Radial and circular slit maps of unbounded multiply connected circle domains

T.K DeLillo, T.A Driscoll, A.R Elcrat, J.A Pfaltzgraff

Abstract

Infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain, or to domains with both radial and circular slit boundary components are derived and implemented numerically and graphically. The formulae are generated by analytic continuation with the reflection principle. Convergence of the infinite products is proved for domains with sufficiently well-separated boundary components. Some recent progress in the numerical implementation of infinite product mapping formulae is presented.

Keywords:

1. Introduction

We develop formulae for conformal maps f:ΩEmbedded Image from unbounded multiply connected circle domains to canonical unbounded slit domains. A circle domain Ω is a domain of connectivity m in the extended complex plane Embedded Image* that contains the point at infinity, and whose m boundary components are circles, Cj, j=1, …, m. A radial or circular slit domain Embedded Image is a domain in Embedded Image*, ∞∈Embedded Image with boundary consisting of m closed segments lying on rays from the origin or m closed circular arcs lying on circles centred at the origin, respectively. Circle, radial slit and circular slit domains are three of the classes of canonical domains in Koebe's classification of multiply connected domains. There are various functional relationships between pairs of slit mappings from different canonical classes (Nehari 1952, Chap. 7), but the circle domains are not related to other canonical classes in such an elementary fashion. Thus, it is of great interest to be able to find explicit formulae for mapping the circle domains onto the radial and the circular slit domains.

We derive our mapping formulae by using the reflection principle to extend the mapping f beyond Ω to a globally defined function. Then, complete knowledge of the zeros and poles of the globally defined function enables one to express f as an infinite product. This is a more direct determination of f than the analogous process in finding the Schwarz–Christoffel mapping formula for general polygonal domains where the reflection process leads to a determination of the derivative of the mapping function (DeLillo et al. 2004). The remaining problem of trying to determine f from an integral of many non-elementary infinite products with unknown accessory parameters is still not solved in a satisfactory general manner (DeLillo et al. 2006). Thus, it is quite interesting to see in the present work that for radial and circular slit mappings, the problem of integrating the derivative of the desired mapping function is eliminated. Schwarz–Christoffel formulae for bounded polygonal domains were derived by Crowdy (2005) and for unbounded polygonal domains by Crowdy (2007) using Schottky–Klein prime functions (see also DeLillo 2006).

The techniques in this paper differ from those employed by Crowdy & Marshall (2006). They follow the approach of Schiffer (1950) giving the radial and circular slit maps in terms of Green's functions by using Schottky–Klein prime functions of the circular domains. By contrast, we use directly the properties that such mappings must have and basic reflection arguments to derive our formulae without recourse to Green's functions. There is a close connection between reflections in circles and the Schottky group, which DeLillo (2006) uses to derive relations between the Schottky–Klein prime functions and (bounded and circular) slit maps in the context of Schwarz–Christoffel mapping.

Green's functions for multiply connected domains are useful in many applications. Crowdy & Marshall (2007) have given Green's functions for circle domains in terms of Schottky–Klein prime functions. Our methods can also be used to give explicit formulae for Green's functions for circle domains. As given by Nehari (1952), the radial and circular slit maps are central components in the construction of Green's functions of a given domain. In a similar fashion, the combined circular/radial slit map given below can be used for the construction of the Robin function, the Green's function for the mixed boundary-value problem. With our approach, the maps to (bounded) circular and radial slit discs and annuli are also needed. However, these maps are closely related to the maps given below. The circular slit disc map is already given by DeLillo (2006). Details will appear in a forthcoming article.

The paper is organized as follows. In §2, we give some preliminaries on reflection in circles. In §3, we give simple derivations of infinite product formulae for maps w=f(z) from circle domains to canonical radial, circular and combined radial/circular slit domains. We prove convergence of the formulae for domains satisfying a separation condition. The evaluation of the (truncated) product formula based on successive reflections is very inefficient. Therefore, in §4, we give an efficient method for calculating the slit maps based on solving a least-squares problem, as given by Trefethen (2005; see also Finn et al. 2003).

2. Preliminaries

We shall introduce notation, recall basic facts about reflections in circles and relate useful information. As already mentioned, w=f(z) is a conformal map f:ΩEmbedded Image from a circle domain to a slit domain of connectivity m, with cj and rj denoting the centres and radii, respectively, of the mutually exterior boundary circles Cj.

The reflection of z through a circle C with centre c and radius r is given byEmbedded Image(2.1)i.e. z and z* are symmetric points with respect to the circle C. If C=Cτ, where τ is an index of a circle, we will denote Embedded Image by ρτ. In our work, the function f that maps Ω on to a canonical slit domain is known to exist by classical uniformization results of Koebe, but no formula is provided by the non-constructive proofs. We develop our mapping formulae by using analytic continuation with the reflection principle to extend f onto Embedded Image Briefly, we begin by continuing f beyond Ω with reflection across its m boundary circles, Cj. We repeat this process, reflecting across the m(m−1) reflections of the original m boundary circles thereby producing m(m−1)2 additional reflected circles. Unlimited iteration of this process produces a global extension, Embedded Image, of f. The values of the extension are obtained at each reflection by reflecting the values of the already defined function across the appropriate boundary slit. Then it can be seen that the global Embedded Image is characterized by its zeros and poles and that the formula for f in terms of infinite products of these zeros and poles follows. It is useful to note that the number of new regions and new boundary components created by the reflections at a given level is m−1 times that at the preceding level.

We need the notation of multi-indices to denote reflected domains and boundary circles. When Ω is reflected through the boundary circle C1, it produces a domain Ω1ρ1(Ω) inside C1 that is bounded by C1 and the reflections of C2, …, Cm, which we denote C12,C13, …, C1m, respectively, i.e. C1j=ρ1(Cj). Similarly, Ωk=ρk(Ω) and Ckj=ρk(Cj), jk. Figure 1 illustrates reflections of circles and centres for m=3 for one level (N=1, below). In general, for a multi-index, ν=ν1ν2 … νn and a quantity Q (point, circle or region),Embedded Image(2.2)

Figure 1

N=1 levels of reflected circles and centres for unbounded case.

The set of multi-indices of length n will be denotedEmbedded Image(2.3)and σ0 (in which case νi=i for ν∈σ0 below). AlsoEmbedded Image(2.4)denotes the set of sequences in σn whose last factor never equals i.

From propostion 1 of DeLillo et al. (2004), we also have that if ν∈σn, n>1, then Embedded Image is a circular domain with outer boundary Cν and m−1 interior boundary circles. Clearly, σn contains m(m−1)n−1 elements, which is consistent with our earlier comment that the number of circular domains Ων at a particular level of reflections, say ν∈σn, is m−1 times the number of domains Embedded Image, Embedded Image, at the preceding level.

In order to state our convergence results, we need the following definition and lemma. The separation parameter of the region isEmbedded Image(2.5)for the assembly of m mutually exterior circles that form the boundary of Ω (cf. Henrici 1986, p. 501). Let Embedded Image denote the circle with centre cj and radius rj/Δ. Then geometrically, 1/Δ is the smallest magnification of the m radii such that at least two Embedded Image's just touch. We will use the following inequality from Henrici (1986; p. 505):

Embedded Image(2.6)

3. Maps to the canonical radial and circular slit domains

In this section, we use simple reflection arguments to derive the mappings of unbounded circle domains to the canonical radial and circular slit domains as well as mapping to a domain with both radial and circular slit boundary components (figures 2 and 3).

Figure 2

Maps from unbounded circle domains to radial and circular slit domains with a Cartesian grid. The products are truncated to N=4 levels of reflection. With N=3 the slits appear to be slightly open as in figure 3.

Figure 3

Map to circular/radial slit domain with N=3 levels of reflection.

(a) Radial slit maps

We will give a detailed derivation and proof of the formula for a conformal mapping f of an m-connected circle domain Ω onto a radial slit domain Embedded Image with f(a)=0 and f(∞)=∞. We begin with a brief outline of the procedure. First, we extend f to a globally defined (many valued) Embedded Image on Embedded Image by repeated use of the reflection principle. When f or Embedded Image is reflected across a circle C, the corresponding extension of w=f(z) across the radial slit γ=f(C) at angle θ is given by reflecting w across γ to Embedded Image. The latter reflection leaves the w-values zero and infinity fixed and hence the zero set of the extended f will be the point a and all of its reflections, and similarly for ∞ and the other the poles of Embedded Image Thus it seems plausible to think that the mapping f:ΩEmbedded Image can be expressed by a formulaEmbedded Image(3.1)where the ak's are the reflections of a across the boundary circles Ck and ck's are the reflections of ∞ across the boundary circles of Ω. Further details including convergence will be proven when the m circles with centres ck satisfy our separation condition in theorem 3.2. Note that f(a)=0 and Embedded Image near ∞.

It is important to note that, although the global Embedded Image is many valued, the differential expression Embedded Image is single valued. Indeed, any two values, Embedded Image and Embedded Image of Embedded Image at a point Embedded Image are related by the composition of an even number of reflections in lines and hence Embedded Image for some AEmbedded Image. The differential expression, Embedded Image, is invariant under maps Embedded Image i.e. Embedded Image. Thus, if one begins with Embedded Image in Ω, the reflection process yielding the many-valued Embedded Image also defines a global analytic function, Embedded Image, that is defined and single valued on Embedded Image. We shall refer to Embedded Image as the singularity function. Our proof will depend on showing thatEmbedded Image(3.2)or in convergent form,Embedded Image(3.3)Our task is to show that, indeed, Embedded Image Note thatEmbedded Image(3.4)We will show that the sums truncated to N levels of reflection,Embedded Image(3.5)converge uniformly to S(z) for Embedded Image as N→∞, provided the circles satisfy our separation condition, that S(z) satisfies an appropriate boundary condition, and that Embedded Image, our main theorem.

Our boundary conditions are given by

Embedded Image

For Embedded Image we have Embedded Image and since f(z) maps to radial slits, we have arg f(z)=const. Therefore,Embedded Image(3.6) ▪

We now state our main theorem for radial slit maps.

Let Embedded Image be an unbounded m-connected radial slit region, Embedded Image, and Ω a conformally equivalent circular domain, Embedded Image. Furthermore, suppose Ω satisfies the separation property Embedded Image for m>1. Then Ω is mapped conformally onto Embedded Image by f with f(a)=0 and f()=∞ if and only ifEmbedded Image(3.7)for some constant C. ▪

The proof, that a map f to a radial slit domain must necessarily be of the form (3.7), follows very closely the proof of theorem 1 by DeLillo et al. (2004). The central idea is to prove that Embedded Image by means of the argument principle. We shall use the following two results whose proofs are given after the present proof in order to keep the essence of the present proof from being obscured by calculation details.

  1. Convergence: Embedded Image uniformly on Embedded Image.

  2. Boundary conditions: Embedded Image.

For Embedded Image, we define the functionsEmbedded Image(3.8)

We first note thatEmbedded Image(3.9)is defined and analytic in Ω since its periods are zero. Indeed Embedded Image, where Embedded Image is a circle concentric with the boundary circle Cτ with radius slightly larger than that of Cτ since the residues add out in pairs. Furthermore, H(z) is analytic in Ω sinceEmbedded Image(3.10)with Embedded Image uniformly on closed subsets of Embedded Image.

The next step is to develop a formula for the antiderivative (up to an additive constant)Embedded Image(3.11)where each logarithm is the branch that vanishes at z=∞, i.e. log 1=0. From the preceding formula, one hasEmbedded Image(3.12)and hence the product formula for P(z),Embedded Image(3.13)

Our theorem, Embedded Image is equivalent to showing that the quotientEmbedded Image(3.14)To accomplish this, we will apply the argument principle to Q(z). First, observe that Embedded Image i.e. Embedded Image, andEmbedded Image(3.15)Then, for Embedded Image, the boundary conditions of lemma 3.1 and theorem 3.4 on Embedded Image and S, respectively, giveEmbedded Image(3.16)By our construction of S(z), Embedded Image is continuous on all Cj. 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 the origin. It is clear by the local behaviour of f and formula (3.13) that Q=f/P is continuous on each Cj and not equal to 0 or ∞ there, since Embedded Image. Thus, for any Embedded Image, Embedded Image, the winding number of Embedded Image around w0, Embedded Image for all j. Let CR be a large circle of radius R centred at the origin and containing w0 and all the Cj's in its interior, and write Embedded Image with the curves oriented so that the region interior to CR and exterior to the Cj's is on the left. Since Q has no poles in the region, by the argument principle (for bounded regions), the number of times Q(z) assumes the value w0 isEmbedded Image(3.17)We now will show that Embedded Image. First,Embedded Image(3.18)Recall that Embedded Image for z near ∞, and that, Embedded Image is a finite constant. It suffices to assume Embedded Image. Then Embedded Image for R sufficiently large and there are constants Embedded Image such thatEmbedded Image(3.19)as Embedded Image. Therefore, Embedded Image and Embedded Image for Embedded Image and Embedded Image Thus, Q assumes values only on the radial segments Embedded Image (or Embedded Image) and hence, by the open mapping property of analytic functions, Q must be constant on Ω.

Finally, we show that a function w=f(z) of the form (3.7) always determines a conformal map to the conformally equivalent slit domain Embedded Image with f(a)=0 when Ω satisfies the separation property: by the basic existence theorem for maps of multiply connected domains, Ω is conformally equivalent, via a map g with g(a)=0, to some radial slit domain Embedded Image′. By the above argument, g(z) must have the form (3.7), and by uniqueness of the conformal maps we must have Embedded Image for some constant C. ▪

In the special case when m=2, there is no restrictive separation hypothesis; since then Embedded Image is equivalent to the fact that the two boundary components are disjoint.

  • (i) Convergence of S(z)

For Embedded Image we writeEmbedded Image(3.20)and hence, in brief notation,Embedded Image(3.21)LetEmbedded Image(3.22)Then, clearly δ>0 holds since the aν's and the Sν's lie inside the circles.

We have the following

For connectivity m≥2, SN(z) converges to S(z) uniformly on Embedded Image satisfying the following estimateEmbedded Image(3.23)for regions satisfying the separation conditionEmbedded Image(3.24)

Note that the number of terms in the Aj(z) sum is Embedded Image This exponential increase in the number of terms is the principal difficulty in establishing convergence. Recall that rνi is the radius of circle Cνi. We bound Aj(z) for Embedded Image by using the facts Embedded Image and the Cauchy–Schwarz inequality, as follows:Embedded Image(3.25)by lemma 2.2 where Embedded Image. Therefore, the series converges if Embedded Image ▪

  • (ii) S(z) satisfies the boundary condition

Here, we prove that S(z) satisfies the boundary conditionEmbedded Image(3.26)as claimed in the proof of the main theorem. We will use the formulaEmbedded Image(3.27)where w and Embedded Image are symmetric points with respect to the unit circle.

The following theorem shows, for general m, that S(z) satisfies the boundary condition for Embedded Image.

If Embedded Image, then for Embedded ImageEmbedded Image(3.28)andEmbedded Image(3.29)

The idea of the proof is, for Embedded Image, to use properties of the reflections (2.2) to group terms in SN(z) related by reflection ρp through Cp with Embedded Image as follows:Embedded Image(3.30)Then, multiplying by Embedded Image, we have in more detail,Embedded Image(3.31)We take the real part of the above expression and using, for instance, Embedded Image and noting that Embedded Image (3.27) givesEmbedded Image(3.32)Taking the real part of (3.31), we see that the first three lines sum to 0. The final m−1 terms, all lying inside circles Ci, Embedded Image, approximate the truncation error and are estimated byEmbedded Image(3.33)This gives our final resultEmbedded Image(3.34) ▪

(b) Circular slit maps

The derivation of the map, Embedded Image from an unbounded circle domain to the conformally equivalent unbounded circular slit domain is similar to that of the radial slit domain. This map is closely related to the Green's function for the Dirichlet boundary-value problem. Once again Embedded Image and Embedded Image with Embedded Image Again, ai is the reflection of a across circle Ci and Embedded Image, the centre of circle Ci, is the reflection of ∞ across Ci. In the w-plane, 0 and ∞ just reflect back and forth to each other. Therefore, when we extend f, we will have Embedded Image and Embedded Image. In this way, we see that all odd numbers of reflections Embedded Image of ai and all even numbers of reflections Embedded Image of ci will be simple zeros, Embedded Image. Likewise, all odd numbers of reflections Embedded Image of ci and all even numbers of reflections Embedded Image of ai will be simple poles, Embedded Image. The infinite product for Embedded Image therefore has the form,Embedded Image(3.35)(where reflections back to a or ∞ are excluded from the product) with Embedded Image, provided the m circles with centres ck satisfy our standard separation criterion.

Now note that, if a circular slit in the w-plane is at radius r1, then w reflects to Embedded Image. Reflection through another circular slit with radius r2 will then take w to Embedded Image and so on. Therefore, an even number successive reflection through circular slits will take Embedded Image to Embedded Image for some A real. As a result, the extended function Embedded Image is invariant under even numbers of reflections and hence is single valued. Here, our singularity function, in non-convergent form, will beEmbedded Image(3.36)or in convergent form,Embedded Image(3.37)Again, our task is to show that Embedded Image Note that, again,Embedded Image(3.38)

We will show that the sums truncated to N levels of reflection,Embedded Image(3.39)converge uniformly to Embedded Image for Embedded Image as Embedded Image, provided the circles satisfy our separation condition, that Embedded Image satisfies an appropriate boundary condition, and that Embedded Image, our main theorem.

Our boundary conditions are given by

Embedded Image

For Embedded Image we have Embedded Image and since Embedded Image maps to circular slits, we have Embedded Imageconst. Therefore,Embedded Image(3.40) ▪

We now state our main theorem for circular slit maps.

Let Embedded Image be an unbounded m-connected circular slit region, Embedded Image, and Ω a conformally equivalent circular domain, Embedded Image. Furthermore, suppose Ω satisfies the separation property Embedded Image for m>1. Then Ω is mapped conformally onto Embedded Image by f with Embedded Image and Embedded Image if and only ifEmbedded Image(3.41)for some constant C.

The proof follows the argument for the radial slit case, using the modified convergence theorems and the following boundary conditions for circular slits. ▪

  1. Convergence of S(z)

    For Embedded Image we writeEmbedded Image(3.42)and hence, in brief notation,Embedded Image(3.43)LetEmbedded Image(3.44)Then, clearly δ<0 holds since the aν's and the Sν's lie inside the circles.

    The convergence of Embedded Image to Embedded Image is identical to theorem 3.3 for the radial case. The details of the proof are nearly identical and we omit them.

  1. S(z) satisfies the boundary condition.

    Here, we prove that Embedded Image satisfies the boundary conditionEmbedded Image(3.45)as claimed in the proof of the main theorem. We will use the formulaEmbedded Image(3.46)where w and Embedded Image are symmetric points with respect to the unit circle.

The following theorem shows, for general m, that Embedded Image satisfies the boundary condition for Embedded Image.

If Embedded Image, then for Embedded ImageEmbedded Image(3.47)andEmbedded Image(3.48)

The idea of the proof is, for Embedded Image, to again use the properties of reflections (2.2) to group terms in Embedded Image related by reflection ρp through Cp with Embedded Image as follows:Embedded Image(3.49)where the plus sign is used if Embedded Image is even and a minus sign if Embedded Image is odd. Then, multiplying by Embedded Image, we have in more detail,Embedded Image(3.50)We take the imaginary part of the above expression and using, for instance, Embedded Image and noting that Embedded Image (3.46) givesEmbedded Image(3.51)Taking the imaginary part of (3.50), we see that the first three lines sum to 0. The final m−1 terms, all lying inside circles Ci, Embedded Image, approximate the truncation error and are estimated byEmbedded Image(3.52)This gives our final resultEmbedded Image(3.53) ▪

(c) Circular and radial slit map

Here we consider the map Embedded Image from the exterior of m discs to the exterior domain bounded by a mixture of radial and circular slits. This map is discussed by Koebe (1916). The mapping formula that we derive here appears to be new. It is not discussed, for instance, in such standard presentations as Nehari (1952) or Schiffer (1950).

Choosing a point Embedded Image, we let Embedded Image and Embedded Image with Embedded Image. Reflections through radial slits will keep 0 and ∞ fixed, whereas that through circular slits will swap 0 and ∞ as in the circular slit map above. Let Embedded Image denote a sequence of reflections with an even number of reflections through circular slits and Embedded Image denote a sequence with an odd number of reflections through circular slits. Then, Embedded Image and Embedded Image are simple zeros of f(z) and Embedded Image and Embedded Image are simple poles. Therefore, we haveEmbedded Image(3.54)(Note that the product over Embedded Image is already included in the reflections of a and ∞ and does not appear explicitly here.) Using arguments like those for the radial and circular slit mappings, one can prove that the separation and convergence theorems hold for the mixed radial and circular slit boundary components. We omit the details of the proof. Figure 3 is a graph of an m=4 case with two circular and two radial slits produced by evaluating a truncated version of (3.54).

Numerical experiments indicate that our convergence criterion for the infinite product formulae is probably not necessary for convergence. We have been unable to find a condition that is both necessary and sufficient.

4. Numerics using least squares

The characterization by means of reflections of the slit maps considered in this paper is natural and leads to straightforward derivations. On the other hand, as the number of circles and slits grows, the required number of reflections for a prescribed accuracy grows exponentially and computation times become impractically large. As an anecdotal example, in one case of the maps like that in figure 4, if m was increased from 3 to 4, the computation time on the third author's laptop increased from 3.5 to 3970 s. Therefore, it is essential to find fast algorithms to compute these maps. We describe such a procedure here.

Figure 4

Map to radial slit domain using least-squares method.

The idea is closely related to an algorithm given by Trefethen (2005) for finding the Green's function for the exterior of discs. We begin by expressing the desired map f asEmbedded Image(4.1)for a function g that is analytic in Ω (and its boundary, according to equations (3.1) and (3.35)). This form imposes the normalizations Embedded Image and Embedded Image

Box 1. MATLAB code for finding the parameters and computing the values of a radial slit map.

Embedded Image

Box 2. Driver for the code in box 1 for the map from an exterior domain bounded by five circles to a radial slit domain.

Embedded Image

. The remainder Embedded Image is then expanded in the formEmbedded Image(4.2)which allows singularities in each of the circles. In practice, we discretize the boundary of Embedded Image by placing N equally spaced points on each of the circles and express the double sum of (4.2) as a matrix–vector product Ax, where each column of the matrix A is the discretization of some Embedded Image and Embedded Image.

The unknown coefficients in x are determined by the fact that Embedded Image is constant on radial slits and Embedded Image is constant on circular slits. Indeed, the key fact is that we can impose these conditions linearly. To do this, we need to break both A and x into its real and imaginary parts. Letting Embedded Image and Embedded Image, we trivially getEmbedded Image(4.3)For concreteness, let us continue the discussion in terms of the radial slit case. The constant values of Embedded Image on each slit are not known in advance. Instead, we ask that pairwise differences of Embedded Image be zero around each circle. DefiningEmbedded Image(4.4)we arrive at the expressionEmbedded Image(4.5)which is an ordinary linear least-squares problem for the unknown coefficients. This problem can be solved very quickly even for fairly large discretizations.

shows a Matlab code based on these ideas. The expression (4.1) and (4.2) for the map is so simple that the function returns a callable object that evaluates to the computed function. illustrates how the code can be used to map points and create level curves for a domain bounded by five circles. This example is given in figure 4. Computing the map parameters (setting up and solving the least-squares system) took approximately 3 s on a 1.4 GHz Pentium-M laptop.

Footnotes

    • Received January 7, 2008.
    • Accepted February 21, 2008.

References

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