# Circular Slits Map of Bounded Multiply Connected Regions

Authors: Ali W. K. Sangawi; Ali H. M. Murid; M. M. S. Nasser
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We present a boundary integral equation method for
the numerical conformal mapping of bounded multiply connected region
onto a circular slit region. The method is based on some uniquely
solvable boundary integral equations with adjoint classical, adjoint
generalized, and modified Neumann kernels. These boundary integral
equations are constructed from a boundary relationship satisfied by
a function analytic on a multiply connected region. Some numerical
examples are presented to illustrate the efficiency of the presented
method.


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## Body

## 1. Introduction

In general, the exact conformal mapping functions are unknown except for some special regions. It is well known that every multiply connected regions can be mapped conformally onto the circle with concentric circular slits, the circular ring with concentric circular slits, the circular slit region, the radial slit region, and the parallel slit region as described in Nehari [1, page 334]. Several methods for numerical approximation for the conformal mapping of multiply connected regions have been proposed in [2–16]. Recently, reformulations of conformal mappings from bounded and unbounded multiply connected regions onto the five canonical slit regions as Riemann-Hilbert problems are discussed in Nasser [12, 13, 17]. An integral equation with the generalized Neumann kernel is then used to solve the RH problem as developed in [18]. The integral equation however involves singular integral which is calculated by Wittich’s method. Murid and Hu [11] formulated an integral equation method based on another form of generalized Neumann kernel for conformal mapping of bounded doubly connected regions onto a disk with circular slit but the kernel of the integral equation involved the unknown circular radii. Discretization of the integral equation yields a system of nonlinear equations which they solved using an optimization method. To overcome this nonlinear problem, Sangawi et al. [19] have developed linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits. In this paper, we describe an integral equation method for computing the conformal mapping function f of bounded multiply connected regions onto a circular slit region. This boundary integral equation is constructed from a boundary relationship that relates the mapping function f on a multiply connected region with f′, θ′(t), and |f|, where θ is the boundary correspondence function.The plan of the paper is as follows. Section2 presents some auxiliary materials. Derivations of two integral equations related to f′ and θ′(t) are given in Sections 3 and 4, respectively. Section 5 presents a method to calculate the modulus of f. In Section 6, we give some examples to illustrate our boundary integral equation method. Finally, Section 7 presents a short conclusion.


## 2. Notations and Auxiliary Material

LetΩ be a bounded multiply connected region of connectivity M+1. The boundary Γ consists of M+1 smooth Jordan curves Γj, j=0,1,…,M, such that Γĵ, ĵ=1,…,M, lies in the interior of Γ0, where the outer curve Γ0 has counterclockwise orientation and the inner curves Γĵ, ĵ=1,…,M, have clockwise orientation. The positive direction of the contour Γ=⋃j=0MΓj is usually that for which Ω is on the left as one traces the boundary (see Figure 1). The curve Γk is parametrized by 2π-periodic twice continuously differentiable complex function zk(t) with nonvanishing first derivative(2.1)zk′(t)=dzk(t)dt≠0,t∈Jk=[0,2π],k=0,1,…,M.Figure 1
Mapping of the bounded multiply connected regionΩ of connectivity M+1 onto a circular slit region.The total parameter domainJ is the disjoint union of M+1 intervals J0,…,JM. We define a parametrization z of the whole boundary Γ on J by(2.2)z(t)={z0(t),t∈J0=[0,2π],⋮zM(t),t∈JM=[0,2π].LetH* be the space of all real Hölder continuous 2π-periodic functions ω(t) of the parameter t on Jk for k=0,1,…,M, that is,(2.3)ω(t)={ω0(t),t∈J0,ω1(t),t∈J1,⋮ωM(t),t∈JM.
Let θ(t) (the boundary corresponding function) be given for t∈J by(2.4)θ(t)={θ0(t),t∈J0,⋮θM(t),t∈JM.
Let μ (a piecewise constant real function) be given for t∈J by(2.5)μ(t)=(μ0,μ1,…,μM)={μ0,t∈J0,⋮μM,t∈JM.
Let Â(t) be a complex continuously differentiable 2π-periodic function for all t∈J. The generalized Neumann kernel formed with Â is defined by(2.6)N̂(t,s)=1πIm⁡(Â(t)Â(s)z′(s)z(s)-z(t)).
The kernel N̂ is continuous with(2.7)N̂(t,t)=1π(12Im⁡z′′(t)z′(t)-Im⁡Â′(t)Â(t)).
Define also the kernel M̂ by(2.8)M̂(t,s)=1πRe(Â(t)Â(s)z′(s)z(s)-z(t)),
which has a cotangent singularity type (see [18] for more detail). The classical Neumann kernel is the generalized Neumann kernel formed with Â(t)=1, that is,(2.9)N(t,s)=1πIm⁡(z′(s)z(s)-z(t)).
The adjoint kernel N*(s,t) of the classical Neumann kernel is given by(2.10)N*(t,s)=N(s,t)=1πIm⁡(z′(t)z(t)-z(s)).
The adjoint function to the function Â is given by(2.11)Ã(t)=z′(t)Â(t)=z′(t).
The generalized Neumann kernel Ñ(s,t) formed with Ã is given by(2.12)Ñ(t,s)=1πIm⁡(Ã(t)Ã(s)z′(s)z(s)-z(t)).
If Â=1, then(2.13)Ñ(t,s)=-N*(t,s).
We define the Fredholm integral operators N,Ñ,N* by(2.14)Nυ(t)=∫JN(t,s)υ(s)ds,t∈J,(2.15)Ñυ(t)=∫JÑ(t,s)υ(s)ds,t∈J,(2.16)N*υ(t)=∫JN(s,t)υ(s)ds,t∈J.
Note that Ñ=-N*, if Â=1.It is known thatλ=1 is an eigenvalue of the kernel N with multiplicity 1 and λ=-1 is an eigenvalue of the kernel N with multiplicity M [18]. We define the piecewise constant functions(2.17)χ[j](ξ)={1,ξ∈Γj,j=0,1,2,…,M.0,otherwise.
Then, we have from [18](2.18)Null(I-N)=span⁡{1},Null(I-N)=span⁡{χ[1],χ[2],…,χ[M]}.Lastly, we define integral operatorsJ and Ĵ by(2.19)Jυ=∫J12π∑j=1Mχ[j](s)χ[j](t)υ(s)ds,Ĵυ=∫J12π∑j=0Mχ[j](s)χ[j](t)υ(s)ds,
which are required for uniqueness of solution in a later section.


## 3. Homogenous and Nonhomogenous Boundary Relationship

### 3.1. Nonhomogeneous Boundary Relationship for Conformal Mapping

Suppose thatc(z), Q(z), and H(z) are complex-valued functions defined on Γ such that c(z)≠0, H(z)≠0, Q(z)≠0, and H(z)¯/(T(z)Q(z))satisfies the Hölder condition on Γ. Then, the interior relationship is defined as follows.A complex-valued functionP(z) is said to satisfy the interior relationship if P(z) is analytic in Ω and satisfies the nonhomogeneous boundary relationship(3.1)P(z)=c(z)T(z)Q(z)¯G(z)¯P(z)¯+H(z)¯,z∈Γ,
where G(z) analytic in Ω, Hölder continuous on Γ, and G(z)≠0 on Γ. The boundary relationship (3.1) also has the following equivalent form:(3.2)G(z)=c(z)¯T(z)Q(z)P(z)2|P(z)|2+G(z)H(z)P(z)¯,z∈Γ.
Let the function LR(z̃) be defined in the region C∪{∞}∖Γ by(3.3)LR(z̃)=12πi∫Γc(z̃)¯H(w)c(w)¯(w-z̃)Q(w)T(w)dw,z̃∈Ω-,
where Ω- is the complement of Ω. The following theorem gives an integral equation for an analytic function satisfying the interior nonhomogeneous boundary relationship (3.1) or (3.2). This theorem generalizes the results of Murid and Razali [9] and can be proved by using the approach used in proving Theorem  3.1 in [20, page 45].Theorem 3.1.
LetU and V be any complex-valued functions that are defined on Γ. If the function P(z) satisfies the interior nonhomogeneous boundary relationship (3.1) or (3.2), then
(3.4)12[V(z)+U(z)T(z)Q(z)¯]P(z)+PV∫ΓK(z,w)P(w)|dw|+c(z)U(z)×[∑aj∈ΩResw=ajP(w)(w-z)G(w)]conj=-U(z)LR-(z)¯,z∈Γ,
where
(3.5)K(z,w)=12πi[c(z)U(z)c(w)(w¯-z¯)Q(w)¯-V(z)T(w)w-z],LR-(z)=-12H(z)Q(z)T(z)+PV12πi∫Γc(z)¯H(w)c(w)¯(w-z)Q(w)T(w)dw.
The symbol “conj” in the superscript denotes complex conjugate, while the minus sign in the superscript denotes limit from the exterior. The sum in (3.4) is over all those zeros a1,a2,…,aM of G that lie inside Ω. If G has no zeros in Ω, then the term containing the residue in (3.4) will not appear.Proof.
Suppose thatP(z) and G(z) are analytic functions in Ω and G has a finite number of zeros at a1,a2,…,aM in Ω. Then, by the calculus of residues, we have
(3.6)12πi∫ΓP(w)(w-z̃)G(w)dw=∑aj∈ΩResw=ajP(w)(w-z̃)G(w),z̃∈Ω-.
Since P and G satisfy the Hölder condition on Γ and G(z)≠0 on Γ, then P/G also satisfies the Hölder condition on Γ. Taking the limit Ω-∋z̃→z∈Γ and applying Sokhotski formula [5], we get
(3.7)-12P(z)G(z)+PV12πi∫ΓP(z)(w-z)G(w)dw=∑aj∈ΩResw=ajP(w)(w-z)G(w),z∈Γ.
By taking conjugate to both sides and using (3.1), we get
(3.8)-12P(z)c(z)Q(z)T(z)¯+12H(z)¯c(z)Q(z)T(z)¯-PV12πi∫ΓP(z)c(w)(w¯-z¯)Q(w)¯dw¯T(w)¯+PV12πi∫ΓH(z)¯dw¯c(w)(w¯-z¯)Q(z)T(z)¯=[∑aj∈ΩResw=ajP(w)(w-z)G(w)]conj,z∈Γ.
Multiplying both sides by -c(z) and the fact that dw=T(w)|dw|, after some arrangement, yield
(3.9)12P(z)Q(z)T(z)¯+PV12πi∫Γc(z)P(z)c(w)(w¯-z¯)Q(w)¯|dw|+c(z)[∑aj∈ΩResw=ajP(w)(w-z)G(w)]conj=-[-12H(z)Q(z)T(z)+PV12πi∫Γc(z)¯H(z)c(w)¯(w-z)Q(z)T(z)dw]conj,z∈Γ.
Applying Sokhotski formulas again to the expression inside the bracket of the right-hand side yields
(3.10)12P(z)Q(z)T(z)¯+PV12πi∫Γc(z)P(z)c(w)(w¯-z¯)Q(w)¯|dw|+c(z)[∑aj∈ΩResw=ajP(w)(w-z)G(w)]conj=-LR-(z)¯,z∈Γ.
Since P(z) is analytic in Ω, then by Cauchy integral formula, we have
(3.11)12πi∫ΓP(z)w-z̃dw=0,z∈Ω-.
Taking the limit ω-∋z̃→z∈Γ and applying Sokhotiski formulas, we get
(3.12)-12P(z)+PV12πi∫ΓT(w)P(z)w-z|dw|=0,z∈Γ.
Multiplying (3.12) by v(z) and subtracting it from (3.10) multiplied by u(z) yield (3.4).


### 3.2.  Homogeneous Boundary Relationship for Conformal Mapping

Letw=f(z) be the analytic function which maps Ω in the z-plane onto a canonical region of the circular slit region in the w-plane. Let 0 and a be a fixed point in Ω such that a≠0. Then, the mapping function is made uniquely determined by assuming that f(a)=0 and f(0)=∞ such that the residue of the function f at 0 is equal to 1 [1]. Hence, the function f can be written in the form(3.13)f(z)=(1z-1a)ezg(z),
where g is analytic in Ω [12, 13]. Note that the boundary value of f can be represented in the form(3.14)f(zp(t))=μpeiθp(t),Γp:z=zp(t),0≤t≤βp,p=0,1,…,M,
where θp is a boundary correspondence function of Γp and μp is the radius of the circular slit. The unit tangent to Γ at z(t) is denoted by T(z(t))=z′(t)/|z′(t)|. Thus, it can be shown that(3.15)f(z)=|f(z)|iT(z)|θp′(t)|θp′(t)f′(z)|f′(z)|,z∈Γ.


## 3.1. Nonhomogeneous Boundary Relationship for Conformal Mapping

Suppose thatc(z), Q(z), and H(z) are complex-valued functions defined on Γ such that c(z)≠0, H(z)≠0, Q(z)≠0, and H(z)¯/(T(z)Q(z))satisfies the Hölder condition on Γ. Then, the interior relationship is defined as follows.A complex-valued functionP(z) is said to satisfy the interior relationship if P(z) is analytic in Ω and satisfies the nonhomogeneous boundary relationship(3.1)P(z)=c(z)T(z)Q(z)¯G(z)¯P(z)¯+H(z)¯,z∈Γ,
where G(z) analytic in Ω, Hölder continuous on Γ, and G(z)≠0 on Γ. The boundary relationship (3.1) also has the following equivalent form:(3.2)G(z)=c(z)¯T(z)Q(z)P(z)2|P(z)|2+G(z)H(z)P(z)¯,z∈Γ.
Let the function LR(z̃) be defined in the region C∪{∞}∖Γ by(3.3)LR(z̃)=12πi∫Γc(z̃)¯H(w)c(w)¯(w-z̃)Q(w)T(w)dw,z̃∈Ω-,
where Ω- is the complement of Ω. The following theorem gives an integral equation for an analytic function satisfying the interior nonhomogeneous boundary relationship (3.1) or (3.2). This theorem generalizes the results of Murid and Razali [9] and can be proved by using the approach used in proving Theorem  3.1 in [20, page 45].Theorem 3.1.
LetU and V be any complex-valued functions that are defined on Γ. If the function P(z) satisfies the interior nonhomogeneous boundary relationship (3.1) or (3.2), then
(3.4)12[V(z)+U(z)T(z)Q(z)¯]P(z)+PV∫ΓK(z,w)P(w)|dw|+c(z)U(z)×[∑aj∈ΩResw=ajP(w)(w-z)G(w)]conj=-U(z)LR-(z)¯,z∈Γ,
where
(3.5)K(z,w)=12πi[c(z)U(z)c(w)(w¯-z¯)Q(w)¯-V(z)T(w)w-z],LR-(z)=-12H(z)Q(z)T(z)+PV12πi∫Γc(z)¯H(w)c(w)¯(w-z)Q(w)T(w)dw.
The symbol “conj” in the superscript denotes complex conjugate, while the minus sign in the superscript denotes limit from the exterior. The sum in (3.4) is over all those zeros a1,a2,…,aM of G that lie inside Ω. If G has no zeros in Ω, then the term containing the residue in (3.4) will not appear.Proof.
Suppose thatP(z) and G(z) are analytic functions in Ω and G has a finite number of zeros at a1,a2,…,aM in Ω. Then, by the calculus of residues, we have
(3.6)12πi∫ΓP(w)(w-z̃)G(w)dw=∑aj∈ΩResw=ajP(w)(w-z̃)G(w),z̃∈Ω-.
Since P and G satisfy the Hölder condition on Γ and G(z)≠0 on Γ, then P/G also satisfies the Hölder condition on Γ. Taking the limit Ω-∋z̃→z∈Γ and applying Sokhotski formula [5], we get
(3.7)-12P(z)G(z)+PV12πi∫ΓP(z)(w-z)G(w)dw=∑aj∈ΩResw=ajP(w)(w-z)G(w),z∈Γ.
By taking conjugate to both sides and using (3.1), we get
(3.8)-12P(z)c(z)Q(z)T(z)¯+12H(z)¯c(z)Q(z)T(z)¯-PV12πi∫ΓP(z)c(w)(w¯-z¯)Q(w)¯dw¯T(w)¯+PV12πi∫ΓH(z)¯dw¯c(w)(w¯-z¯)Q(z)T(z)¯=[∑aj∈ΩResw=ajP(w)(w-z)G(w)]conj,z∈Γ.
Multiplying both sides by -c(z) and the fact that dw=T(w)|dw|, after some arrangement, yield
(3.9)12P(z)Q(z)T(z)¯+PV12πi∫Γc(z)P(z)c(w)(w¯-z¯)Q(w)¯|dw|+c(z)[∑aj∈ΩResw=ajP(w)(w-z)G(w)]conj=-[-12H(z)Q(z)T(z)+PV12πi∫Γc(z)¯H(z)c(w)¯(w-z)Q(z)T(z)dw]conj,z∈Γ.
Applying Sokhotski formulas again to the expression inside the bracket of the right-hand side yields
(3.10)12P(z)Q(z)T(z)¯+PV12πi∫Γc(z)P(z)c(w)(w¯-z¯)Q(w)¯|dw|+c(z)[∑aj∈ΩResw=ajP(w)(w-z)G(w)]conj=-LR-(z)¯,z∈Γ.
Since P(z) is analytic in Ω, then by Cauchy integral formula, we have
(3.11)12πi∫ΓP(z)w-z̃dw=0,z∈Ω-.
Taking the limit ω-∋z̃→z∈Γ and applying Sokhotiski formulas, we get
(3.12)-12P(z)+PV12πi∫ΓT(w)P(z)w-z|dw|=0,z∈Γ.
Multiplying (3.12) by v(z) and subtracting it from (3.10) multiplied by u(z) yield (3.4).


## 3.2.  Homogeneous Boundary Relationship for Conformal Mapping

Letw=f(z) be the analytic function which maps Ω in the z-plane onto a canonical region of the circular slit region in the w-plane. Let 0 and a be a fixed point in Ω such that a≠0. Then, the mapping function is made uniquely determined by assuming that f(a)=0 and f(0)=∞ such that the residue of the function f at 0 is equal to 1 [1]. Hence, the function f can be written in the form(3.13)f(z)=(1z-1a)ezg(z),
where g is analytic in Ω [12, 13]. Note that the boundary value of f can be represented in the form(3.14)f(zp(t))=μpeiθp(t),Γp:z=zp(t),0≤t≤βp,p=0,1,…,M,
where θp is a boundary correspondence function of Γp and μp is the radius of the circular slit. The unit tangent to Γ at z(t) is denoted by T(z(t))=z′(t)/|z′(t)|. Thus, it can be shown that(3.15)f(z)=|f(z)|iT(z)|θp′(t)|θp′(t)f′(z)|f′(z)|,z∈Γ.


## 4.  Integral Equation Method for ComputingF′(Z)

Note that the value ofθp′(t) may be positive or negative since each circular slit f(Γp) is traversed twice. Thus, |θp′|/θp′=±1. Hence, the boundary relationship (3.15) can be written as(4.1)f(z)=±T(z)|f(z)|if′(z)|f′(z)|,z∈Γ.
To eliminate the ± sign, we square both sides of the boundary relationship (4.1) to get(4.2)f(z)2=-T(z)2|f(z)|2f′(z)2|f′(z)|2,z∈Γ.
Then, the function E(z) defined by(4.3)D(z)=z2f′(z)=z2f(z)[zg′(z)+g(z)]-ezg(z)
is analytic in Ω.Combining (4.3), (4.2), and (3.13), we obtain the following boundary relationship:(4.4)ze2zh(z)a2=-z¯|z|2(a-z)2|f(z)|2T(z)2D(z)2|D(z)|2,z∈Γ.
Comparison of (4.4) and (3.2) leads to a choice of P(z)=D(z), c(z)=-z|z|2|f(z)|2/(a¯-z¯)2, Q(z)=T(z), G(z)=ze2zh(z)/a2, H(z)=0. Setting U(z)=T(z)¯Q(z)¯ and V(z)=1, Theorem 3.1 yields(4.5)T(z)D(z)+PV12πi∫Γ[z|z|2|f(z)|2(a-w)2¯T(z)¯w|w|2|f(w)|2(a-z)2¯(w¯-z¯)-T(z)w-z]T(w)D(w)|dw|=z|z|2|f(z)|2(a¯-z¯)2T(z)¯[∑aj∈ΩResw=aja2D(w)(w-z)we2wh(w)]conj,z∈Γ.
Note that a2D(w)/(w-z)w2 has a simple pole at w=0. To evaluate the residue in (4.5), we use the fact that if L(z)=d(z)/q(z) where d(z) and q(z) are analytic at z0 and d(z0)≠0, q(z0)=0 and q′(z0)≠0, which means z0 is a simple pole of L(z), then(4.6)Resw=z0L(w)=d(z0)q′(z0).
Applying (4.6) to the residue in (4.5) and after several algebraic manipulations, we obtain(4.7)∑aj∈ΩResw=aja2D(w)(w-z)we2wg(w)=a2z.
Thus, integral equation (4.5) becomes (4.8)F(Z)+∫ΓN+(z,w)F(w)|dw|=a2¯z2|f(z)|2(a¯-z¯)2T(z)¯,z∈Γ,
where(4.9)F(z)=T(z)D(z),D(z)=z2f′(z),N+(z,w)=12πi[T(z)z-w-z|z|2|f(z)|2(a-w)2¯T(z)¯w|w|2|f(w)|2(a-z)2¯(z¯-w¯)],N+(t,t)=12π|z′(t)|Im⁡z′′(t)z′(t)+1πi|z′(t)|[z′(t)¯z(t)¯-a¯-Re(z′(t)z(t))]-12πi|z′(t)|z′(t)z(t).
By using single valuedness of the mapping function f leads to the following condition:(4.10)12π∫-ΓqF(w)w2|dw|=0,q=0,1,…,M.
By means of Cauchy’s integral formula, we can get the following condition:(4.11)12π∫ΓF(w)w|dw|=-i.
Thus, the integral equation (4.8) with the conditions (4.10) and (4.11) should give a unique solution provided the parameters μp, p=0,1,…,M that appear in N+(z,w) are known.Integral equation methods for computingμp and θp′ are discussed in the next two sections.


## 5.  Integral Equation for Computing|f(z)|

Note that, from (3.13) and (3.14), we get the following equation:(5.1)z(t)g(z(t))=log⁡|f(z(t))|-log⁡|1z(t)-1a|-iarg(1z(t)-1a)+θp(t).
Since g(z) is analytic in Ω, thus(5.2)Â(t)g(z(t))=γ(t)+h(t)+iυ,
from (5.1) and (5.2), yields(5.3)Â(t)=z(t),(5.4)γ(t)=-log⁡(1z(t)-1a),(5.5)h(t)=log⁡μ(t)=(log⁡μ0,log⁡μ1,…,log⁡μM).The following theorem from [22] gives a method for calculating h(t), and hence μp=|f(zp)|.Theorem 5.1 (see [22, Theorem 5]).
The functionh is given by h=(h0,h1,…,hM), where
(5.6)hj=(γ,ϕ[j])=12π∫Jγ(t)ϕ[j](t)dt,
and where ϕ[j] is the unique solution of the following integral equation
(5.7)(I+N̂*+Ĵ)ϕ[j]=-χ[j],j=0,1,…,M,
where the kernel N̂*(s,t) is the adjoint kernel of the kernel N̂(s,t) which is formed with Â(t)=z(t).By obtainingh0,h1,…,hM from (5.6), in view of (5.5), we obtain(5.8)μj=ehj,j=0,1,…,M.


## 6.  Integral Equation Method for Computingθp′(t)

This section gives another application of Theorem3.1 for computing f′/f. Let f be the mapping function as described in Section 3.2. Note that (4.2) can be written in the following form:(6.1)|f′(z)f(z)|2=-T(z)2(f′(z)f(z))2,z∈Γ.
Taking the derivative of both sides of (3.13) together with some elementary calculations yields(6.2)f′(z)f(z)+az(a-z)=zg′(z)+g(z).
Let E(z)=(f′(z)/f(z))+(a/z(a-z))=zg′(z)+g(z) be analytic in Ω. Then,(6.3)f′(z)f(z)=E(z)+az(z-a),z∈Γ.
Equations (6.1) and (6.3) together with some elementary calculations yield(6.4)E(z)=-T(z)2¯E(z)¯-a¯T(z)2¯z¯(z¯-a¯)-az(z-a),z∈Γ.Comparison of (6.4) and (3.1) leads to a choice of P(z)=E(z), c(z)=-1, Q(z)=T(z), G(z)=1, H(z)=-(aT(z)2/z(z-a))-(¯a/¯z(z¯-a¯)). Setting U(z)=T(z)¯Q(z)¯ and V(z)=1, Theorem 3.1 yields(6.5)E(z)T(z)+PV12πi∫Γ[T(z)¯w¯-z¯-T(z)w-z]E(w)T(w)|dw|=-T(z)¯LR-(z)¯,z∈Γ,
where(6.6)T(z)LR-(z)=-12[-aT(z)z(z-a)-a¯T(z)¯z¯(z¯-a¯)]+T(z)PV12πi∫Γaw(w-z)(w-a)dw-T(z)PV12πi∫ΓAa¯T(w)2¯w¯(w¯-a¯)(w-z)dw,z∈Γ.
Then, it follows from [5, page 91] that(6.7)PV12πi∫Γaw(w-z)(w-a)dw=-12az(z-a).
From (6.5),(6.6), (6.7), and (6.3), we obtain the integral equation(6.8)f′(z)f(z)T(z)+PV12πi∫Γ[T(z)z-w-T(z)¯z¯-w¯]f′(w)f(w)T(w)|dw|=2iIm⁡[aT(z)z(z-a)],z∈Γ.
In the above integral equation, let z=z(t) and w=z(s). Then, by multiplying both sides of (6.8) by |z′(t)| and using the fact that(6.9)f′(z)f(z)z′(t)=iθp′(t),z∈Γ,
the above integral equation can also be written as(6.10)θp′(t)+∫JN(s,t)θp′(s)ds=2Im⁡[az′(t)z(t)(z(t)-a)].
Since N(s,t)=N*(t,s), the integral equation can be written as an integral equation in operator form(6.11)(I+N*)θp′=ψ̃,
where(6.12)ψ̃=2Im⁡[az′(t)z(t)(z(t)-a)].
However, λ=-1 is an eigenvalue of N* with multiplicity M, by [18, Theorem  12]. Therefore, the integral equation (6.11) is not uniquely solvable. To overcome this problem, note that(6.13)∫Jjθp′(t)dt=0,j=1,2,…,M,
which implies(6.14)Jθp′=0.
By adding (6.14) to (6.11), we obtain the integral equation(6.15)(I+N*+J)θp′=ψ̃.
The integral equation (6.15) is uniquely solvable in view of the following theorem which can be proved by using the approach used in proving [22, Theorem  4].Theorem 6.1.
(6.16)Null(I+N*+J)={0}.Proof.
Letυ∈Null(I+N*+J), that is, υ is a solution of the integral equation
(6.17)(I+N*+J)υ=0.
Then, it follows from the definition of the operator J, (2.18), and the Fredholm alternative theorem that
(6.18)J=J*=J2,Range(J)=span⁡{χ[1],…,χ[M]}=Null(I+N),Null(J)=(span⁡{χ[1],…,χ[M]})⊥=Null(I+N)⊥=Range(I+N*).
Hence, we have NJ=-J and JN*=J*N*=(NJ)*=-J. By multiplying (6.17) by J, we obtain
(6.19)Jυ=0,(I+N*)υ=0.
Thus,
(6.20)υ∈Null(J)∩Null(I+N*)=Range(I+N*)∩Null(I+N*).
Since Â=1, thus the index of the function Â is given by (see [18] for the definition of the index)
(6.21)κj=0,j=0,1,…,m,κ=0.
The space S+ defined in [18, Equation (30)] is then given by S+=span{1}. Then, it follows from [18, Equation (92)]  that the dimension of the space S̃+ defined in [18, Equation (32)] is given by dim(S̃+)=M. Similarly, it follows from [18, Equation (105)] that
(6.22)dim⁡(Null(I+N*))=dim⁡(Null(I-Ñ))=M.
Thus, it follows from [18, Lemma 20(b)] that Null(I+N*)=S̃+ and the space R̃+∩S̃- in [18, Lemma 20(a)] contains only the zero function, that is, R̃+∩S̃-={0}. Thus, it follows from [18, Equation (103)] (applied to the adjoint function Ã(t)=Â(t)/z′(t) instead of Â(t)) and from [18, Equation (100)] that
(6.23)Range(I+N*)∩Null(I+N*)={0}.
Hence, it follows from (6.20) that υ=0.By solving the integral equation (6.15), we get θp(t). And solving the integral equation (5.7), we get ϕ[j], j=0,1,…,M, which gives hj through (5.6) which in turn gives μj through (5.8). By solving integral equation (4.8), (4.10), and (4.11) with the known values of μj, we get F(z). From the definition of F(z), we get(6.24)f′(z(t))=F(z(t))z2(t)z′(t).
Finally, from (3.14) and (6.24), the approximate boundary value of f(z) is given by(6.25)f(z)=|f(z)|iT(z)|θp′(t)||θp′(t)|f′(z)|f′(z)|,z∈Γ.
The approximate interior value of the function f(z) is calculated by the Cauchy integral formula(6.26)f(z)=a-zaz12πi∫Γawf(w)a-w1w-zdw,z∈Γ.
For points z which are not close to the boundary, the integral in (6.26) is approximated by the trapezoidal rule. However, for the points z closed to the boundary Γ, the numerical integration in (6.26) is nearly singular. This difficulty is overcome by using the fact that (1/2πi)∫Γ(1/(w-z))dw=1, and rewrite f(z) as(6.27)f(z)=((a-z)/az)(1/2πi)∫Γ(awf(w)/(a-w))(1/(w-z))dw∫Γ(1/(w-z))dw,z∈Ω.
This idea has the advantage that the denominator in this formula compensates for the error in the numerator (see [23]). The integrals in (6.27) are approximated by the trapezoidal rule.


## 7.  Numerical Examples

Since the functionzp(t) is 2π-periodic, a reliable procedure for solving the integral equations (6.15), (5.7), and (4.8) with the conditions (4.10) and (4.11) numerically is by using the Nyström’s method with the trapezoidal rule [24]. The trapezoidal rule is the most accurate method for integrating periodic functions numerically [25, page 134–142]. Thus, solving the integral equations numerically reduces to solving linear systems of the form(7.1)AX=B.
The above linear system (7.1) is uniquely solvable for sufficiently large number of collocation points on each boundary component, since the integral equations (6.15), (5.7), and (4.8) with the conditions (4.10) and (4.11) are uniquely solvable [26]. The computational details are similar to [6, 11–13].For numerical experiments, we have used some test regions of connectivity two, three, four, and five based on the examples given in [2, 4, 7, 12, 13, 15, 27–29]. All the computations were done using MATLAB 7.8.0.347(R2009a)(double precision floating point number). The number of points used in the discretization of each boundary component Γj is n.


### 7.1.  Regions of Connectivity One

In this section, we have used three test regions of connectivity one. Only the first test region has known exact mapping function. The results for sup norm error between the exact values off, μ1 and approximate values fn, μ1n are shown in Table 1.Table 1
Error norm (unit circle).
n∥μ-μn∥∞∥f-fn(t)∥∞81.8×10-052.2×10-02163.7×10-105.0×10-06328.8×10-163.4×10-14Example 7.1.
Consider a regionΩ bounded by the unit circle
(7.2)Γ:{z(t)=eit},a=-0.2+0.2i,
Then, the exact mapping function is given by [1, page 340]
(7.3)g(z)=(a-z)az(1-a¯z),r=1|a|.
Figure 2 shows the region and its image based on our method. See Table 1 for results.Mapping a regionΩ bounded by unit circle onto a circular slit region.
(a)(b)Example 7.2.
Consider the elliptical region bounded by the ellipse(7.4)Γ:{z(t)=4cos⁡⁡t+2isin⁡⁡t},a=-0.2-0.2i.
Figure 3 shows the region and its image based on our method. See Table 2 for our computed value of μ0.Table 2
The numerical values ofμ0 for Example 7.2.
nμ0163.5383174719052323.5355590602433643.5355585660566128—Mapping for Example7.2.
(a)(b)Example 7.3.
Consider a regionΩ bounded by
(7.5)Γ:{z(t)=(10+3cos⁡⁡3t)eit},a=0.1-0.6i.
Figure 4 shows the region and its image based on our method. See Table 3 for comparison between our computed values of μ0 with those computed values μ0n of Nasser [12, 13].Table 3
Error norm for Example7.3.
n∥μ0-μ0n∥∞81.0×10-02167.2×10-05321.1×10-08644.6×10-15Mapping an original region and its image.
(a)(b)


### 7.2.  Regions of Connectivity Two

In this section, we have used two test regions of connectivity two whose exact mapping functions are unknown. The first and second test regions are circular frame, and the third test region is bounded by an ellipse and circle. Figures5–7 show the region and its image based on our method, and approximate values of μ0 and μ1 are shown in Tables 4–6.Table 4
Error norm for Example7.4.
2n∥μ0-μ0n∥∞∥μ1-μ1n∥∞323.2×10-035.8×10-03642.4×10-065.1×10-061281.7×10-123.5×10-122568.8×10-162.2×10-15Table 5
Error norm for Example7.5.
2n∥μ0-μ0n∥∞∥μ1-μ1n∥∞641.5×10-036.2×10-041284.9×10-078.5×10-102567.1×10-143.5×10-14Table 6
Error norm for Example7.6.
2n∥μ0-μ0n∥∞∥μ1-μ1n∥∞642.3×10-032.4×10-031287.4×10-079.5×10-072567.3×10-149.9×10-14Mapping a regionΩ bounded by two circles onto a circular slit region.
(a)(b)Mapping a regionΩ bounded by an ellipse and a circle onto a circular slit region.
(a)(b)Mapping a regionΩ bounded by two ellipses onto a circular slit region.
(a)(b)Example 7.4 (circular frame).
Consider a pair of circles [28]
(7.6)Γ0:{z(t)=eit},Γ1:{z(t)=-0.6+0.2e-it},t:0≤t≤2π,a=0.25+0.25i,
such that the region bounded by Γ0 and Γ1 is the region between a unit circle and a circle centered at −0.6 with radius 0.2. Then, Figure 5 shows the region and its image based on our method. See Table 4 for comparison between our computed values of μ0 and μ1 with those computed values μ0n and μ1n of Nasser [12, 13].Example 7.5 (ellipse with one circle).
Consider a regionΩ bounded by an ellipse and a circle
(7.7)Γ0:{z(t)=4cos⁡⁡t+isin⁡⁡t},Γ1:{z(t)=-1+0.25e-it},t:0≤t≤2π,a=-1.4,
such that the region bounded by Γ0 and Γ1 is the region between an ellipse and a circle centered at −1 with radius 0.25. Then, Figure 6 shows the region and its image based on our method. See Table 5 for comparison between our computed values of μ0 and μ1 with those computed values μ0n and μ1n of Nasser [12, 13].Example 7.6 (two ellipses).
Consider a regionΩ bounded by pair of ellipses
(7.8)Γ0:{z(t)=4cos⁡⁡t+isin⁡⁡t},Γ1:{z(t)=1+0.7cos⁡⁡t-0.3isin⁡⁡t},t:0≤t≤2π,a=2.3.
Figure 7 shows the region and its image based on our method. See Table 6 for comparison between our computed values of μ0 and μ1 with those computed values μ0n and μ1n of Nasser [12, 13].


### 7.3. Regions of Connectivity Three

In this section, we have used three test regions of connectivity three. The first test region is bounded by three ellipses, the second test region is bounded by an ellipse and two circles, and the third test region is a circular region. The results for sup norm error between the our numerical values ofμ0, μ1, μ2 and the computed values of μ0n, μ1n, μ2n obtained from [12, 13] are shown in Tables 7–9.Table 7
Error norm for Example7.7.
3n∥μ0-μ0n∥∞∥μ1-μ1n∥∞∥μ2-μ2n∥∞485.1×10-041.3×10-034.7×10-04962.8×10-067.5×10-063.9×10-061922.4×10-106.3×10-103.1×10-103845.5×10-172.7×10-164.9×10-16Table 8
Error norm for Example7.8.
3n∥μ0-μ0n∥∞∥μ1-μ1n∥∞∥μ2-μ2n∥∞961.6×10-051.0×10-034.9×10-031922.7×10-062.8×10-068.6×10-073841.2×10-111.4×10-111.2×10-11Table 9
The numerical values ofμ0, μ1, and μ2 for Example 7.9.
3nμ0μ1μ2961.1448447121121.3334475601141.7117792226481921.1448440806441.3334469442821.711778670173384—1.333446944281—Example 7.7 (three ellipses).
LetΩ be the region bounded by
(7.9)Γ0:{z(t)=10cos⁡⁡t+6isin⁡⁡t},Γ1:{z(t)=-4-2i+3cos⁡⁡t-2isin⁡⁡t},Γ2:{z(t)=4+2cos⁡⁡t-3isin⁡⁡t},0≤t≤2π,a=7.
Figure 8 shows the region and its image based on our method. See Table 7 for comparison between our computed values of μ0, μ1, and μ2 with those computed values of Nasser [12].Mapping a regionΩ bounded by three ellipses onto a circular slit region.
(a)(b)Example 7.8 (ellipse with two circles).
LetΩ be the region bounded by [7, 13, 15]
(7.10)Γ0:{z(t)=4cos⁡⁡t+isin⁡⁡t},Γ1:{z(t)=1.2+0.3(cos⁡⁡t-isin⁡⁡t)},Γ2:{z(t)=-1+0.6(cos⁡⁡t-isin⁡⁡t)},0≤t≤2π,a=-2.5-0.1i.
Figure 9 shows the region and its image based on our method. See Table 8 for comparison between our computed values of μ0, μ1, and μ2 with those computed values of Nasser [13].Mapping a regionΩ bounded by an ellipse and two circles onto a circular slit region.
(a)(b)Example 7.9 (three circles).
LetΩ be the region bounded by
(7.11)Γ0:{z(t)=2eit},Γ1:{z(t)=1.2+0.3e-it},Γ2:{z(t)=-1+0.6e-it},0≤t≤2π,a=0.5-1.25i.
Figure 10 shows the region and its image based on our method. See Table 9 for our computed values of μ0, μ1, and μ2.Mapping a regionΩ bounded by three circles onto a circular slit region.
(a)(b)


### 7.4.  Regions of Connectivity Four and Five

In this section, we have used four test regions for multiply connected regions whose exact mapping functions are unknown. The results for sup norm error for first and third regions between the our numerical values ofμ0, μ1, μ2, μ3, μ4 and the computed values of μ0n, μ1n, μ2n, μ3n, μ4n obtained from [12] are shown in Tables 10 and 12.Table 10
Error norm for Example7.10.
4n∥μ0-μ0n∥∞∥μ1-μ1n∥∞∥μ2-μ2n∥∞∥μ3-μ3n∥∞646.7×10-057.2×10-059.9×10-052.2×10-051286.4×10-095.0×10-081.8×10-094.5×10-082566.8×10-131.0×10-129.8×10-139.7×10-135121.3×10-161.2×10-153.0×10-164.4×10-16Example 7.10.
LetΩ be the region bounded by [12]
(7.12)Γ0:{z(t)=(10+3cos⁡⁡3t)eit},Γ1:{z(t)=-3.5+6i+0.5e-iπ/4(eit+4e-it)},Γ2:{z(t)=5+0.5eiπ/4(eit+4e-it)},Γ3:{z(t)=-3.5-6i+0.5eiπ/4(eit+4e-it)},0≤t≤2π,a=8.5+0.1i.
Figure 11 shows the region and its image based on our method. See Table 10 for comparison between our computed values of μ0, μ1, μ2, and μ3 with those computed values of Nasser [12].Mapping for Example7.10.
(a)(b)Example 7.11 (ellipse with three circles).
LetΩ be the region bounded by
(7.13)Γ0:{z(t)=2cos⁡⁡t+1.5isin⁡⁡t},Γ1:{z(t)=1+0.25(cos⁡⁡t-isin⁡⁡t)},Γ2:{z(t)=-1+0.25(cos⁡⁡t-isin⁡⁡t)},Γ3:{z(t)=0.75i+0.25(cos⁡⁡t-isin⁡⁡t)},0≤t≤2π,a=0.25-0.25i.
Figure 12 shows the region and its image based on our method. See Table 11 for our computed values of μ0, μ1, μ2, and μ3.Table 11
The numerical values ofμ0, μ1, μ2, and μ3 for Example 7.11.
4nμ0μ1μ2μ3642.973169983112.501705004113.453737116183.691252055101282.967572775022.499230616053.450410676503.699041617292562.967563610862.499227351003.450406178453.699051243065122.967563610852.499227350993.450406178443.69905124308Table 12
Error norm for Example7.12.
5n∥μ0-μ0n∥∞∥μ1-μ1n∥∞∥μ2-μ2n∥∞∥μ3-μ3n∥∞∥μ4-μ4n∥∞804.2×10-054.5×10-054.5×10-054.4×10-054.3×10-051601.1×10-073.2×10-083.2×10-086.6×10-086.6×10-083201.6×10-135.7×10-145.7×10-141.2×10-131.2×10-134009.9×10-1609.9×10-1600Mapping a regionΩ bounded by an ellipse and three circles onto a circular slit region.
(a)(b)Example 7.12 (ellipse with four circles).
LetΩ be the region bounded by
(7.14)Γ0:{z(t)=0.2+8cos⁡⁡t+6isin⁡⁡t},Γ1:{z(t)=3+2i+cos⁡⁡t-isin⁡⁡t},Γ2:{z(t)=-3+2i+cos⁡⁡t-isin⁡⁡t},Γ3:{z(t)=-3-2i+cos⁡⁡t-isin⁡⁡t},Γ4:{z(t)=3-2i+cos⁡⁡t-isin⁡⁡t},0≤t≤2π,a=4i.
Figure 13 shows the region and its image based on our method. See Table 12 for comparison between our computed values of μ0, μ1, μ2, μ3, and μ4 with those computed values of Nasser [12].Mapping a regionΩ bounded by an ellipse and four circles onto a circular slit region.
(a)(b)Example 7.13 (five ellipses).
LetΩ be the region bounded by
(7.15)Γ0:{z(t)=-1.5i+6cos⁡⁡t+8isin⁡⁡t},Γ1:{z(t)=3+0.5i+1.5cos⁡⁡t-isin⁡⁡t},Γ2:{z(t)=-3+0.5i+1.5cos⁡⁡t-isin⁡⁡t},Γ3:{z(t)=-3i+0.7cos⁡⁡t-1.7isin⁡⁡t},Γ4:{z(t)=-6i+1.7cos⁡⁡t-0.7isin⁡⁡t},0≤t≤2π,a=0.4i.
Figure 14 shows the region and its image based on our method. See Table 13 for our computed values of μ0, μ1, μ2, μ3, and μ4.Table 13
The numerical values ofμ0, μ1, μ2, μ3, and μ4 for Example 7.13.
5nμ0μ1μ2μ3μ41600.40817694610.54702547510.54702547510.68508792890.52586419023200.40810975910.54705051810.54705051810.68504663600.52580668214000.40810978850.54705050710.54705050710.68504665370.5258067072Mapping a regionΩ bounded by five ellipses onto a circular slit region.
(a)(b)


## 7.1.  Regions of Connectivity One

In this section, we have used three test regions of connectivity one. Only the first test region has known exact mapping function. The results for sup norm error between the exact values off, μ1 and approximate values fn, μ1n are shown in Table 1.Table 1
Error norm (unit circle).
n∥μ-μn∥∞∥f-fn(t)∥∞81.8×10-052.2×10-02163.7×10-105.0×10-06328.8×10-163.4×10-14Example 7.1.
Consider a regionΩ bounded by the unit circle
(7.2)Γ:{z(t)=eit},a=-0.2+0.2i,
Then, the exact mapping function is given by [1, page 340]
(7.3)g(z)=(a-z)az(1-a¯z),r=1|a|.
Figure 2 shows the region and its image based on our method. See Table 1 for results.Mapping a regionΩ bounded by unit circle onto a circular slit region.
(a)(b)Example 7.2.
Consider the elliptical region bounded by the ellipse(7.4)Γ:{z(t)=4cos⁡⁡t+2isin⁡⁡t},a=-0.2-0.2i.
Figure 3 shows the region and its image based on our method. See Table 2 for our computed value of μ0.Table 2
The numerical values ofμ0 for Example 7.2.
nμ0163.5383174719052323.5355590602433643.5355585660566128—Mapping for Example7.2.
(a)(b)Example 7.3.
Consider a regionΩ bounded by
(7.5)Γ:{z(t)=(10+3cos⁡⁡3t)eit},a=0.1-0.6i.
Figure 4 shows the region and its image based on our method. See Table 3 for comparison between our computed values of μ0 with those computed values μ0n of Nasser [12, 13].Table 3
Error norm for Example7.3.
n∥μ0-μ0n∥∞81.0×10-02167.2×10-05321.1×10-08644.6×10-15Mapping an original region and its image.
(a)(b)


## 7.2.  Regions of Connectivity Two

In this section, we have used two test regions of connectivity two whose exact mapping functions are unknown. The first and second test regions are circular frame, and the third test region is bounded by an ellipse and circle. Figures5–7 show the region and its image based on our method, and approximate values of μ0 and μ1 are shown in Tables 4–6.Table 4
Error norm for Example7.4.
2n∥μ0-μ0n∥∞∥μ1-μ1n∥∞323.2×10-035.8×10-03642.4×10-065.1×10-061281.7×10-123.5×10-122568.8×10-162.2×10-15Table 5
Error norm for Example7.5.
2n∥μ0-μ0n∥∞∥μ1-μ1n∥∞641.5×10-036.2×10-041284.9×10-078.5×10-102567.1×10-143.5×10-14Table 6
Error norm for Example7.6.
2n∥μ0-μ0n∥∞∥μ1-μ1n∥∞642.3×10-032.4×10-031287.4×10-079.5×10-072567.3×10-149.9×10-14Mapping a regionΩ bounded by two circles onto a circular slit region.
(a)(b)Mapping a regionΩ bounded by an ellipse and a circle onto a circular slit region.
(a)(b)Mapping a regionΩ bounded by two ellipses onto a circular slit region.
(a)(b)Example 7.4 (circular frame).
Consider a pair of circles [28]
(7.6)Γ0:{z(t)=eit},Γ1:{z(t)=-0.6+0.2e-it},t:0≤t≤2π,a=0.25+0.25i,
such that the region bounded by Γ0 and Γ1 is the region between a unit circle and a circle centered at −0.6 with radius 0.2. Then, Figure 5 shows the region and its image based on our method. See Table 4 for comparison between our computed values of μ0 and μ1 with those computed values μ0n and μ1n of Nasser [12, 13].Example 7.5 (ellipse with one circle).
Consider a regionΩ bounded by an ellipse and a circle
(7.7)Γ0:{z(t)=4cos⁡⁡t+isin⁡⁡t},Γ1:{z(t)=-1+0.25e-it},t:0≤t≤2π,a=-1.4,
such that the region bounded by Γ0 and Γ1 is the region between an ellipse and a circle centered at −1 with radius 0.25. Then, Figure 6 shows the region and its image based on our method. See Table 5 for comparison between our computed values of μ0 and μ1 with those computed values μ0n and μ1n of Nasser [12, 13].Example 7.6 (two ellipses).
Consider a regionΩ bounded by pair of ellipses
(7.8)Γ0:{z(t)=4cos⁡⁡t+isin⁡⁡t},Γ1:{z(t)=1+0.7cos⁡⁡t-0.3isin⁡⁡t},t:0≤t≤2π,a=2.3.
Figure 7 shows the region and its image based on our method. See Table 6 for comparison between our computed values of μ0 and μ1 with those computed values μ0n and μ1n of Nasser [12, 13].


## 7.3. Regions of Connectivity Three

In this section, we have used three test regions of connectivity three. The first test region is bounded by three ellipses, the second test region is bounded by an ellipse and two circles, and the third test region is a circular region. The results for sup norm error between the our numerical values ofμ0, μ1, μ2 and the computed values of μ0n, μ1n, μ2n obtained from [12, 13] are shown in Tables 7–9.Table 7
Error norm for Example7.7.
3n∥μ0-μ0n∥∞∥μ1-μ1n∥∞∥μ2-μ2n∥∞485.1×10-041.3×10-034.7×10-04962.8×10-067.5×10-063.9×10-061922.4×10-106.3×10-103.1×10-103845.5×10-172.7×10-164.9×10-16Table 8
Error norm for Example7.8.
3n∥μ0-μ0n∥∞∥μ1-μ1n∥∞∥μ2-μ2n∥∞961.6×10-051.0×10-034.9×10-031922.7×10-062.8×10-068.6×10-073841.2×10-111.4×10-111.2×10-11Table 9
The numerical values ofμ0, μ1, and μ2 for Example 7.9.
3nμ0μ1μ2961.1448447121121.3334475601141.7117792226481921.1448440806441.3334469442821.711778670173384—1.333446944281—Example 7.7 (three ellipses).
LetΩ be the region bounded by
(7.9)Γ0:{z(t)=10cos⁡⁡t+6isin⁡⁡t},Γ1:{z(t)=-4-2i+3cos⁡⁡t-2isin⁡⁡t},Γ2:{z(t)=4+2cos⁡⁡t-3isin⁡⁡t},0≤t≤2π,a=7.
Figure 8 shows the region and its image based on our method. See Table 7 for comparison between our computed values of μ0, μ1, and μ2 with those computed values of Nasser [12].Mapping a regionΩ bounded by three ellipses onto a circular slit region.
(a)(b)Example 7.8 (ellipse with two circles).
LetΩ be the region bounded by [7, 13, 15]
(7.10)Γ0:{z(t)=4cos⁡⁡t+isin⁡⁡t},Γ1:{z(t)=1.2+0.3(cos⁡⁡t-isin⁡⁡t)},Γ2:{z(t)=-1+0.6(cos⁡⁡t-isin⁡⁡t)},0≤t≤2π,a=-2.5-0.1i.
Figure 9 shows the region and its image based on our method. See Table 8 for comparison between our computed values of μ0, μ1, and μ2 with those computed values of Nasser [13].Mapping a regionΩ bounded by an ellipse and two circles onto a circular slit region.
(a)(b)Example 7.9 (three circles).
LetΩ be the region bounded by
(7.11)Γ0:{z(t)=2eit},Γ1:{z(t)=1.2+0.3e-it},Γ2:{z(t)=-1+0.6e-it},0≤t≤2π,a=0.5-1.25i.
Figure 10 shows the region and its image based on our method. See Table 9 for our computed values of μ0, μ1, and μ2.Mapping a regionΩ bounded by three circles onto a circular slit region.
(a)(b)


## 7.4.  Regions of Connectivity Four and Five

In this section, we have used four test regions for multiply connected regions whose exact mapping functions are unknown. The results for sup norm error for first and third regions between the our numerical values ofμ0, μ1, μ2, μ3, μ4 and the computed values of μ0n, μ1n, μ2n, μ3n, μ4n obtained from [12] are shown in Tables 10 and 12.Table 10
Error norm for Example7.10.
4n∥μ0-μ0n∥∞∥μ1-μ1n∥∞∥μ2-μ2n∥∞∥μ3-μ3n∥∞646.7×10-057.2×10-059.9×10-052.2×10-051286.4×10-095.0×10-081.8×10-094.5×10-082566.8×10-131.0×10-129.8×10-139.7×10-135121.3×10-161.2×10-153.0×10-164.4×10-16Example 7.10.
LetΩ be the region bounded by [12]
(7.12)Γ0:{z(t)=(10+3cos⁡⁡3t)eit},Γ1:{z(t)=-3.5+6i+0.5e-iπ/4(eit+4e-it)},Γ2:{z(t)=5+0.5eiπ/4(eit+4e-it)},Γ3:{z(t)=-3.5-6i+0.5eiπ/4(eit+4e-it)},0≤t≤2π,a=8.5+0.1i.
Figure 11 shows the region and its image based on our method. See Table 10 for comparison between our computed values of μ0, μ1, μ2, and μ3 with those computed values of Nasser [12].Mapping for Example7.10.
(a)(b)Example 7.11 (ellipse with three circles).
LetΩ be the region bounded by
(7.13)Γ0:{z(t)=2cos⁡⁡t+1.5isin⁡⁡t},Γ1:{z(t)=1+0.25(cos⁡⁡t-isin⁡⁡t)},Γ2:{z(t)=-1+0.25(cos⁡⁡t-isin⁡⁡t)},Γ3:{z(t)=0.75i+0.25(cos⁡⁡t-isin⁡⁡t)},0≤t≤2π,a=0.25-0.25i.
Figure 12 shows the region and its image based on our method. See Table 11 for our computed values of μ0, μ1, μ2, and μ3.Table 11
The numerical values ofμ0, μ1, μ2, and μ3 for Example 7.11.
4nμ0μ1μ2μ3642.973169983112.501705004113.453737116183.691252055101282.967572775022.499230616053.450410676503.699041617292562.967563610862.499227351003.450406178453.699051243065122.967563610852.499227350993.450406178443.69905124308Table 12
Error norm for Example7.12.
5n∥μ0-μ0n∥∞∥μ1-μ1n∥∞∥μ2-μ2n∥∞∥μ3-μ3n∥∞∥μ4-μ4n∥∞804.2×10-054.5×10-054.5×10-054.4×10-054.3×10-051601.1×10-073.2×10-083.2×10-086.6×10-086.6×10-083201.6×10-135.7×10-145.7×10-141.2×10-131.2×10-134009.9×10-1609.9×10-1600Mapping a regionΩ bounded by an ellipse and three circles onto a circular slit region.
(a)(b)Example 7.12 (ellipse with four circles).
LetΩ be the region bounded by
(7.14)Γ0:{z(t)=0.2+8cos⁡⁡t+6isin⁡⁡t},Γ1:{z(t)=3+2i+cos⁡⁡t-isin⁡⁡t},Γ2:{z(t)=-3+2i+cos⁡⁡t-isin⁡⁡t},Γ3:{z(t)=-3-2i+cos⁡⁡t-isin⁡⁡t},Γ4:{z(t)=3-2i+cos⁡⁡t-isin⁡⁡t},0≤t≤2π,a=4i.
Figure 13 shows the region and its image based on our method. See Table 12 for comparison between our computed values of μ0, μ1, μ2, μ3, and μ4 with those computed values of Nasser [12].Mapping a regionΩ bounded by an ellipse and four circles onto a circular slit region.
(a)(b)Example 7.13 (five ellipses).
LetΩ be the region bounded by
(7.15)Γ0:{z(t)=-1.5i+6cos⁡⁡t+8isin⁡⁡t},Γ1:{z(t)=3+0.5i+1.5cos⁡⁡t-isin⁡⁡t},Γ2:{z(t)=-3+0.5i+1.5cos⁡⁡t-isin⁡⁡t},Γ3:{z(t)=-3i+0.7cos⁡⁡t-1.7isin⁡⁡t},Γ4:{z(t)=-6i+1.7cos⁡⁡t-0.7isin⁡⁡t},0≤t≤2π,a=0.4i.
Figure 14 shows the region and its image based on our method. See Table 13 for our computed values of μ0, μ1, μ2, μ3, and μ4.Table 13
The numerical values ofμ0, μ1, μ2, μ3, and μ4 for Example 7.13.
5nμ0μ1μ2μ3μ41600.40817694610.54702547510.54702547510.68508792890.52586419023200.40810975910.54705051810.54705051810.68504663600.52580668214000.40810978850.54705050710.54705050710.68504665370.5258067072Mapping a regionΩ bounded by five ellipses onto a circular slit region.
(a)(b)


## 8. Conclusion

In this paper, we have constructed new boundary integral equations for conformal mapping of multiply regions onto a circular slit region. We have also constructed a new method to find the values of modulus off(z). The advantage of our method is that our boundary integral equations are all linear. Several mappings of the test regions of connectivity one, two, three, four, and five were computed numerically using the proposed method. After the boundary values of the mapping function are computed, the interior mapping function is calculated by the means of Cauchy integral formula. The numerical examples presented have illustrated that our boundary integral equation method has high accuracy.



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*Source: 970928-2012-04-17.xml*