Conformal Mapping of Unbounded Multiply Connected Regions onto Canonical Slit Regions

Abstract

We present a boundary integral equation method for conformal mapping of unbounded multiply connected regions onto five types of canonical slit regions. For each canonical region, three linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region. The integral equations are uniquely solvable. The kernels involved in these integral equations are the modified Neumann kernels and the adjoint generalized Neumann kernels.

1. Introduction

In this paper, we present a new method for numerical conformal mapping of unbounded multiply connected regions onto five types of canonical slit regions. A canonical region in conformal mapping is known as a set of finitely connected regions 𝑆 such that each finitely connected nondegenerate region is conformally equivalent to a region in 𝑆. With regard to conformal mapping of multiply connected regions, there are several types of canonical regions as listed in [1–4]. The five types of canonical slit regions are disk with concentric circular slits 𝑈_𝑑, annulus with concentric circular slits 𝑈_𝑎, circular slit regions 𝑈_𝑐, radial slit regions 𝑈_𝑟, and parallel slit regions 𝑈_𝑝. One major setback in conformal mapping is that only for certain regions exact conformal maps are known. One way to deal with this limitation is by numerical computation. Trefethen [5] has discussed several methods for computing conformal mapping numerically. Amano [6] and DeLillo et al. [7] have successfully map unbounded regions onto circular and radial slit regions. Boundary integral equation related to a boundary relationship satisfied by a function which is analytic in a simply or doubly connected region bounded by closed smooth Jordan curves has been given by Murid [8] and Murid and Razali [9]. Special realizations of this integral equation are the integral equations related to the Szegö kernel, Bergmann kernel, Riemann map, and Ahlfors map. The kernels arise in these integral equations are the Neumann kernel and the Kerzman-Stein kernel. Murid and Hu [10] managed to construct a boundary integral equation for numerical conformal mapping of a bounded multiply connected region onto a unit disk with slits. However, the integral equation involves unknown radii which lead to a system of nonlinear algebraic equations upon discretization of the integral equation. Nasser [11–13] produced another technique for numerical conformal mapping of bounded and unbounded multiply connected regions by expressing the mapping function in terms of the solution of a uniquely solvable Riemann-Hilbert problem. This uniquely solvable Riemann Hilbert problem can be solved by means of boundary integral equation with the generalized Neumann kernel. Recently, Sangawi et al. [14–17] have constructed new linear boundary integral equations for conformal mapping of bounded multiply region onto canonical slit regions, which improves the work of Murid and Hu [10] where in [10], the system of algebraic equations are nonlinear. In this paper, we extend the work of [14–17] for numerical conformal mapping of unbounded multiply connected regions onto all five types of canonical slit regions. These boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region.

The plan of this paper is as follows: Section 2 presents some auxiliary material. Section 3 presents a boundary integral equation related to a boundary relationship. In Sections 4–8, we present the derivations for numerical conformal mapping for all five types of canonical regions. In Section 9, we give some examples to illustrate the effectiveness of our method. Finally, Section 10 presents a short conclusion.

2. Auxiliary Material

Let Ω⁻ be an unbounded multiply connected region of connectivity 𝑚. The boundary Γ consists of 𝑚 smooth Jordan curves Γ_𝑗, 𝑗=1,2,…,𝑚 and will be denoted by Γ=Γ₁∪Γ₂∪⋯∪Γ_𝑚. The boundaries Γ_𝑗 are assumed in clockwise orientation (see Figure 1). The curve Γ_𝑗 is parameterized by 2𝜋-periodic twice continuously differentiable complex function 𝜂_𝑗(𝑡) with nonvanishing first derivative, that is, 𝜂^′_𝑗(𝑡)=𝑑𝜂_𝑗(𝑡)𝑑𝑡≠0,𝑡∈𝐽_𝑗=[0,2𝜋],𝑘=1,…,𝑚. (2.1) The total parameter domain 𝐽 is the disjoint union of 𝑚 intervals 𝐽₁,…,𝐽_𝑚. We define a parameterization 𝜂 of the whole boundary Γ on 𝐽 by 𝜂(𝑡)=𝜂₁(𝑡),𝑡∈𝐽₁=[],⋮𝜂0,2𝜋_𝑚(𝑡),𝑡∈𝐽_𝑚=[0,2𝜋]. (2.2)

Figure 1 

An unbounded multiply connected region Ω− with connectivity 𝑚.

Let Φ(𝑧) be the conformal mapping function that maps Ω⁻ onto 𝑈⁻, where 𝑈⁻ represents any canonical region mentioned above, 𝑧₁ is a prescribed point located inside Γ₁, 𝑧_𝑚 is a prescribed point inside Γ_𝑚 and 𝛽 is a prescribed point located in Ω⁻. In this paper, we determine the mapping function Φ(𝑧) by computing the derivatives of the mapping function Φ′(𝜂(𝑡)) and two real functions on 𝐽, that is, the unknown function 𝜑(𝑡) and a piecewise constant real function 𝑅(𝑡). Let 𝐻 be the space of all real Hölder continuous 2𝜋-periodic functions and 𝑆 be the subspace of 𝐻 which contains the piecewise real constant functions 𝑅(𝑡). The piecewise real constant function 𝑅(𝑡) can be written as 𝑅(𝑡)=𝑅₁,𝑡∈𝐽₁=[],⋮𝑅0,2𝜋_𝑚,𝑡∈𝐽_𝑚=[0,2𝜋], (2.3) briefly written as 𝑅(𝑡)=(𝑅₁,…,𝑅_𝑚). Let 𝐴(𝑡) be a complex continuously differentiable 2𝜋-periodic function for all 𝑡∈𝐽. We define two real kernels formed with 𝐴 as [18] 𝑁()=1𝑠,𝑡𝜋𝐴(𝑠)Im𝐴(𝑡)𝜂′(𝑡)𝜂(𝑡)(𝑠),𝑀()=1−𝜂𝑠,𝑡𝜋𝐴(𝑠)Re𝐴(𝑡)𝜂^′(𝑡)𝜂(𝑡)(𝑠).−𝜂 (2.4) The kernel 𝑁(𝑠,𝑡) is known as the generalized Neumann kernel formed with complex-valued functions 𝐴 and 𝜂. The kernel 𝑁(𝑠,𝑡) is continuous with 𝑁()=1𝑡,𝑡𝜋𝜂Im^′′(𝑡)𝜂^′(𝑡)−𝐴^′(𝑡)𝐴(𝑡). (2.5) The kernel 𝑀(𝑠,𝑡) has a cotangent singularity 𝑀()1𝑠,𝑡=−2𝜋cot𝑠−𝑡2+𝑀₁(𝑠,𝑡), (2.6) where the kernel 𝑀₁(𝑠,𝑡) is continuous with 𝑀₁()=1𝑡,𝑡𝜋1Re𝜂2𝜋^′′(𝑡)𝜂^′(𝑡)−𝐴^′(𝑡)𝐴(𝑡). (2.7) The adjoint function 𝐴 of 𝐴 is defined by 𝜂𝐴=^′(𝑡)𝐴(𝑡). (2.8) The generalized Neumann kernel 𝑁(𝑠,𝑡) and the real kernel 𝑀 formed with 𝐴 are defined by 𝑁()=1𝑠,𝑡𝜋𝐴(𝑠)Im𝐴(𝑡)𝜂^′(𝑡)𝜂(𝑡)(𝑠),𝑀()=1−𝜂𝑠,𝑡𝜋𝐴(𝑠)Re𝐴(𝑡)𝜂^′(𝑡)𝜂(𝑡)(𝑠).−𝜂 (2.9) Then, 𝑁(𝑠,𝑡)=−𝑁^∗(𝑠,𝑡),𝑀(𝑠,𝑡)=−𝑀^∗(𝑠,𝑡), (2.10) where 𝑁^∗(𝑠,𝑡)=𝑁(𝑡,𝑠) is the adjoint kernel of the generalized Neumann kernel 𝑁(𝑠,𝑡). We define the Fredholm integral operators 𝐍^∗ by 𝐍^∗𝜐(𝑡)=_𝐽𝑁^∗(𝑡,𝑠)𝜐(𝑠)𝑑𝑠,𝑡∈𝐽. (2.11) Integral operators 𝑀^∗, 𝑁, and 𝑀 are defined in a similar way. Throughout this paper, we will assume the functions 𝐴 and 𝐴 are given by 𝐴(𝑡)=1,𝐴(𝑡)=𝜂^′(𝑡). (2.12) It is known that 𝜆=1 is not an eigenvalue of the kernel 𝑁 and 𝜆=−1 is an eigenvalue of the kernel 𝑁 with multiplicity 𝑚 [18]. The eigenfunctions of 𝑁 corresponding to the eigenvalue 𝜆=−1 are {𝜒^[1],𝜒^[2],…,𝜒^[𝑚]}, where 𝜒^[𝑗](𝜉)=1,𝜉∈Γ_𝑗,0,otherwise.𝑗=1,2,…,𝑚. (2.13) We also define an integral operator 𝐉 by (see [14]) 𝐉𝜇(𝑠)∶=_𝐽12𝜋^𝑚_𝑗=0𝜒^[𝑗](𝑠)𝜒^[𝑗](𝑡)𝜇(𝑡)𝑑𝑡. (2.14) The following theorem gives us a method for calculating the piecewise constant real function ℎ(𝑡) in connection with conformal mapping later. This theorem can be proved by using the approach as in [19, Theorem  5].

Theorem 2.1. Let i=√−1, 𝛾, 𝜇∈𝐻 and ℎ∈𝑆 such that 𝐴𝑓=𝛾+ℎ+i𝜇 (2.15) are the boundary values of a function 𝑓(𝑧) analytic in Ω⁻. Then the function ℎ=(ℎ₁,ℎ₂,…,ℎ_𝑚) has each element given by ℎ_𝑗=𝛾,𝜌^[𝑗]=12𝜋_Γ𝛾(𝑡)𝜌^[𝑡]𝑑𝑡, (2.16) where 𝜌^[𝑡] is the unique solution of the integral equation 𝐈+𝐍^∗+𝐉𝜌^[𝑗]=−𝜒^[𝑗],𝑗=1,2,…,𝑚. (2.17)

3. The Homogeneous Boundary Relationship

Suppose we are given a function 𝐷(𝑧) which is analytic in Ω⁻, continuous on Ω⁻∪Γ, Hölder continuous on Γ and 𝐷(∞) is finite. The boundary Γ_𝑗 is assumed to be a smooth Jordan curve. The unit tangent to Γ at the point 𝜂(𝑡)∈Γ will be denoted by 𝑇(𝜂(𝑡))=𝜂^′(𝑡)/|𝜂^′(𝑡)|. Suppose further that 𝐷(𝜂(𝑡)) satisfies the exterior homogeneous boundary relationship 𝐷𝜂(𝑡)=𝑐(𝑡)𝑇𝜂(𝑡)²𝐷𝜂(𝑡)𝑃𝜂(𝑡,) (3.1) where 𝑐(𝑡) and 𝑃 are complex-valued functions with the following properties: (1)𝑃(𝑧) is analytic in Ω⁻ and does not have zeroes on Ω⁻∪Γ, (2)𝑃(∞)≠0, 𝐷(∞) is finite,(3)𝑐(𝑡)≠0, 𝑃(𝜂(𝑡))≠0. Note that the boundary relationship (3.1) also has the following equivalent form: 𝑃𝜂(𝑡)=𝑐(𝑡)𝑇𝜂(𝑡)²𝐷𝜂(𝑡)²𝐷𝜂(𝑡)². (3.2) Under these assumptions, an integral equation for 𝐷(𝜂(𝑡)) can be constructed by means of the following theorem.

Theorem 3.1. If the function 𝐷(𝜂(𝑡)) satisfies the exterior homogeneous boundary relationship (3.1), then 𝜙(𝑡)+_𝐽𝐾(𝑠,𝑡)𝜙(𝑠)𝑑𝑠=𝜈(𝑡), (3.3) where 𝜙(𝑡)=𝐷𝜂(𝑡)𝜂^′(𝑡),𝐾()=1𝑠,𝑡𝜂2𝜋i^′(𝑡)𝜂(𝑡)(𝑠)−𝑐(𝑡)−𝜂𝑐(𝑠)𝜂′(𝑡)𝜂(𝑡)−𝜂(𝑠),𝜈(𝑡)=𝐷(∞)𝜂^′(𝑡)+𝑐(𝑡)𝜂^′(𝑡)𝐷(∞)𝑃(∞). (3.4)

Proof. Consider the integral 𝐼₁(𝜂(𝑡)), 𝐼₁𝜂(𝑡)=12𝜋i_𝐽𝐷𝜂(𝑠)𝜂(𝑠)−𝜂(𝑡)𝑑𝑠. (3.5) Since the boundary is in clockwise orientation and 𝐷 is analytic in Ω⁻, then by [20, p. 2] we have 𝐼₁𝜂(𝑡)=𝐷𝜂(𝑡)2−𝐷(∞). (3.6) Now, let the integral 𝐼₂(𝜂(𝑡)) be defined as 𝐼₂𝜂(𝑡)=12𝜋i_𝐽𝑐(𝑡)𝑇𝜂(𝑡)²𝐷𝜂(𝑠)𝑐(𝑠)𝜂(𝑠)−𝜂(𝑡)𝑇𝜂(𝑠||.)𝑑𝑠 (3.7) Using the boundary relationship (3.1) and the fact that 𝑇(𝜂(𝑡))|𝑑𝑡|=𝑑𝑡 and 𝑃(𝜂(𝑡)) does not contain zeroes, then by [20, p. 2] we obtain 𝐼₂𝜂(𝑡)=−𝑐(𝑡)𝑇𝜂(𝑡)²𝐷𝜂(𝑡)𝑃𝜂(𝑡)+𝑐(𝑡)𝑇𝜂(𝑡)²𝐷(∞)𝑃(∞). (3.8) Next, by taking 𝐼₂(𝜂(𝑡))−𝐼₁(𝜂(𝑡)) with further arrangement yields 𝐷𝜂(𝑡)+12𝜋i_𝐽1𝜂(𝑡)(𝑠)−𝑐(𝑡)−𝜂𝑐(𝑠)𝑇𝜂(𝑡)²𝜂(𝑡)(𝑠)𝐷𝜂(𝑠)||(∞)(𝑡)−𝜂𝑑𝑠=𝐷+𝑐𝑇𝜂²𝐷(∞)𝑃(∞). (3.9) Then multiplying (3.9) with 𝑇(𝜂(𝑡)) and |𝜂^′(𝑡)|, subsequently yields (3.3).

Theorem 3.2. The kernel 𝐾(𝑠,𝑡) is continuous with 𝐾()=1𝑡,𝑡𝜂2𝜋Im^′′(𝑡)𝜂^′(𝑡)−1𝑐2𝜋i^′(𝑡)𝑐(𝑡). (3.10)

Proof. Let the kernel 𝐾(𝑠,𝑡) be written as 𝐾(𝑠,𝑡)=𝐾₁(𝑠,𝑡)+𝐾₂(𝑠,𝑡), (3.11) where 𝐾₁()=1𝑠,𝑡𝜂2𝜋i^′(𝑡)𝜂(𝑡)(𝑠)−−𝜂𝜂^′(𝑡)𝜂(𝑡)(𝑠),𝐾−𝜂₂()=1𝑠,𝑡−𝑐(𝑡)2𝜋i𝑐(𝑠)𝜂′(𝑡)𝜂(𝑡)(𝑠)+−𝜂𝜂^′(𝑡)𝜂(𝑡)(𝑠)=1−𝜂2𝜋i𝜂^′(𝑡)𝑐(𝑠)𝑐(𝑠)−𝑐(𝑡)𝜂(𝑡)(𝑠).−𝜂 (3.12) Notice that 𝐾₁(𝑠,𝑡) is the classical Neumann kernel with 𝐾₁(𝑡,𝑡)=1/(2𝜋)Im((𝜂^′′(𝑡))/(𝜂^′(𝑡))). Now for 𝐾₂(𝑠,𝑡), as we take the limit 𝑠→𝑡 we have, 𝐾₂()=1𝑡,𝑡2𝜋ilim_𝑠→𝑡𝜂^′(𝑡)𝑐(𝑠)lim_𝑠→𝑡𝑐(𝑠)−𝑐(𝑡)𝜂(𝑡)(𝑠)1−𝜂=−𝑐2𝜋i^′(𝑡)𝑐(𝑡). (3.13) Hence, by combining 𝐾₁(𝑡,𝑡) and 𝐾₂(𝑡,𝑡), we obtain 𝐾()=1𝑡,𝑡𝜂2𝜋Im^′′(𝑡)𝜂^′(𝑡)−1i𝑐′(𝑡)𝑐(𝑡). (3.14)

Note that when 𝑐(𝑡)=1, the kernel 𝐾(𝑠,𝑡) reduces to the classical Neumann kernel.

We define the Fredholm integral operator 𝐊 by (𝑡)=1𝐊𝜐2𝜋i_𝐽𝐾(𝑠,𝑡)𝜐(𝑠)𝑑𝑠,𝑡∈𝐽. (3.15) Hence, (3.3) becomes (𝐈+𝐊)𝜙(𝑡)=𝜈(𝑡). (3.16) The solvability of the integral equation (3.16) depends on the possibility of 𝜆=−1 being an eigenvalue of the kernel 𝐾(𝑠,𝑡). For the numerical examples considered in this paper, 𝜆=−1 is always an eigenvalue of the kernel 𝐾(𝑠,𝑡). Although there is no theoretical proof yet, numerical evidence suggests that 𝜆=−1 is an eigenvalue of 𝐾(𝑠,𝑡). If the multiplicity of the eigenvalue 𝜆=−1 is 𝑚, then one need to add 𝑚 conditions to the integral equation to ensure the integral equation is uniquely solvable.

4. Exterior Unit Disk with Circular Slits

The canonical region 𝑈_𝑑 is the exterior unit disk along with 𝑚−1 arcs of circles. We assume that 𝑤=Φ(𝑧) maps the curve Γ₁ onto the unit circle |𝑤|=1, the curve Γ_𝑗, where 𝑗=2,3,…,𝑚, onto circular slit on the circle |𝑤|=𝑅_𝑗, where 𝑅₂,𝑅₃,…,𝑅_𝑚 are unknown real constants. The circular slits are traversed twice. The boundary values of the mapping function Φ are given by Φ𝜂(𝑡)=𝑅(𝑡)𝑒^i𝜃(𝑡), (4.1) where 𝜃(𝑡) represents the boundary correspondence function and 𝑅(𝑡)=(1,𝑅₂,…,𝑅_𝑚). By differentiating (4.1) with respect to 𝑡 and dividing the result obtained by its modulus, we have Φ^′𝜂(𝑡)𝜂^′(𝑡)Φ^′𝜂(𝑡)𝜂^′(𝑡𝜃)=isign^′(𝑡)𝑒^i𝜃(𝑡). (4.2) Using the fact that unit tangent 𝑇(𝜂(𝑡))=𝜂′(𝑡)/|𝜂′(𝑡)| and 𝑒^i𝜃(𝑡)=Φ(𝜂(𝑡))/𝑅(𝑡), it can be shown that Φ𝜂(𝑡)=sign𝜃^′(𝑡)𝑅(𝑡)i𝑇𝜂(𝑡)Φ′𝜂(𝑡)Φ′𝜂(𝑡). (4.3) Boundary relationship (4.3) is useful for computing the boundary values of Φ(𝑧) provided 𝜃^′(𝑡),𝑅(𝑡), and Φ′(𝜂(𝑡)) are all known. By taking logarithmic derivative on (4.1), we obtain 𝜂^′(𝑡)Φ^′𝜂(𝑡)Φ𝜂(𝑡)=i𝜃^′(𝑡). (4.4) The mapping function Φ(𝑧) can be uniquely determined by assuming Φ(∞)=∞,𝑐=Φ^′(∞)=lim_𝑧→∞Φ(𝑧)𝑧>0. (4.5) Thus, the mapping function can be expressed as [12] Φ(𝑧)=𝑐𝑧−𝑧₁𝑒^𝐹(𝑧), (4.6) where 𝐹(𝑧) is an analytic function and 𝐹(∞)=0. By taking logarithm on both sides of (4.6), we obtain 𝐹𝜂(𝑡)𝑅(𝑡)=ln𝑐+i𝜃−log𝜂(𝑡)−𝑧₁. (4.7) Hence (4.7) satisfies boundary values (2.15) in Theorem 2.1 with 𝐴(𝑡)=1, ℎ(𝑡)=1ln𝑐𝑅,ln²𝑐𝑅,…,ln^𝑚𝑐,𝛾(𝑡)=−ln𝜂(𝑡)−𝑧₁. (4.8) Hence, the values of 𝑅_𝑗 can be calculated by 𝑅_𝑗=𝑒^{ℎ𝑗−ℎ1}for𝑗=1,2,…,𝑚. (4.9) To find 𝜃′(𝑡), we began by differentiating (4.7) and comparing with (4.4) which yields i𝜃^′(𝑡)𝜂(𝑡)(𝑡)−(𝑡)=𝐹′𝜂′𝜂′𝜂(𝑡)−𝑧₁. (4.10) In view of 𝐴=𝜂′(𝑡) and letting 𝑓(𝑧)=𝐹′(𝑧)−1/(𝑧−𝑧₁), where 𝑓(𝑧) is analytic function in Ω⁻, (4.10) becomes 𝐴𝑓𝜂(𝑡)=i𝜃′(𝑡). (4.11) By [18, Theorem 2(c)], we obtain (𝐈−𝐍)𝜃′=0 which implies 𝐈+𝐍^∗𝜃^′=0. (4.12) However, this integral equation is not uniquely solvable according to [18, Theorem  12]. To overcome this, since the image of the curve Γ₁ is clockwise oriented and the images of the curves Γ_𝑗,𝑗=2,3,…,𝑚 are slits so we have 𝜃₁(2𝜋)−𝜃₁(0)=−2𝜋 and 𝜃_𝑗(2𝜋)−𝜃_𝑗(0)=0, which implies 𝐉𝜃^′(𝑡)=ℎ(𝑡)=(−1,0,…,0). (4.13) By adding this condition to (4.12), the unknown function 𝜃^′(𝑡) is the unique solution of the integral equation 𝐈+𝐍^∗+𝐉𝜃′(𝑡)=ℎ(𝑡). (4.14) Next, the presence of sign(𝜃′(𝑡)) in (4.3) can be eliminated by squaring both sides of (4.3), that is, Φ𝜂(𝑡)²=−𝑅(𝑡)²𝑇𝜂(𝑡)²Φ^′𝜂(𝑡)²Φ^′𝜂(𝑡)². (4.15) Upon comparing (4.15) with (3.2), this leads to a choice of 𝑃(𝜂(𝑡))=Φ(𝜂(𝑡))², 𝑃(∞)=∞, 𝐷(𝜂(𝑡))=Φ′(𝜂(𝑡)), 𝐷(∞)=𝑐, 𝑐(𝑡)=−𝑅(𝑡)². Hence, 𝜙(𝑡)=Φ^′𝜂(𝑡)𝜂^′(𝑡) (4.16) satisfies the integral equation (3.16) with 𝜈(𝑡)=Φ^′(∞)𝜂^′(𝑡). (4.17) Numerical evidence shows that 𝜆=−1 is an eigenvalue of 𝐾(𝑠,𝑡) of multiplicity 𝑚, which means one needs to add 𝑚 conditions. Since Φ(𝑧) is assumed to be single-valued, it is also required that the unknown mapping function Φ′(𝑧) satisfies [4] _𝐽𝑗𝜙(𝑡)𝑑𝑡=_Γ𝑗Φ′𝜂𝑑𝜂=0,𝑗=1,2,…,𝑚, (4.18) that is, 𝐉𝜙=0. (4.19) Then, 𝜙(𝑡) is the unique solution of the following integral equation (𝐈+𝐊+𝐉)𝜙(𝑡)=𝜈(𝑡). (4.20) By obtaining 𝜙(𝑡), the derivatives of the mapping function, Φ′(𝑡) can be found using Φ^′(𝑡)=𝜙(𝑡)𝜂^′(𝑡). (4.21) By obtaining the values of 𝑅(𝑡), 𝜃′(𝑡) and Φ′(𝜂(𝑡)), the boundary value of Φ(𝜂(𝑡)) can be calculated by using (4.3).

5. Annulus with Circular Slits

The canonical region 𝑈_𝑎 consists of an annulus centered at the origin together with 𝑚−2 circular arcs. We assume that Φ(𝑧) maps the curve Γ₁ onto the unit circle |Φ|=1, the curve Γ_𝑚 onto the circle |Φ|=𝑅_𝑚 and Γ_𝑗 onto circular slit |Φ|=𝑅_𝑗, where 𝑗=2,3,…,𝑚−1. The slit are traversed twice. The boundary values of the mapping function Φ are given by Φ𝜂(𝑡)=𝑅(𝑡)𝑒^i𝜃(𝑡), (5.1) where 𝜃(𝑡) represents the boundary correspondence function and 𝑅(𝑡)=(1,𝑅₂,…,𝑅_𝑚) is a piecewise real constant function. By using the same reasoning as in Section 4, we get Φ𝜂_𝑗(𝑡)=sign𝜃^′(𝑡)𝑅_𝑗(𝑡)i𝑇𝜂_𝑗(𝑡)Φ′𝜂_𝑗(𝑡)Φ′𝜂_𝑗(𝑡)𝜂_′,(𝑡)Φ^′𝜂(𝑡)Φ𝜂(𝑡)=i𝜃^′(𝑡). (5.2)(5.3) The mapping function Φ(𝑧) can be uniquely determined by assuming 𝑐=Φ(∞)>0. Thus, the mapping function can be expressed as [12] Φ(𝑧)=𝑐𝑧−𝑧_𝑚𝑧−𝑧₁𝑒^𝐹(𝑧). (5.4) By taking logarithm on both sides of (5.4), we obtain 𝐹𝜂(𝑡)𝑅(𝑡)=ln𝑐𝜂(𝑡)+i𝜃−log−𝑧_𝑚𝜂(𝑡)−𝑧₁. (5.5) Hence, (5.5) satisfies boundary values (2.15) in Theorem 2.1 with 𝐴(𝑡)=1, ℎ(𝑡)=1ln𝑐𝑅,ln²𝑐𝑅,…,ln^𝑚𝑐(𝑡)𝜂(𝑡),𝛾=−ln−𝑧_𝑚𝜂(𝑡)−𝑧₁. (5.6) By obtaining ℎ_𝑗, the values of 𝑅_𝑗 can be computed by 𝑅_𝑗=e^{ℎ𝑗−ℎ1},𝑗=2,3,…,𝑚. (5.7) By differentiating (5.5) and comparing with (5.3) yields i𝜃^′(𝑡)=𝐹^′𝜂(𝑡)𝜂^′(𝑡)+𝜂^′(𝑡)𝜂(𝑡)−𝑧_𝑚−𝜂^′(𝑡)𝜂(𝑡)−𝑧₁. (5.8) In view of 𝐴=𝜂′(𝑡) and 𝑓(𝜂(𝑡))=𝐹′(𝜂(𝑡))+1/(𝜂(𝑡)−𝑧_𝑚)−1/(𝜂(𝑡)−𝑧₁), where 𝑓(𝑧) is analytic in Ω⁻, (5.8) is equivalent to 𝐴𝑓𝜂(𝑡)=i𝜃^′(𝑡). (5.9) By [18, Theorem 2(c)], we obtain 𝐈+𝐍^∗𝜃^′(𝑡)=0. (5.10) Note that the image of the curve Γ₁ is counterclockwise oriented, Γ_𝑚 is clockwise oriented and the images of the curves Γ_𝑗,𝑗=2,3,…,𝑚−1 are slits so we have 𝜃₁(2𝜋)−𝜃₁(0)=2𝜋, 𝜃_𝑚(2𝜋)−𝜃_𝑚(0)=−2𝜋 and 𝜃_𝑗(2𝜋)−𝜃_𝑗(0)=0, which implies 𝐉𝜃^′(𝑡)=ℎ(𝑡)=(1,0,…,−1). (5.11) Hence, the unknown function 𝜃′(𝑡) is the unique solution of the integral equation 𝐈+𝐍^∗+𝐉𝜃^′(𝑡)=ℎ(𝑡). (5.12) Next, the presence of sign(𝜃′(𝑡)) in (5.2) can be eliminated by squaring both sides of the equation, that is, Φ𝜂(𝑡)²=−𝑅(𝑡)²𝑇𝜂(𝑡)²Φ^′𝜂(𝑡)²Φ^′𝜂(𝑡)². (5.13) Comparing (5.13) with (3.2) leads to a choice of 𝑃(𝜂(𝑡))=Φ(𝜂(𝑡))², 𝑃(∞)=𝑐², 𝐷(𝜂(𝑡))=Φ′(𝜂(𝑡)), 𝐷(∞)=0, 𝑐(𝑡)=−𝑅(𝑡)². Hence, 𝜙(𝑡)=Φ^′𝜂(𝑡)𝜂^′(𝑡) (5.14) satisfies the integral equation (3.16) with 𝜈(𝑡)=(0,…,0). (5.15) Numerical evidence shows that 𝜆=−1 is an eigenvalue of 𝐾(𝑠,𝑡) of multiplicity 𝑚+1, which implies one needs to add 𝑚+1 conditions. Since Φ(𝑧) is assumed to be single-valued, hence it is also required that the unknown mapping function Φ′(𝑧) satisfies [4] _𝐽𝑗𝜙(𝑠)𝑑𝑠=_Γ𝑗Φ^′𝜂𝑑𝜂=0,𝑗=1,2,…,𝑚, (5.16) that is, 𝐉𝜙=0. (5.17) Since we assume the mapping function Φ(𝑧) can be uniquely determined by 𝑐=Φ(∞)>0, hence by [20] Φ(∞)1=𝑐=2𝜋_𝐽𝜂^′(𝑡)𝜃^′(𝑡)𝜂−𝑧₁𝜙(𝑡)𝑑𝑡. (5.18) If we define the Fredholm operator 𝐆 as (𝑠)=1𝐆𝜇2𝜋_𝐽𝜂′(𝑡)𝜃^′(𝑡)𝜂−𝑧₁𝜇(𝑡)𝑑𝑡, (5.19) then 𝜙(𝑡) is the unique solution of the following integral equation: (𝐈+𝐊+𝐉+𝐆)𝜙(𝑡)=𝑐. (5.20) By obtaining 𝜙(𝜂(𝑡)), the derivatives of the mapping function, Φ′(𝑡) can be obtained by Φ^′(𝑡)=𝜙𝜂(𝑡)𝜂^′(𝑡). (5.21)

6. Circular Slits

The canonical region 𝑈_𝑐 consists of 𝑚 slits along the circle |Φ|=𝑅_𝑗 where 𝑗=1,2,…,𝑚 and 𝑅₁,𝑅₂,…,𝑅_𝑚 are unknown real constants. The boundary values of the mapping function Φ are given by Φ𝜂(𝑡)=𝑅(𝑡)𝑒^i𝜃(𝑡), (6.1) where 𝜃(𝑡) represents the boundary correspondence function and 𝑅(𝑡)=(1,𝑅₂,…,𝑅_𝑚). By using the same reasoning as in Section 4, we get Φ𝜂_𝑗(𝑡)=sign𝜃^′(𝑡)𝑅_𝑗(𝑡)i𝑇𝜂_𝑗(𝑡)Φ^′𝜂_𝑗(𝑡)Φ^′𝜂_𝑗(𝑡),𝜂^′(𝑡)Φ^′𝜂(𝑡)Φ𝜂(𝑡)=i𝜃^′(𝑡). (6.2)(6.3) The mapping function Φ(𝑧) can be uniquely determined by assuming Φ(𝛽)=0, Φ(∞)=∞ and Φ′(∞)=1. Thus, the mapping function Φ can be expressed as [12] Φ(𝑧)=𝑧−𝛽𝑒^𝐹(𝑧). (6.4) By taking logarithm on both sides of (6.4), we obtain 𝐹𝜂(𝑡)=ln𝑅(𝑡)+i𝜃−log𝜂(𝑡)−𝛽. (6.5) Hence, (6.5) satisfies boundary values in Theorem 2.1 with 𝐴(𝑡)=1, ℎ(𝑡)=ln𝑅₁,ln𝑅₂,…,ln𝑅_𝑚,𝛾(𝑡)=−ln𝜂(𝑡)−𝛽. (6.6) By obtaining ℎ_𝑗, the values of 𝑅_𝑗 can be obtained by 𝑅_𝑗=𝑒^ℎ𝑗. (6.7) By differentiating (6.5) and comparing with (6.3) yields i𝜃^′(𝑡)=𝐹′𝜂(𝑡)𝜂^′(𝑡)+𝜂^′(𝑡)𝜂(𝑡).−𝛽 (6.8) In view of 𝐴=𝜂′(𝑡), 𝑓(𝑡)=𝐹′(𝜂(𝑡)) and 𝑔(𝑡)=1/(𝜂(𝑡)−𝛽) where 𝑓(𝑧) is analytic in Ω⁻ and 𝑔(𝑧) is analytic in Ω⁺, we rewrite (6.8) as 𝐴𝑓𝜂(𝑡)=i𝜃^′(𝑡)−𝐴𝑔𝜂(𝑡). (6.9) Let 𝐴𝑔(𝑡)=𝜓+i𝜑. Then by [18, Theorems 2(c) and 2(d)], we obtain 𝐈+𝐍^∗𝜃^′(𝑡)−𝜑(𝑡)=𝐌𝜓(𝑡),𝐈−𝐍^∗𝜑(𝑡)=𝐌𝜓(𝑡). (6.10)(6.11) Subtracting (6.11) from (6.10), we get 𝐈+𝐍^∗𝜃^′(𝑡)=2𝜑(𝑡). (6.12) Since the images of the curve Γ₁,Γ₂,…,Γ_𝑚 are slits, we have 𝜃_𝑗(2𝜋)−𝜃_𝑗(0)=0. Thus 𝐉𝜃^′(𝑡)=(0,0,…,0). (6.13) Hence the unknown function 𝜃′(𝑡) is the unique solution of the integral equation 𝐈+𝐍^∗+𝐉𝜃^′(𝑡)=2𝜑(𝑡), (6.14) where 𝜑(𝑡)𝐴(𝑡)𝑔𝜂(𝑡)𝜂=Im=Im^′(𝑡)𝜂(𝑡).−𝛽 (6.15) By squaring both sides of (6.2) and dividing the result by (𝜂(𝑡)−𝛽)², we obtain Φ𝜂(𝑡)²𝜂(𝑡)−𝛽²𝑅(𝑡)=−²𝜂(𝑡)−𝛽²𝑇𝜂(𝑡)²Φ^′𝜂(𝑡)²Φ^′𝜂(𝑡)². (6.16) Upon comparing (6.16) with (3.2) leads to a choice of 𝑃(𝜂(𝑡))=Φ(𝜂(𝑡))²/(𝜂(𝑡)−𝛽)², 𝑃(∞)=1, 𝐷(𝜂(𝑡))=Φ′(𝜂(𝑡)), 𝐷(∞)=1, 𝑐(𝑡)=−𝑅(𝑡)²/(𝜂(𝑡)−𝛽²). Hence, 𝜙(𝑡)=Φ^′𝜂(𝑡)𝜂^′(𝑡) (6.17) satisfies the integral equation (3.16) with 𝜈(𝑡)𝑅(𝑡)=−²𝜂(𝑡)−𝛽²𝜂^′(𝑡)+𝜂^′(𝑡). (6.18) Numerical evidence shows that 𝜆=−1 is an eigenvalue of 𝐾(𝑠,𝑡) of multiplicity 𝑚, thus one needs to add 𝑚 conditions. Since Φ(𝑧) is assumed to be single-valued, it is also required that the unknown mapping function Φ′(𝑧) satisfies [4] _𝐽𝑗𝜙(𝑡)𝑑𝑡=_Γ𝑗Φ^′𝜂𝑑𝜂=0,𝑗=1,2,…,𝑚, (6.19) that is, 𝐉𝜙=0. (6.20) Then, 𝜙(𝑡) is the unique solution of the integral equation (𝐈+𝐊+𝐉)𝜙(𝑡)=𝜈(𝑡). (6.21) By obtaining 𝜙(𝜂(𝑡)), the derivatives of the mapping function, Φ′(𝑡) can be obtained by (𝑡)=𝜙𝜂(𝑡)Φ′𝜂^′(𝑡). (6.22)

7. Radial Slits

The canonical region 𝑈_𝑟 consists of 𝑚 slits along 𝑚 segments of the rays with arg(Φ)=𝜃_𝑗,𝑗=1,2,…,𝑚. Then, the boundary values of the mapping function Φ are given by Φ𝜂(𝑡)=𝑟(𝑡)𝑒^i𝜃(𝑡)=𝑒^𝑅(𝑡)𝑒^i𝜃(𝑡), (7.1) where the boundary correspondence function 𝜃(𝑡)=(𝜃₁,𝜃₂,…,𝜃_𝑚) now becomes real constant function and 𝑅(𝑡) is an unknown function. By taking logarithmic derivative on (7.1), we obtain (𝑡)Φ𝜂′^′𝜂(𝑡)Φ𝜂(𝑡)=𝑅^′(𝑡). (7.2) It can be shown that the mapping function Φ(𝑧) can be determined using Φ𝜂(𝑡)=𝜂^′(𝑡)Φ^′𝜂(𝑡)𝑅^′(𝑡). (7.3) The mapping function Φ(𝑧) can be uniquely determined by assuming Φ(𝛽)=0, Φ(∞)=∞ and Φ′(∞)=1. Thus, the mapping function Φ(𝑧) can be expressed as [12] Φ(𝑧)=𝑧−𝛽𝑒^i𝐹(𝑧). (7.4) By taking logarithm on both sides of (7.4) and multiplying the result by −i, we obtain 𝐹𝜂(𝑡)=𝜃−i𝑅(𝑡)+ilog𝜂(𝑡)−𝛽. (7.5) Hence, (7.5) satisfies boundary values in Theorem 2.1 with 𝐴(𝑡)=1, ℎ(𝑡)=𝜃₁,𝜃₂,…,𝜃_𝑚,𝛾(𝑡)=−Arg𝜂(𝑡)−𝛽. (7.6) By obtaining ℎ(𝑡), one can obtain the values of 𝜃(𝑡) by 𝜃_𝑗=ℎ_𝑗for𝑗=1,2,…,𝑚. (7.7) Then, by differentiating (7.5) and comparing with (7.2) yields i𝑅^′(𝑡)=−𝐹^′𝜂(𝑡)𝜂^′(𝑡)𝜂+i^′(𝑡)𝜂(𝑡).−𝛽 (7.8) In view of 𝐴=𝜂′(𝑡), 𝑓(𝑡)=𝐹′(𝜂(𝑡)) and 𝑔(𝑡)=i/(𝜂(𝑡)−𝛽) where 𝑓(𝑧) is analytic in Ω⁻ and 𝑔(𝑧) is analytic in Ω⁺, we rewrite (7.8) as 𝐴𝑓𝜂(𝑡)=i𝑅′(𝑡)−𝐴𝑔𝜂(𝑡). (7.9) Let 𝐴𝑔(𝑡)=𝜓+i𝜑. Then by [18, Theorems 2(c) and 2(d)], we obtain 𝐈+𝐍^∗𝜑(𝑡)−𝑅^′(𝑡)=−𝐌𝜓(𝑡),𝐈−𝐍^∗𝜑(𝑡)=𝐌𝜓(𝑡). (7.10) Adding these equations, we get 𝐈+𝐍^∗𝑅^′(𝑡)=2𝜑(𝑡). (7.11) Since the images of the curve Γ₁,Γ₂,…,Γ_𝑚 are slits, we have 𝑅_𝑗(2𝜋)−𝑅_𝑗(0)=0. Therefore 𝐉𝑅^′(𝑡)=(0,0,…,0). (7.12) Hence, the unknown function 𝑅′(𝑡) is the unique solution of the integral equation 𝐈+𝐍^∗+𝐉𝑅^′(𝑡)=2𝜑(𝑡), (7.13) where 𝜑(𝑡)=Im𝐴(𝑡)𝑔𝜂(𝑡)=Imi𝜂^′(𝑡)𝜂(𝑡).−𝛽 (7.14) The boundary relationship (7.3) can be rewritten as Φ^′𝜂(𝑡)=±𝑒^i𝜃𝑗𝑇𝜂(𝑡)Φ^′𝜂(𝑡). (7.15) Squaring both sides of (7.15) yields Φ^′𝜂(𝑡)=𝑒^2i𝜃𝑗𝑇𝜂(𝑡)²Φ^′𝜂(𝑡). (7.16) Upon comparing (7.16) with (3.1) leads to a choice of 𝑃(𝜂(𝑡))=1, 𝑃(∞)=1, 𝐷(𝜂(𝑡))=Φ′(𝜂(𝑡)), 𝐷(∞)=1, 𝑐(𝑡)=𝑒^i2𝜃𝑗. Hence, 𝜙(𝑡)=Φ^′𝜂(𝑡)𝜂^′(𝑡) (7.17) satisfies the integral equation (3.16) with 𝜈(𝑡)=𝑒^{2i𝜃𝑗(𝑡)}𝜂^′(𝑡)+𝜂^′(𝑡). (7.18) Numerical evidence shows that 𝜆=−1 is an eigenvalue of 𝐾(𝑠,𝑡) of multiplicity 𝑚, which suggests one needs to add 𝑚 conditions. Since Φ(𝑧) is assumed to be single-valued, hence it is also required that the unknown mapping function Φ′(𝑧) satisfies [4] _𝐽𝑗𝜙(𝑡)𝑑𝑡=_Γ𝑗Φ^′𝜂𝑑𝜂=0,𝑗=1,2,…,𝑚, (7.19) that is, 𝐉𝜙=0. (7.20) Then, 𝜙(𝑡) is the solution of the following integral equation (𝐈+𝐊+𝐉)𝜙(𝑡)=𝜈(𝑡). (7.21) By obtaining 𝜙(𝜂(𝑡)), the derivatives of the mapping function Φ′(𝑡) can be found using Φ^′(𝑡)=𝜙𝜂(𝑡)𝜂^′(𝑡). (7.22)

8. Parallel Slits

The canonical region 𝑈_𝑝 consists of a 𝑚 parallel straight slits on the 𝑤-plane. Let 𝐵=𝑒^{i(𝜋/2−𝜃)}, then the boundary values of the mapping function Φ are given by 𝐵Φ𝜂(𝑡)=𝑅(𝑡)+i𝛿(𝑡), (8.1) where 𝜃 is the angle of intersection between the lines Re[𝐵Φ]=𝑅_𝑗 and the real axis. 𝑅(𝑡)=(𝑅₁(𝑡),𝑅₂(𝑡),…,𝑅_𝑀(𝑡)) is a piecewise real constant function and 𝛿(𝑡) is an unknown function. It can be shown that (8.1) can be written as 𝐵Φ^′𝜂(𝑡)𝜂^′(𝑡)=i𝛿^′(𝑡). (8.2) The mapping function Φ(𝑧) can be uniquely determined by assuming Φ(∞)=∞ and lim_𝑧→∞Φ(𝑧)−𝑧=0. Thus, the mapping function Φ can be expressed as [12] Φ(𝑧)=𝑧+𝐵𝐹(𝑧), (8.3) where 𝐹(𝑧) is an analytic function with 𝐹(∞)=0. By multiplying both sides of (8.3) with 𝐵, we obtain 𝐹𝜂(𝑡)=𝐵Φ𝜂(𝑡)−𝐵𝜂(𝑡). (8.4) Hence, (8.4) satisfies the boundary values in Theorem 2.1 with 𝐴(𝑡)=1, ℎ(𝑡)=𝑅₁,𝑅₂,…,𝑅_𝑚𝛾(𝑡)=−𝐵𝜂(𝑡). (8.5) Differentiating (8.4) and comparing the result with (8.2) yield i𝛿^′(𝑡)=𝐵𝜂^′(𝑡)+𝜂^′(𝑡)𝐹^′𝜂(𝑡). (8.6) In view of 𝐴=𝜂′(𝑡), 𝑓(𝑡)=𝐹′(𝜂(𝑡)) and 𝑔(𝑡)=𝐵, where 𝑓(𝑧) is analytic in Ω⁻ and 𝑔(𝑧) is analytic in Ω⁺, we rewrite (8.6) as 𝐴𝑓𝜂(𝑡)=i𝛿^′(𝑡)−𝐴𝑔𝜂(𝑡). (8.7) Assuming 𝐴𝑔(𝑡)=𝜓+i𝜑, then by [18, Theorems 2(c) and 2(d)], we obtain 𝐈+𝐍^∗𝛿^′(𝑡)−𝜑(𝑡)=𝐌𝜓(𝑡),𝐈−𝐍^∗𝜑(𝑡)=𝐌𝜓(𝑡). (8.8) These two equations yields 𝐈+𝐍^∗𝛿^′(𝑡)=2𝜑(𝑡). (8.9) Note that, the images of the curves Γ₁,Γ₂,…,Γ_𝑚 are slits, so we have 𝛿_𝑗(2𝜋)−𝛿_𝑗(0)=0, which implies 𝐉𝛿^′(𝑡)=(0,0,…,0). (8.10) Hence, the unknown function 𝛿^′(𝑡) is the unique solution of the integral equation 𝐈+𝐍^∗+𝐉𝛿^′(𝑡)=2𝜑(𝑡), (8.11) where 𝜑(𝑡)=Im𝐴(𝑡)𝑔𝜂(𝑡)=Im𝜂^′(𝑡)𝐵. (8.12) From (8.1), we introduce an analytic function 𝜗(𝜂(𝑡)) such that 𝜗𝜂(𝑡)=𝑒^𝐹(𝑧)=𝑒^{−𝐵𝜂(𝑡)}𝑒^{𝑅(𝑡)+i𝛿(𝑡)}, (8.13) where 𝜗(∞)=1. By differentiating (8.13) with respect to 𝑡, we obtain 𝜗^′𝜂(𝑡)𝜂^′(𝑡)=i𝛿^′(𝑡)−𝐵𝜂^′(𝑡)𝜗𝜂(𝑡). (8.14) Let 𝜎(𝑡) be an analytic function such that it has the following representation 𝜎𝜂(𝑡)=𝜗^′𝜂(𝑡)+𝐵𝜗𝜂(𝑡), (8.15) where 𝜎(∞)=𝐵. From (8.13)–(8.15) it can be shown that, the function 𝜗(𝜂(𝑡)) can be rewritten as 𝜗𝜂(𝑡)=𝑒^𝑅𝑗𝑒^{(−Re[𝐵𝜂(𝑡)])}sign𝛿^′(𝑡)i𝑇𝜂(𝑡)𝜎𝜂(𝑡)𝜎𝜂(𝑡). (8.16) By squaring both sides of (8.16), the sign 𝛿′(𝑡) is eliminated, that is, 𝜗𝜂(𝑡)²=−𝑒^2𝑅𝑗𝑒^{(−2Re[𝐵𝜂(𝑡)])}𝑇𝜂(𝑡)²𝜎𝜂(𝑡)²𝜎(𝜂(𝑡))². (8.17) Comparing (8.17) with (3.2) leads to a choice of 𝑃(𝑧)=𝜗(𝑧)², 𝑃(∞)=1, 𝐷(𝑧)=𝜎(𝑧), 𝐷(∞)=𝐵, 𝑐(𝑡)=−𝑒^2𝑅𝑗𝑒^{(−2Re[𝐵𝜂(𝑡)])}. Hence, 𝜙(𝑡)=𝜎𝜂(𝑡)𝜂^′(𝑡) (8.18) satisfies the integral equation (3.16) with 𝜈(𝑡)=−𝑒^2𝑅𝑗𝑒^{(−2Re[𝐵𝜂(𝑡)])}𝜂′(𝑡)𝐵+𝐵𝜂′(𝑡). (8.19) Numerical evidence shows that 𝜆=−1 is an eigenvalue of 𝐾(𝑠,𝑡) of multiplicity 𝑚, which suggests one needs to add 𝑚 conditions. Since 𝑒^{𝐵𝜂𝑗(𝑡)}𝜗(𝜂_𝑗(𝑡))|₀^2𝜋=0, hence we can have 𝑚 additional conditions for the integral equation above as in the following: _𝐽𝑗𝑑𝑒𝑑𝑡^{𝐵𝜂𝑗(𝑡)}𝜗_𝑗𝜂_𝑗(𝑡)𝑑𝑡=0,_02𝜋𝑒^{𝐵𝜂𝑗(𝑡)}𝜗^′_𝑗𝜂_𝑗(𝑡)+𝐵𝜗𝜂_𝑗(𝑡)𝜂^′(𝑡)𝑑𝑡=0,_02𝜋𝑒^{𝐵𝜂𝑗(𝑡)}𝜎𝜂_𝑗(𝑡)𝜂′(𝑡)𝑑𝑡=0,_02𝜋𝑒^{𝐵𝜂𝑗(𝑡)}𝜙𝜂_𝑗(𝑡)𝑑𝑡=0,𝑞=1,2,…,𝑚. (8.20) We define Fredholm operator 𝐋 as 𝐋𝜇(𝑠)=_𝐽𝑗𝑒^{𝐵𝜂𝑗(𝑡)}𝜇(𝑡)𝑑𝑡. (8.21) Then, 𝜙(𝑡) is the solution of the following integral equation: (𝐈+𝐊+𝐋)𝜙(𝑡)=𝜈(𝑡). (8.22) Hence, by obtaining 𝜙(𝜂(𝑡)), the function 𝜎(𝑡) can be found by 𝜎(𝑡)=𝜙𝜂(𝑡)𝜂^′(𝑡). (8.23) This allows the value for 𝜗(𝜂(𝑡)) to be calculated from (8.16), which in turn allows the boundary values for the mapping function Φ(𝜂(𝑡)) to be calculated by Φ𝜂(𝑡)=𝜂(𝑡)+𝐵log𝜗𝜂(𝑡). (8.24)

9. Numerical Examples

Since the boundaries Γ_𝑗 are parameterized by 𝜂(𝑡) which are 2𝜋-periodic functions, the reliable method to solve the integral equations are by means of Nyström method with trapezoidal rule [21]. Each boundary will be discretized by 𝑛 number of equidistant points. The resulting linear systems are then solved by using Gaussian elimination. For numerical examples, we choose regions with connectivities one, two, three and four. For the region with connectivity one, we compare our result with the analytic solution given in [12]. All the computations were done by using MATLAB R2008a software.

Example 9.1. Consider an unbounded region Ω⁻ bounded by a unit circle Γ₁(𝑡)=𝑒^−i𝑡,(0≤𝑡≤2𝜋). (9.1) We choose the special point 𝛽=2.5+1.5i. The exact mapping function for 𝑈_𝑐,𝑈_𝑟, and 𝑈_𝑝 are given respectively by [12] 𝑤_𝑐=𝑧−𝛽+𝛽−𝑧1−,𝑤𝛽𝑧_𝑟=𝑧−𝛽−𝑧−𝛽,𝑤𝛽𝑧_𝑝𝑒=𝑧+^2i𝜃𝑧. (9.2) For this example, we compare the error for each boundary value between our method and the exact mapping function. See Table 1 for Error Norm of ||𝑤_𝑗−𝑤_𝑗||_∞.

Table 1 

Error norm ‖𝑤_𝑗−𝑤_𝑗‖_∞ for Example 9.1.

Example 9.2. Consider an unbounded region Ω⁻ bounded by a circle and an ellipse Γ₁(𝑡)=2+i+𝑒^−i𝑡,(),Γ0≤𝑡≤2𝜋₂(𝑡)=−2+cos𝑡−2isin𝑡,(0≤𝑡≤2𝜋). (9.3)
Figure 2 shows the region and its five canonical images by using our proposed method.

Figure 2 

The original region Ω− and its canonical images with 𝜃𝑝=𝜋/3 for parallel slits.

Example 9.3. Consider an unbounded region Ω⁻ bounded by 3 circles Γ₁(𝑡)=2+𝑒^−i𝑡,(),Γ0≤𝑡≤2𝜋₂(𝑡)=−1+i√3+0.5𝑒^−i𝑡,(),Γ0≤𝑡≤2𝜋₃(𝑡)=−1−i√3+1.5𝑒^−i𝑡,(0≤𝑡≤2𝜋). (9.4)
This example has also been considered in [6, 12]. Figure 3 shows the regions and its five canonical images by using our proposed method. See Table 3 for numerical comparison between our parameter values (see Table 2) those in [12]. Note that our method has considered exterior unit disk with slits as a canonical region while [12] has considered interior unit disk with slits. Thus, in computing the error for 𝑈_𝑑, we need to change the values for 𝑈_𝑑 to 1/|Φ(𝑧)|. See Table 4 for Error Norm of max_1≤𝑗≤3||𝑤_𝑗−𝑤_𝑗||_∞. We also compared the condition number of our linear system for each 𝑛 with [6, 12], see Figures 4 and 5. The results show that for our integral equations for finding Φ′(𝑧) that is, (4.20), (5.20), (6.21), (7.21) and (8.22), the condition numbers are almost constant except for (5.20). This is because the kernel involves 𝜃′(𝑡), which varies with the number of collocation points, 𝑛. However, this does not have any effect on the accuracy of the method.

Table 2 

The values for approximated parameters in Example 9.3 with 𝑛=256.

Table 3 

Error norm max_1≤𝑗≤3  ‖𝑅_𝑗−𝑅_𝑗‖_∞ of our method with [12] for Example 9.3.

Table 4 

Error norm max_1≤𝑗≤3  ‖𝑤_𝑗−𝑤_𝑗‖_∞ of our method with [12] for Example 9.3.

Figure 3 

The original region Ω− and its canonical images with 𝜃𝑝=𝜋/2 for parallel slits.

Figure 4 

Condition numbers of the matrices for our method for 𝑈𝑑,𝑈𝑎,𝑈𝑐,𝑈𝑟 and 𝑈𝑝.

Figure 5 

Condition number of the matrices for generalized Neumann kernel (G.N.K), adjoint generalized Neumann kernel (Adj.G. N. K), charge simulation for circular slits (C.S. circular) and charge simulation for radial slits (C.S. radial).

Example 9.4. Consider an unbounded region Ω⁻ of 4-connectivity with boundaries Γ₁(𝑡)=3+2i+𝑒^−i𝑡,(),Γ0≤𝑡≤2𝜋₂(𝑡)=−3+2i+𝑒^−i𝑡,(),Γ0≤𝑡≤2𝜋₃(𝑡)(),Γ=−3−2i+0.7cos𝑡−1.4isin𝑡,0≤𝑡≤2𝜋₄(𝑡)=3−2i+0.7cos𝑡−1.4isin𝑡,(0≤𝑡≤2𝜋). (9.5)(9.6)(9.7)(9.8)
Figure 6 shows the region and its five canonical images by using our proposed method.

Figure 6 

The original region Ω− and its canonical images with 𝜃𝑝=𝜋/2 for parallel slits.

10. Conclusion

In this paper, we have constructed a unified method for numerical conformal mapping of unbounded multiply connected regions onto canonical slit regions. The advantage of this method is that the integral equations are all linear which overcomes the nonlinearity problem encountered in [10]. From the numerical experiments, we can conclude that our method works on any finite connectivity with high accuracy. By computing the boundary values of the mapping function, the exterior points will be calculated by means of Cauchy’s integral formula.

Acknowledgments

This work was supported in part by the Malaysian Ministry of Higher Education (MOHE) through the Research Management Centre (RMC), Universiti Teknologi Malaysia (GUP Q.J130000.7126.01H75). This support is gratefully acknowledged. The authors would like to thank the anonymous referees for their valuable comments and suggestions on the manuscript which improved the presentation of the paper.