5,

z1,2 = c2 + r2ei3π/8, z2,2 = c2 + r2e

i5π/8, z3,2 = c2 + r2ei11π/8, z4,2 = c2 + r2e

i13π/8,

c3 = 3 + 3i, r3 = 0.2, β1,3 = β2,3 = 1, z1,3 = c3 + r3eiπ/8, z2,3 = c3 + r3e

i9π/8


−1 0 1 2 3 4 5−1

0

1

2

3

4

5

Fig.2. Triply connected S-C image of Fig.3 domain

−1 0 1 2 3 4 5−1

0

1

2

3

4

5

Fig.3. Reflected circles in the exterior of the preimageof Fig.3


Figure 3 illustrates the extremely rapid rate of decrease in the size of the circlesdue to the (visually) large separation of the three original boundary circles. Thisrapid change is confirmed by the numerical results in the following table showingthat the decrease in the areas of the reflected circles is much faster than predictedby our theoretical estimate with ∆.

N ∆4N∑3

i=1 r2i

∑|ν|=N+1 r2

ν

0 .45 .451 .12 · 10−1 .91 · 10−3

2 .31 · 10−3 .19 · 10−5

Table 1 Rapid decrease of areas of reflected circles

Remark. Since our main theorem shows that f ′′/f ′ = S, we have that S (z) =[1/z3

]at ∞. As a consequence, the coefficient of 1/z2 in the expansion at ∞ of

S (z) must be zero. Thus, from the series expansion in Definition 3, one has thenecessary side condition on the prevertex and turning parameters

m∑

i=1

Ki∑

k=1

∞∑j=0

ν∈σj(i)

βk,i (zk,νi − sνi) = limN→∞

m∑

i=1

Ki∑

k=1

N∑j=0

ν∈σj(i)

βk,i (zk,νi − sνi) = 0.

This side condition will be necessary to complete the system of equations for thenumerical solution of the parameter problem. The simpler, well-known side condi-tion

∑K1 βkzk = 0 appears in the S-C mapping of the exterior of the unit disk to

the exterior of a single polygon for the same reason. In our numerical example theside condition has not been imposed, but is satisfied with an error of about 0.01.This is roughly the level at which plots of the images of the circles fail to closeindicating a ”small lack of single-valuedness”.

Remarks. Conformal mappings between general multiply connected unboundedregions can be obtained by combining the methods of this paper with those of [4].We mentioned earlier that parallel slit domains are a natural type of polygonaldomain to use in conjunction with our results. For example, if one has a parallelslit domain (in the w - plane) with slits parallel to the real axis then a uniformpotential flow parallel to the real axis can be transplanted back to a conformallyequivalent circle domain with our S-C mapping w = f (z). The streamlines ofthe corresponding flow around the preimage circles in the z − plane are the curvesIm f (z) = constant. Combining this with the numerical methods of [4] one thenhas the streamlines of uniform flow around a finite set of rather general obstaclesin the plane. This is just one primitive example of many applications of the S-Cconformal map of multiply connected domains. Other areas of applications includeproblems in electrostatic fields, c.f., [1], and the complex analysis problem of (nu-merically) determining the conformal invariants of a multiply connected polygonaldomain (since one can read the invariants of a circular domain from the center andradius values).

Acknowledgement: The work of the first two authors was partially supportedby NSF EPSCoR grant No. 9874732.


References

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[13] Komatu, Y., Conformal mapping of polygonal domains, J. Math. Soc. Japan, 2, 1–2, (1950)133–147.

[14] Komatu, Y., Topics from the classical theory of conformal mapping, Complex Analysis,Banach Center Publications, Warsaw, 11 (1983) 165–183.

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Department of Mathematics and Statistics, Wichita State University, Wichita, KS67620-0033

E-mail address: [email protected]

Department of Mathematics and Statistics, Wichita State University, Wichita, KS67620-0033


Department of Mathematics, CB 3250, University of North Carolina, Chapel Hill,NC 27599 3250


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SCHWARZ-CHRISTOFFEL MAPPING OF MULTIPLY CONNECTED … · SCHWARZ-CHRISTOFFEL MAPPING OF MULTIPLY CONNECTED DOMAINS T.K. DELILLO, A.R. ELCRAT, AND J.A. PFALTZGRAFF Abstract. A Schwarz-Christoﬀel