Abstract and Applied Analysis

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Volume 2012 |Article ID 970928 | https://doi.org/10.1155/2012/970928

Ali W. K. Sangawi, Ali H. M. Murid, M. M. S. Nasser, "Circular Slits Map of Bounded Multiply Connected Regions", Abstract and Applied Analysis, vol. 2012, Article ID 970928, 26 pages, 2012. https://doi.org/10.1155/2012/970928

Circular Slits Map of Bounded Multiply Connected Regions

Academic Editor: Karl Joachim Wirths
Received14 Dec 2011
Revised25 Jan 2012
Accepted26 Jan 2012
Published17 Apr 2012

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.

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 [2โ€“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 Wittichโ€™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 ๐‘“ 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 ๐‘“ on a multiply connected region with ๐‘“โ€ฒ, ๐œƒ๎…ž(๐‘ก), and |๐‘“|, where ๐œƒ is the boundary correspondence function.

The plan of the paper is as follows. Section 2 presents some auxiliary materials. Derivations of two integral equations related to ๐‘“โ€ฒ and ๐œƒ๎…ž(๐‘ก) are given in Sections 3 and 4, respectively. Section 5 presents a method to calculate the modulus of ๐‘“. 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

Let ฮฉ be a bounded multiply connected region of connectivity ๐‘€+1. The boundary ฮ“ consists of ๐‘€+1 smooth Jordan curves ฮ“๐‘—, ๐‘—=0,1,โ€ฆ,๐‘€, such that ฮ“ฬ‚๐‘—, ฬ‚๐‘—=1,โ€ฆ,๐‘€, lies in the interior of ฮ“0, where the outer curve ฮ“0 has counterclockwise orientation and the inner curves ฮ“ฬ‚๐‘—, ฬ‚๐‘—=1,โ€ฆ,๐‘€, have clockwise orientation. The positive direction of the contour โ‹ƒฮ“=๐‘€๐‘—=0ฮ“๐‘— is usually that for which ฮฉ is on the left as one traces the boundary (see Figure 1). The curve ฮ“๐‘˜ is parametrized by 2๐œ‹-periodic twice continuously differentiable complex function ๐‘ง๐‘˜(๐‘ก) with nonvanishing first derivative๐‘ง๎…ž๐‘˜(๐‘ก)=๐‘‘๐‘ง๐‘˜(๐‘ก)๐‘‘๐‘กโ‰ 0,๐‘กโˆˆ๐ฝ๐‘˜=[]0,2๐œ‹,๐‘˜=0,1,โ€ฆ,๐‘€.(2.1)

The total parameter domain ๐ฝ is the disjoint union of ๐‘€+1 intervals ๐ฝ0,โ€ฆ,๐ฝ๐‘€. We define a parametrization ๐‘ง of the whole boundary ฮ“ on ๐ฝ byโŽงโŽชโŽจโŽชโŽฉ๐‘ง๐‘ง(๐‘ก)=0(๐‘ก),๐‘กโˆˆ๐ฝ0=[],โ‹ฎ๐‘ง0,2๐œ‹๐‘€(๐‘ก),๐‘กโˆˆ๐ฝ๐‘€=[].0,2๐œ‹(2.2)

Let ๐ปโˆ— be the space of all real Hรถlder continuous 2๐œ‹-periodic functions ๐œ”(๐‘ก) of the parameter ๐‘ก on ๐ฝ๐‘˜ for ๐‘˜=0,1,โ€ฆ,๐‘€, that is,โŽงโŽชโŽจโŽชโŽฉ๐œ”๐œ”(๐‘ก)=0(๐‘ก),๐‘กโˆˆ๐ฝ0,๐œ”1(๐‘ก),๐‘กโˆˆ๐ฝ1,โ‹ฎ๐œ”๐‘€(๐‘ก),๐‘กโˆˆ๐ฝ๐‘€.(2.3) Let ๐œƒ(๐‘ก) (the boundary corresponding function) be given for ๐‘กโˆˆ๐ฝ byโŽงโŽชโŽจโŽชโŽฉ๐œƒ๐œƒ(๐‘ก)=0(๐‘ก),๐‘กโˆˆ๐ฝ0,โ‹ฎ๐œƒ๐‘€(๐‘ก),๐‘กโˆˆ๐ฝ๐‘€.(2.4) Let ๐œ‡ (a piecewise constant real function) be given for ๐‘กโˆˆ๐ฝ by๎€ท๐œ‡๐œ‡(๐‘ก)=0,๐œ‡1,โ€ฆ,๐œ‡๐‘€๎€ธ=โŽงโŽชโŽจโŽชโŽฉ๐œ‡0,๐‘กโˆˆ๐ฝ0,โ‹ฎ๐œ‡๐‘€,๐‘กโˆˆ๐ฝ๐‘€.(2.5) Let ๎๐ด(๐‘ก) be a complex continuously differentiable 2๐œ‹-periodic function for all ๐‘กโˆˆ๐ฝ. The generalized Neumann kernel formed with ๎๐ด is defined by๎1๐‘(๐‘ก,๐‘ )=๐œ‹๎ƒฉ๎Im๐ด(๐‘ก)๎๐‘ง๐ด(๐‘ )๎…ž(๐‘ )๎ƒช๐‘ง(๐‘ )โˆ’๐‘ง(๐‘ก).(2.6) The kernel ๎๐‘ is continuous with๎1N(๐‘ก,๐‘ก)=๐œ‹๎ƒฉ12๐‘งImโ€ฒ๎…ž(๐‘ก)๐‘ง๎…ž๎๐ด(๐‘ก)โˆ’Im๎…ž(๐‘ก)๎๎ƒช๐ด(๐‘ก).(2.7) Define also the kernel ๎‚Š๐‘€ by๎‚Š1๐‘€(๐‘ก,๐‘ )=๐œ‹๎ƒฉ๎Re๐ด(๐‘ก)๎๐‘ง๐ด(๐‘ )๎…ž(๐‘ )๎ƒช๐‘ง(๐‘ )โˆ’๐‘ง(๐‘ก),(2.8) which has a cotangent singularity type (see [18] for more detail). The classical Neumann kernel is the generalized Neumann kernel formed with ๎๐ด(๐‘ก)=1, that is,1๐‘(๐‘ก,๐‘ )=๐œ‹๎‚ต๐‘งIm๎…ž(๐‘ )๎‚ถ๐‘ง(๐‘ )โˆ’๐‘ง(๐‘ก).(2.9) The adjoint kernel ๐‘โˆ—(๐‘ ,๐‘ก) of the classical Neumann kernel is given by๐‘โˆ—1(๐‘ก,๐‘ )=๐‘(๐‘ ,๐‘ก)=๐œ‹๎‚ต๐‘งIm๎…ž(๐‘ก)๎‚ถ๐‘ง(๐‘ก)โˆ’๐‘ง(๐‘ ).(2.10) The adjoint function to the function ๎๐ด is given by๎‚๐‘ง๐ด(๐‘ก)=๎…ž(๐‘ก)๎๐ด(๐‘ก)=๐‘ง๎…ž(๐‘ก).(2.11) The generalized Neumann kernel ๎‚๐‘(๐‘ ,๐‘ก) formed with ๎‚๐ด is given by๎‚1๐‘(๐‘ก,๐‘ )=๐œ‹๎ƒฉ๎‚Im๐ด(๐‘ก)๎‚๐‘ง๐ด(๐‘ )๎…ž(๐‘ )๎ƒช๐‘ง(๐‘ )โˆ’๐‘ง(๐‘ก).(2.12) If ๎๐ด=1, then๎‚๐‘(๐‘ก,๐‘ )=โˆ’๐‘โˆ—(๐‘ก,๐‘ ).(2.13) We define the Fredholm integral operators ๎‚๐,๐,๐โˆ— by๎€œ๐๐œ(๐‘ก)=๐ฝ๎‚๎€œ๐‘(๐‘ก,๐‘ )๐œ(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ๐ฝ,(2.14)๐๐œ(๐‘ก)=๐ฝ๎‚๐๐‘(๐‘ก,๐‘ )๐œ(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ๐ฝ,(2.15)โˆ—๎€œ๐œ(๐‘ก)=๐ฝ๐‘(๐‘ ,๐‘ก)๐œ(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ๐ฝ.(2.16) Note that ๎‚๐=โˆ’๐โˆ—, if ๎๐ด=1.

It is known that ๐œ†=1 is an eigenvalue of the kernel ๐‘ with multiplicity 1 and ๐œ†=โˆ’1 is an eigenvalue of the kernel ๐‘ with multiplicity ๐‘€ [18]. We define the piecewise constant functions๐œ’[๐‘—]๎‚ป(๐œ‰)=1,๐œ‰โˆˆฮ“๐‘—,๐‘—=0,1,2,โ€ฆ,๐‘€.0,otherwise.(2.17) Then, we have from [18]๎€ฝ๐œ’Null(๐ˆโˆ’๐)=span{1},Null(๐ˆโˆ’๐)=span[1],๐œ’[2],โ€ฆ,๐œ’[๐‘€]๎€พ.(2.18)

Lastly, we define integral operators ๐‰ and ฬ‚๐‰ by๎€œ๐‰๐œ=๐ฝ12๐œ‹๐‘€๎“๐‘—=1๐œ’[๐‘—](๐‘ )๐œ’[๐‘—]ฬ‚โˆซ(๐‘ก)๐œ(๐‘ )๐‘‘๐‘ ,๐‰๐œ=๐ฝ12๐œ‹๐‘€โˆ‘๐‘—=0๐œ’[๐‘—](๐‘ )๐œ’[๐‘—](๐‘ก)๐œ(๐‘ )๐‘‘๐‘ ,(2.19) 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 that ๐‘(๐‘ง), ๐‘„(๐‘ง), and ๐ป(๐‘ง) are complex-valued functions defined on ฮ“ such that ๐‘(๐‘ง)โ‰ 0, ๐ป(๐‘ง)โ‰ 0, ๐‘„(๐‘ง)โ‰ 0, and ๐ป(๐‘ง)/(๐‘‡(๐‘ง)๐‘„(๐‘ง))satisfies the Hรถlder condition on ฮ“. Then, the interior relationship is defined as follows.

A complex-valued function ๐‘ƒ(๐‘ง) is said to satisfy the interior relationship if ๐‘ƒ(๐‘ง) is analytic in ฮฉ and satisfies the nonhomogeneous boundary relationship๐‘ƒ(๐‘ง)=๐‘(๐‘ง)๐‘‡(๐‘ง)๐‘„(๐‘ง)๐บ(๐‘ง)๐‘ƒ(๐‘ง)+๐ป(๐‘ง),๐‘งโˆˆฮ“,(3.1) where ๐บ(๐‘ง) analytic in ฮฉ, Hรถlder continuous on ฮ“, and ๐บ(๐‘ง)โ‰ 0 on ฮ“. The boundary relationship (3.1) also has the following equivalent form:๐บ(๐‘ง)=๐‘(๐‘ง)๐‘‡(๐‘ง)๐‘„(๐‘ง)๐‘ƒ(๐‘ง)2||||๐‘ƒ(๐‘ง)2+๐บ(๐‘ง)๐ป(๐‘ง)๐‘ƒ(๐‘ง),๐‘งโˆˆฮ“.(3.2) Let the function ๐ฟ๐‘…(ฬƒ๐‘ง) be defined in the region ๐ถโˆช{โˆž}โงตฮ“ by๐ฟ๐‘…1(ฬƒ๐‘ง)=๎€œ2๐œ‹iฮ“๐‘(ฬƒ๐‘ง)๐ป(๐‘ค)๐‘(๐‘ค)(๐‘คโˆ’ฬƒ๐‘ง)๐‘„(๐‘ค)๐‘‡(๐‘ค)๐‘‘๐‘ค,ฬƒ๐‘งโˆˆฮฉโˆ’,(3.3) where ฮฉโˆ’ is the complement of ฮฉ. 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 Theoremโ€‰โ€‰3.1 in [20, page 45].

Theorem 3.1. Let ๐‘ˆ and ๐‘‰ be any complex-valued functions that are defined on ฮ“. If the function ๐‘ƒ(๐‘ง) satisfies the interior nonhomogeneous boundary relationship (3.1) or (3.2), then 12๎ƒฌ๐‘‰(๐‘ง)+๐‘ˆ(๐‘ง)๐‘‡๎ƒญ๎€œ(๐‘ง)๐‘„(๐‘ง)๐‘ƒ(๐‘ง)+PVฮ“||||ร—๎ƒฌโˆ‘๐พ(๐‘ง,๐‘ค)๐‘ƒ(๐‘ค)๐‘‘๐‘ค+๐‘(๐‘ง)๐‘ˆ(๐‘ง)๐‘Ž๐‘—โˆˆฮฉRes๐‘ค=๐‘Ž๐‘—๐‘ƒ(๐‘ค)(๎ƒญ๐‘คโˆ’๐‘ง)๐บ(๐‘ค)conj=โˆ’๐‘ˆ(๐‘ง)๐ฟโˆ’๐‘…(๐‘ง),๐‘งโˆˆฮ“,(3.4) where 1๐พ(๐‘ง,๐‘ค)=๎ƒฌ2๐œ‹i๐‘(๐‘ง)๐‘ˆ(๐‘ง)๐‘๎€ท(๐‘ค)๐‘คโˆ’๐‘ง๎€ธ๐‘„โˆ’(๐‘ค)๐‘‰(๐‘ง)๐‘‡(๐‘ค)๎ƒญ,๐ฟ๐‘คโˆ’๐‘งโˆ’๐‘…(๐‘ง)=โˆ’12๐ป(๐‘ง)1๐‘„(๐‘ง)๐‘‡(๐‘ง)+PVโˆซ2๐œ‹iฮ“๐‘(๐‘ง)๐ป(๐‘ค)๐‘(๐‘ค)(๐‘คโˆ’๐‘ง)๐‘„(๐‘ค)๐‘‡(๐‘ค)๐‘‘๐‘ค.(3.5) The symbol โ€œconjโ€ 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 ๐‘Ž1,๐‘Ž2,โ€ฆ,๐‘Ž๐‘€ of ๐บ that lie inside ฮฉ. If ๐บ has no zeros in ฮฉ, then the term containing the residue in (3.4) will not appear.

Proof. Suppose that ๐‘ƒ(๐‘ง) and ๐บ(๐‘ง) are analytic functions in ฮฉ and ๐บ has a finite number of zeros at ๐‘Ž1,๐‘Ž2,โ€ฆ,๐‘Ž๐‘€ in ฮฉ. Then, by the calculus of residues, we have 1๎€œ2๐œ‹๐‘–ฮ“๐‘ƒ(๐‘ค)๎“(๐‘คโˆ’ฬƒ๐‘ง)๐บ(๐‘ค)๐‘‘๐‘ค=๐‘Ž๐‘—โˆˆฮฉRes๐‘ค=๐‘Ž๐‘—๐‘ƒ(๐‘ค)(๐‘คโˆ’ฬƒ๐‘ง)๐บ(๐‘ค),ฬƒ๐‘งโˆˆฮฉโˆ’.(3.6) Since ๐‘ƒ and ๐บ satisfy the Hรถlder condition on ฮ“ and ๐บ(๐‘ง)โ‰ 0 on ฮ“, then ๐‘ƒ/๐บ also satisfies the Hรถlder condition on ฮ“. Taking the limit ฮฉโˆ’โˆ‹ฬƒzโ†’๐‘งโˆˆฮ“ and applying Sokhotski formula [5], we get โˆ’12๐‘ƒ(๐‘ง)1๐บ(๐‘ง)+PV๎€œ2๐œ‹๐‘–ฮ“๐‘ƒ(๐‘ง)๎“(๐‘คโˆ’๐‘ง)๐บ(๐‘ค)๐‘‘๐‘ค=๐‘Ž๐‘—โˆˆฮฉRes๐‘ค=๐‘Ž๐‘—๐‘ƒ(๐‘ค)(๐‘คโˆ’๐‘ง)๐บ(๐‘ค),๐‘งโˆˆฮ“.(3.7) By taking conjugate to both sides and using (3.1), we get โˆ’12๐‘ƒ(๐‘ง)๐‘(๐‘ง)+1๐‘„(๐‘ง)T(๐‘ง)2๐ป(๐‘ง)๐‘(๐‘ง)1๐‘„(๐‘ง)๐‘‡(๐‘ง)โˆ’PV๎€œ2๐œ‹๐‘–ฮ“๐‘ƒ(๐‘ง)๎€ท๐‘(๐‘ค)๐‘คโˆ’๐‘ง๎€ธ๐‘„(๐‘ค)๐‘‘๐‘ค1๐‘‡(๐‘ค)+PV๎€œ2๐œ‹๐‘–ฮ“๐ป(๐‘ง)๐‘‘๐‘ค๎€ท๐‘(๐‘ค)๐‘คโˆ’๐‘ง๎€ธ=โŽกโŽขโŽขโŽฃ๎“๐‘„(๐‘ง)๐‘‡(๐‘ง)๐‘Ž๐‘—โˆˆฮฉRes๐‘ค=๐‘Ž๐‘—๐‘ƒ(๐‘ค)โŽคโŽฅโŽฅโŽฆ(๐‘คโˆ’๐‘ง)๐บ(๐‘ค)conj,๐‘งโˆˆฮ“.(3.8) Multiplying both sides by โˆ’๐‘(๐‘ง) and the fact that ๐‘‘๐‘ค=๐‘‡(๐‘ค)|๐‘‘๐‘ค|, after some arrangement, yield 12๐‘ƒ(๐‘ง)1๐‘„(๐‘ง)๐‘‡(๐‘ง)+PV๎€œ2๐œ‹๐‘–ฮ“๐‘(๐‘ง)๐‘ƒ(๐‘ง)๎€ท๐‘(๐‘ค)๐‘คโˆ’๐‘ง๎€ธ||||โŽกโŽขโŽขโŽฃ๎“๐‘„(๐‘ค)๐‘‘๐‘ค+๐‘(๐‘ง)๐‘Ž๐‘—โˆˆฮฉRes๐‘ค=๐‘Ž๐‘—๐‘ƒ(๐‘ค)โŽคโŽฅโŽฅโŽฆ(๐‘คโˆ’๐‘ง)๐บ(๐‘ค)conj๎ƒฌโˆ’1=โˆ’2๐ป(๐‘ง)1๐‘„(๐‘ง)๐‘‡(๐‘ง)+PV๎€œ2๐œ‹๐‘–ฮ“๐‘(๐‘ง)๐ป(๐‘ง)๐‘๎ƒญ(๐‘ค)(๐‘คโˆ’๐‘ง)๐‘„(๐‘ง)๐‘‡(๐‘ง)๐‘‘๐‘คconj,๐‘งโˆˆฮ“.(3.9) Applying Sokhotski formulas again to the expression inside the bracket of the right-hand side yields 12๐‘ƒ(๐‘ง)1๐‘„(๐‘ง)๐‘‡(๐‘ง)+PV๎€œ2๐œ‹๐‘–ฮ“๐‘(๐‘ง)๐‘ƒ(๐‘ง)๎€ท๐‘(๐‘ค)๐‘คโˆ’๐‘ง๎€ธ||||โŽกโŽขโŽขโŽฃ๎“๐‘„(๐‘ค)๐‘‘๐‘ค+๐‘(๐‘ง)๐‘Ž๐‘—โˆˆฮฉRes๐‘ค=๐‘Ž๐‘—๐‘ƒ(๐‘ค)โŽคโŽฅโŽฅโŽฆ(๐‘คโˆ’๐‘ง)๐บ(๐‘ค)conj=โˆ’๐ฟโˆ’๐‘…(๐‘ง),๐‘งโˆˆฮ“.(3.10) Since ๐‘ƒ(๐‘ง) is analytic in ฮฉ, then by Cauchy integral formula, we have 1๎€œ2๐œ‹๐‘–ฮ“๐‘ƒ(๐‘ง)๐‘คโˆ’ฬƒ๐‘ง๐‘‘๐‘ค=0,๐‘งโˆˆฮฉโˆ’.(3.11) Taking the limit ๐œ”โˆ’โˆ‹ฬƒ๐‘งโ†’๐‘งโˆˆฮ“ and applying Sokhotiski formulas, we get โˆ’121๐‘ƒ(๐‘ง)+PV๎€œ2๐œ‹๐‘–ฮ“๐‘‡(๐‘ค)๐‘ƒ(๐‘ง)||||๐‘คโˆ’๐‘ง๐‘‘๐‘ค=0,๐‘งโˆˆฮ“.(3.12) Multiplying (3.12) by ๐‘ฃ(๐‘ง) and subtracting it from (3.10) multiplied by ๐‘ข(๐‘ง) yield (3.4).

3.2. Homogeneous Boundary Relationship for Conformal Mapping

Let ๐‘ค=๐‘“(๐‘ง) be the analytic function which maps ฮฉ in the ๐‘ง-plane onto a canonical region of the circular slit region in the ๐‘ค-plane. Let 0 and ๐‘Ž be a fixed point in ฮฉ such that ๐‘Žโ‰ 0. Then, the mapping function is made uniquely determined by assuming that ๐‘“(๐‘Ž)=0 and ๐‘“(0)=โˆž such that the residue of the function ๐‘“ at 0 is equal to 1 [1]. Hence, the function ๐‘“ can be written in the form๎‚€1๐‘“(๐‘ง)=๐‘งโˆ’1๐‘Ž๎‚๐‘’๐‘ง๐‘”(๐‘ง),(3.13) where ๐‘” is analytic in ฮฉ [12, 13]. Note that the boundary value of ๐‘“ can be represented in the form๐‘“๎€ท๐‘ง๐‘๎€ธ(๐‘ก)=๐œ‡๐‘๐‘’i๐œƒ๐‘(๐‘ก),ฮ“๐‘โˆถ๐‘ง=๐‘ง๐‘(๐‘ก),0โ‰ค๐‘กโ‰ค๐›ฝ๐‘,๐‘=0,1,โ€ฆ,๐‘€,(3.14) where ๐œƒ๐‘ is a boundary correspondence function of ฮ“๐‘ and ๐œ‡๐‘ is the radius of the circular slit. The unit tangent to ฮ“ at ๐‘ง(๐‘ก) is denoted by ๐‘‡(๐‘ง(๐‘ก))=๐‘งโ€ฒ(๐‘ก)/|๐‘ง๎…ž(๐‘ก)|. Thus, it can be shown that||||๐‘“(๐‘ง)=๐‘“(๐‘ง)๐‘–||๐œƒ๐‘‡(๐‘ง)๎…ž๐‘(||๐‘ก)๐œƒ๎…ž๐‘๐‘“(๐‘ก)๎…ž(๐‘ง)||๐‘“๎…ž||(๐‘ง),๐‘งโˆˆฮ“.(3.15)

4. Integral Equation Method for Computing ๐น๎…ž(๐‘)

Note that the value of ๐œƒ๎…ž๐‘(๐‘ก) may be positive or negative since each circular slit ๐‘“(ฮ“๐‘) is traversed twice. Thus, |๐œƒ๎…ž๐‘|/๐œƒ๎…ž๐‘=ยฑ1. Hence, the boundary relationship (3.15) can be written as||||๐‘“(๐‘ง)=ยฑ๐‘‡(๐‘ง)๐‘“(๐‘ง)๐‘–๐‘“๎…ž(๐‘ง)||๐‘“๎…ž||(๐‘ง),๐‘งโˆˆฮ“.(4.1) To eliminate the ยฑ sign, we square both sides of the boundary relationship (4.1) to get๐‘“(๐‘ง)2=โˆ’๐‘‡(๐‘ง)2||||๐‘“(๐‘ง)2๐‘“โ€ฒ(๐‘ง)2||๐‘“๎…ž||(๐‘ง)2,๐‘งโˆˆฮ“.(4.2) Then, the function ๐ธ(๐‘ง) defined by๐ท(๐‘ง)=๐‘ง2๐‘“๎…ž(๐‘ง)=๐‘ง2๎€บ๐‘“(๐‘ง)๐‘ง๐‘”๎…ž๎€ป(๐‘ง)+๐‘”(๐‘ง)โˆ’๐‘’๐‘ง๐‘”(๐‘ง)(4.3) is analytic in ฮฉ.

Combining (4.3), (4.2), and (3.13), we obtain the following boundary relationship:๐‘ง๐‘’2๐‘งโ„Ž(๐‘ง)๐‘Ž2=โˆ’๐‘ง|๐‘ง|2(๐‘Žโˆ’๐‘ง)2||||๐‘“(๐‘ง)2๐‘‡(๐‘ง)2๐ท(๐‘ง)2||||๐ท(๐‘ง)2,๐‘งโˆˆฮ“.(4.4) Comparison of (4.4) and (3.2) leads to a choice of ๐‘ƒ(๐‘ง)=๐ท(๐‘ง), ๐‘(๐‘ง)=โˆ’๐‘ง|๐‘ง|2|๐‘“(๐‘ง)|2/(๐‘Žโˆ’๐‘ง)2, ๐‘„(๐‘ง)=๐‘‡(๐‘ง), ๐บ(๐‘ง)=๐‘ง๐‘’2๐‘งโ„Ž(๐‘ง)/๐‘Ž2, ๐ป(๐‘ง)=0. Setting ๐‘ˆ(๐‘ง)=๐‘‡(๐‘ง)๐‘„(๐‘ง) and ๐‘‰(๐‘ง)=1, Theorem 3.1 yields1๐‘‡(๐‘ง)๐ท(๐‘ง)+PV๎€œ2๐œ‹iฮ“โŽกโŽขโŽขโŽฃ๐‘ง|๐‘ง|2||||๐‘“(๐‘ง)2(๐‘Žโˆ’๐‘ค)2๐‘‡(๐‘ง)๐‘ค|๐‘ค|2||||๐‘“(๐‘ค)2(๐‘Žโˆ’๐‘ง)2๎€ท๐‘คโˆ’๐‘ง๎€ธโˆ’๐‘‡(๐‘ง)โŽคโŽฅโŽฅโŽฆ||||=๐‘คโˆ’๐‘ง๐‘‡(๐‘ค)๐ท(๐‘ค)๐‘‘๐‘ค๐‘ง|๐‘ง|2||||๐‘“(๐‘ง)2๎€ท๐‘Žโˆ’๐‘ง๎€ธ2๐‘‡โŽกโŽขโŽขโŽฃ๎“(๐‘ง)๐‘Ž๐‘—โˆˆฮฉRes๐‘ค=๐‘Ž๐‘—๐‘Ž2๐ท(๐‘ค)(๐‘คโˆ’๐‘ง)๐‘ค๐‘’2๐‘คโ„Ž(๐‘ค)โŽคโŽฅโŽฅโŽฆconj,๐‘งโˆˆฮ“.(4.5) Note that ๐‘Ž2๐ท(๐‘ค)/(๐‘คโˆ’๐‘ง)๐‘ค2 has a simple pole at ๐‘ค=0. To evaluate the residue in (4.5), we use the fact that if ๐ฟ(๐‘ง)=๐‘‘(๐‘ง)/๐‘ž(๐‘ง) where ๐‘‘(๐‘ง) and ๐‘ž(๐‘ง) are analytic at ๐‘ง0 and ๐‘‘(๐‘ง0)โ‰ 0, ๐‘ž(๐‘ง0)=0 and ๐‘žโ€ฒ(๐‘ง0)โ‰ 0, which means ๐‘ง0 is a simple pole of ๐ฟ(๐‘ง), thenRes๐‘ค=๐‘ง0๐‘‘๎€ท๐‘ง๐ฟ(๐‘ค)=0๎€ธ๐‘žโ€ฒ๎€ท๐‘ง0๎€ธ.(4.6) Applying (4.6) to the residue in (4.5) and after several algebraic manipulations, we obtain๎“๐‘Ž๐‘—โˆˆฮฉRes๐‘ค=๐‘Ž๐‘—๐‘Ž2๐ท(๐‘ค)(๐‘คโˆ’๐‘ง)๐‘ค๐‘’2๐‘ค๐‘”(๐‘ค)=๐‘Ž2๐‘ง.(4.7) Thus, integral equation (4.5) becomes ๎€œ๐น(๐‘)+ฮ“๐‘+||||=(๐‘ง,๐‘ค)๐น(๐‘ค)๐‘‘๐‘ค๐‘Ž2๐‘ง2||||๐‘“(๐‘ง)2๎€ท๐‘Žโˆ’๐‘ง๎€ธ2๐‘‡(๐‘ง),๐‘งโˆˆฮ“,(4.8) where๐น(๐‘ง)=๐‘‡(๐‘ง)๐ท(๐‘ง),๐ท(๐‘ง)=๐‘ง2๐‘“โ€ฒ๐‘(๐‘ง),+1(๐‘ง,๐‘ค)=โŽกโŽขโŽขโŽฃ2๐œ‹๐‘–๐‘‡(๐‘ง)โˆ’๐‘งโˆ’๐‘ค๐‘ง|๐‘ง|2||||๐‘“(๐‘ง)2(๐‘Žโˆ’๐‘ค)2๐‘‡(๐‘ง)๐‘ค|๐‘ค|2||||๐‘“(๐‘ค)2(๐‘Žโˆ’๐‘ง)2๎€ท๐‘งโˆ’๐‘ค๎€ธโŽคโŽฅโŽฅโŽฆ,๐‘+1(๐‘ก,๐‘ก)=||๐‘ง2๐œ‹๎…ž(||๐‘ง๐‘ก)Im๎…ž๎…ž(๐‘ก)๐‘ง๎…ž+1(๐‘ก)||๐‘ง๐œ‹๐‘–๎…ž(||๎ƒฌ๐‘ก)๐‘ง๎…ž(๐‘ก)๐‘ง(๐‘ก)โˆ’๐‘Ž๎‚ต๐‘งโˆ’Re๎…ž(๐‘ก)๎‚ถ๎ƒญโˆ’1๐‘ง(๐‘ก)||๐‘ง2๐œ‹๐‘–๎…ž(||๐‘ง๐‘ก)๎…ž(๐‘ก).๐‘ง(๐‘ก)(4.9) By using single valuedness of the mapping function ๐‘“ leads to the following condition:1๎€œ2๐œ‹โˆ’ฮ“๐‘ž๐น(๐‘ค)๐‘ค2||||๐‘‘๐‘ค=0,๐‘ž=0,1,โ€ฆ,๐‘€.(4.10) By means of Cauchyโ€™s integral formula, we can get the following condition:1๎€œ2๐œ‹ฮ“๐น(๐‘ค)๐‘ค||||๐‘‘๐‘ค=โˆ’๐‘–.(4.11) Thus, the integral equation (4.8) with the conditions (4.10) and (4.11) should give a unique solution provided the parameters ๐œ‡๐‘, ๐‘=0,1,โ€ฆ,๐‘€ that appear in ๐‘+(๐‘ง,๐‘ค) are known.

Integral equation methods for computing ๐œ‡๐‘ and ๐œƒ๎…ž๐‘ are discussed in the next two sections.

5. Integral Equation for Computing |๐‘“(๐‘ง)|

Note that, from (3.13) and (3.14), we get the following equation:๐‘ง||๐‘“||||||1(๐‘ก)๐‘”(๐‘ง(๐‘ก))=log(๐‘ง(๐‘ก))โˆ’logโˆ’1๐‘ง(๐‘ก)๐‘Ž||||๎‚ต1โˆ’๐‘–argโˆ’1๐‘ง(๐‘ก)๐‘Ž๎‚ถ+๐œƒ๐‘(๐‘ก).(5.1) Since ๐‘”(๐‘ง) is analytic in ฮฉ, thus๎๐ด(๐‘ก)๐‘”(๐‘ง(๐‘ก))=๐›พ(๐‘ก)+โ„Ž(๐‘ก)+๐‘–๐œ,(5.2) from (5.1) and (5.2), yields๎๎‚ต1๐ด(๐‘ก)=๐‘ง(๐‘ก),(5.3)๐›พ(๐‘ก)=โˆ’logโˆ’1๐‘ง(๐‘ก)๐‘Ž๎‚ถ๎€ท,(5.4)โ„Ž(๐‘ก)=log๐œ‡(๐‘ก)=log๐œ‡0,log๐œ‡1,โ€ฆ,log๐œ‡๐‘€๎€ธ.(5.5)

The following theorem from [22] gives a method for calculating โ„Ž(๐‘ก), and hence ๐œ‡๐‘=|๐‘“(๐‘ง๐‘)|.

Theorem 5.1 (see [22, Theorem 5]). The function โ„Ž is given by โ„Ž=(โ„Ž0,โ„Ž1,โ€ฆ,โ„Ž๐‘€), where โ„Ž๐‘—=๎€ท๐›พ,๐œ™[๐‘—]๎€ธ=1๎€œ2๐œ‹J๐›พ(๐‘ก)๐œ™[๐‘—](๐‘ก)๐‘‘๐‘ก,(5.6) and where ๐œ™[๐‘—] is the unique solution of the following integral equation ๎‚€๎๐๐ˆ+โˆ—+ฬ‚๐‰๎‚๐œ™[๐‘—]=โˆ’๐œ’[๐‘—],๐‘—=0,1,โ€ฆ,๐‘€,(5.7) where the kernel ๎๐‘โˆ—(๐‘ ,๐‘ก) is the adjoint kernel of the kernel ๎๐‘(๐‘ ,๐‘ก) which is formed with ๎๐ด(๐‘ก)=๐‘ง(๐‘ก).

By obtaining โ„Ž0,โ„Ž1,โ€ฆ,โ„Ž๐‘€ from (5.6), in view of (5.5), we obtain๐œ‡๐‘—=๐‘’โ„Ž๐‘—,๐‘—=0,1,โ€ฆ,๐‘€.(5.8)

6. Integral Equation Method for Computing ๐œƒโ€ฒ๐‘(๐‘ก)

This section gives another application of Theorem 3.1 for computing ๐‘“โ€ฒ/๐‘“. Let ๐‘“ be the mapping function as described in Section 3.2. Note that (4.2) can be written in the following form:||||๐‘“๎…ž(๐‘ง)f||||(๐‘ง)2=โˆ’๐‘‡(๐‘ง)2๎ƒฉ๐‘“โ€ฒ(๐‘ง)๐‘“๎ƒช(๐‘ง)2,๐‘งโˆˆฮ“.(6.1) Taking the derivative of both sides of (3.13) together with some elementary calculations yields๐‘“๎…ž(๐‘ง)+๐‘Ž๐‘“(๐‘ง)๐‘ง(๐‘Žโˆ’๐‘ง)=๐‘ง๐‘”๎…ž(๐‘ง)+๐‘”(๐‘ง).(6.2) Let ๐ธ(๐‘ง)=(๐‘“๎…ž(๐‘ง)/๐‘“(๐‘ง))+(๐‘Ž/๐‘ง(๐‘Žโˆ’๐‘ง))=๐‘ง๐‘”๎…ž(๐‘ง)+๐‘”(๐‘ง) be analytic in ฮฉ. Then,๐‘“๎…ž(๐‘ง)๐‘Ž๐‘“(๐‘ง)=๐ธ(๐‘ง)+๐‘ง(๐‘งโˆ’๐‘Ž),๐‘งโˆˆฮ“.(6.3) Equations (6.1) and (6.3) together with some elementary calculations yield๐ธ(๐‘ง)=โˆ’๐‘‡(๐‘ง)2๐ธ(๐‘ง)โˆ’๐‘Ž๐‘‡(๐‘ง)2๐‘ง๎€ท๐‘งโˆ’๐‘Ž๎€ธโˆ’๐‘Ž๐‘ง(๐‘งโˆ’๐‘Ž),๐‘งโˆˆฮ“.(6.4)

Comparison of (6.4) and (3.1) leads to a choice of ๐‘ƒ(๐‘ง)=๐ธ(๐‘ง), ๐‘(๐‘ง)=โˆ’1, ๐‘„(๐‘ง)=๐‘‡(๐‘ง), ๐บ(๐‘ง)=1, ๐ป(๐‘ง)=โˆ’(๐‘Ž๐‘‡(๐‘ง)2/๐‘ง(๐‘งโˆ’๐‘Ž))โˆ’(ฬ„๐‘Ž/ฬ„๐‘ง(๐‘งโˆ’๐‘Ž)). Setting ๐‘ˆ(๐‘ง)=๐‘‡(๐‘ง)๐‘„(๐‘ง) and ๐‘‰(๐‘ง)=1, Theorem 3.1 yields1๐ธ(๐‘ง)๐‘‡(๐‘ง)+PV๎€œ2๐œ‹๐‘–ฮ“๎ƒฌ๐‘‡(๐‘ง)๐‘คโˆ’๐‘งโˆ’๐‘‡(๐‘ง)๎ƒญ||||๐‘คโˆ’๐‘ง๐ธ(๐‘ค)๐‘‡(๐‘ค)๐‘‘๐‘ค=โˆ’๐‘‡(๐‘ง)๐ฟโˆ’๐‘…(๐‘ง),๐‘งโˆˆฮ“,(6.5) where๐‘‡(๐‘ง)๐ฟโˆ’๐‘…1(๐‘ง)=โˆ’2๎ƒฌโˆ’๐‘Ž๐‘‡(๐‘ง)โˆ’๐‘ง(๐‘งโˆ’๐‘Ž)๐‘Ž๐‘‡(๐‘ง)๐‘ง๎€ท๐‘งโˆ’๐‘Ž๎€ธ๎ƒญ1+๐‘‡(๐‘ง)PV๎€œ2๐œ‹๐‘–ฮ“๐‘Ž1๐‘ค(๐‘คโˆ’๐‘ง)(๐‘คโˆ’๐‘Ž)๐‘‘๐‘คโˆ’๐‘‡(๐‘ง)PV๎€œ2๐œ‹๐‘–ฮ“๐ด๐‘Ž๐‘‡(๐‘ค)2๐‘ค๎€ท๐‘คโˆ’๐‘Ž๎€ธ(๐‘คโˆ’๐‘ง)๐‘‘๐‘ค,๐‘งโˆˆฮ“.(6.6) Then, it follows from [5, page 91] that1PV๎€œ2๐œ‹๐‘–ฮ“๐‘Ž1๐‘ค(๐‘คโˆ’๐‘ง)(๐‘คโˆ’๐‘Ž)๐‘‘๐‘ค=โˆ’2๐‘Ž.๐‘ง(๐‘งโˆ’๐‘Ž)(6.7) From (6.5),(6.6), (6.7), and (6.3), we obtain the integral equation๐‘“โ€ฒ(๐‘ง)1๐‘“(๐‘ง)๐‘‡(๐‘ง)+PV๎€œ2๐œ‹๐‘–ฮ“๎ƒฌ๐‘‡(๐‘ง)โˆ’๐‘งโˆ’๐‘ค๐‘‡(๐‘ง)๐‘งโˆ’๐‘ค๎ƒญ๐‘“โ€ฒ(๐‘ค)||||๎‚ธ๐‘“(๐‘ค)๐‘‡(๐‘ค)๐‘‘๐‘ค=2๐‘–Im๐‘Ž๐‘‡(๐‘ง)๎‚น๐‘ง(๐‘งโˆ’๐‘Ž),๐‘งโˆˆฮ“.(6.8) In the above integral equation, let ๐‘ง=๐‘ง(๐‘ก) and ๐‘ค=๐‘ง(๐‘ ). Then, by multiplying both sides of (6.8) by |๐‘ง๎…ž(๐‘ก)| and using the fact that๐‘“๎…ž(๐‘ง)๐‘“(๐‘ง)๐‘งโ€ฒ(๐‘ก)=๐‘–๐œƒ๎…ž๐‘(๐‘ก),๐‘งโˆˆฮ“,(6.9) the above integral equation can also be written as๐œƒ๎…ž๐‘๎€œ(๐‘ก)+๐ฝ๐‘(๐‘ ,๐‘ก)๐œƒ๎…ž๐‘๎‚ธ(๐‘ )๐‘‘๐‘ =2Im๐‘Ž๐‘ง๎…ž(๐‘ก)๎‚น๐‘ง(๐‘ก)(๐‘ง(๐‘ก)โˆ’๐‘Ž).(6.10) Since ๐‘(๐‘ ,๐‘ก)=๐‘โˆ—(๐‘ก,๐‘ ), the integral equation can be written as an integral equation in operator form๎€ท๐ˆ+๐โˆ—๎€ธ๐œƒ๎…ž๐‘=๎‚๐œ“,(6.11) where๎‚ธ๎‚๐œ“=2Im๐‘Ž๐‘งโ€ฒ(๐‘ก)๎‚น๐‘ง(๐‘ก)(๐‘ง(๐‘ก)โˆ’๐‘Ž).(6.12) However, ๐œ†=โˆ’1 is an eigenvalue of ๐‘โˆ— with multiplicity ๐‘€, by [18, Theoremโ€‰โ€‰12]. Therefore, the integral equation (6.11) is not uniquely solvable. To overcome this problem, note that๎€œ๐ฝ๐‘—๐œƒ๎…ž๐‘(๐‘ก)๐‘‘๐‘ก=0,๐‘—=1,2,โ€ฆ,๐‘€,(6.13) which implies๐‰๐œƒ๎…ž๐‘=0.(6.14) By adding (6.14) to (6.11), we obtain the integral equation๎€ท๐ˆ+๐โˆ—๎€ธ๐œƒ+๐‰๎…ž๐‘=๎‚๐œ“.(6.15) 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, Theoremโ€‰โ€‰4].

Theorem 6.1. ๎€ทNull๐ˆ+๐โˆ—๎€ธ=+๐‰{0}.(6.16)

Proof. Let ๐œโˆˆNull(๐ˆ+๐โˆ—+๐‰), that is, ๐œ is a solution of the integral equation ๎€ท๐ˆ+๐โˆ—๎€ธ+๐‰๐œ=0.(6.17) Then, it follows from the definition of the operator ๐‰, (2.18), and the Fredholm alternative theorem that ๐‰=๐‰โˆ—=๐‰2,๎€ฝ๐œ’Range(๐‰)=span[1],โ€ฆ,๐œ’[๐‘€]๎€พ๎€ท๎€ฝ๐œ’=Null(๐ˆ+๐),Null(๐‰)=span[1],โ€ฆ,๐œ’[๐‘€]๎€พ๎€ธโŸ‚=Null(๐ˆ+๐)โŸ‚๎€ท=Range๐ˆ+๐โˆ—๎€ธ.(6.18) Hence, we have ๐๐‰=โˆ’๐‰ and ๐‰๐โˆ—=๐‰โˆ—๐โˆ—=(๐๐‰)โˆ—=โˆ’๐‰. By multiplying (6.17) by ๐‰, we obtain ๎€ท๐‰๐œ=0,๐ˆ+๐โˆ—๎€ธ๐œ=0.(6.19) Thus, ๎€ท๐œโˆˆNull(๐‰)โˆฉNull๐ˆ+๐โˆ—๎€ธ๎€ท=Range๐ˆ+๐โˆ—๎€ธ๎€ทโˆฉNull๐ˆ+๐โˆ—๎€ธ.(6.20) Since ๎๐ด=1, thus the index of the function ๎๐ด is given by (see [18] for the definition of the index) ๐œ…๐‘—=0,๐‘—=0,1,โ€ฆ,๐‘š,๐œ…=0.(6.21) The space ๐‘†+ defined in [18, Equation (30)] is then given by ๐‘†+=span{1}. Then, it follows from [18, Equation (92)] that the dimension of the space ๎‚๐‘†+ defined in [18, Equation (32)] is given by ๎‚๐‘†dim(+)=๐‘€. Similarly, it follows from [18, Equation (105)] that ๎€ท๎€ทdimNull๐ˆ+๐โˆ—๎‚€๎‚€๎‚๐๎€ธ๎€ธ=dimNull๐ˆโˆ’๎‚๎‚=๐‘€.(6.22) Thus, it follows from [18, Lemma 20(b)] that Null(๐ˆ+๐โˆ—๎‚๐‘†)=+ and the space ๎‚๐‘…+โˆฉ๎‚๐‘†โˆ’ in [18, Lemma 20(a)] contains only the zero function, that is, ๎‚๐‘…+โˆฉ๎‚๐‘†โˆ’={0}. Thus, it follows from [18, Equation (103)] (applied to the adjoint function ๎‚๎๐ด(๐‘ก)=๐ด(๐‘ก)/๐‘งโ€ฒ(๐‘ก) instead of ๎๐ด(๐‘ก)) and from [18, Equation (100)] that ๎€ทRange๐ˆ+๐โˆ—๎€ธ๎€ทโˆฉNull๐ˆ+๐โˆ—๎€ธ={0}.(6.23) Hence, it follows from (6.20) that ๐œ=0.

By solving the integral equation (6.15), we get ๐œƒ๐‘(๐‘ก). And solving the integral equation (5.7), we get ๐œ™[๐‘—], ๐‘—=0,1,โ€ฆ,๐‘€, which gives โ„Ž๐‘— through (5.6) which in turn gives ๐œ‡๐‘— through (5.8). By solving integral equation (4.8), (4.10), and (4.11) with the known values of ๐œ‡๐‘—, we get ๐น(๐‘ง). From the definition of ๐น(๐‘ง), we get๐‘“โ€ฒ(๐‘ง(๐‘ก))=๐น(๐‘ง(๐‘ก))๐‘ง2(๐‘ก)๐‘ง๎…ž.(๐‘ก)(6.24) Finally, from (3.14) and (6.24), the approximate boundary value of ๐‘“(๐‘ง) is given by||||๐‘“(๐‘ง)=๐‘“(๐‘ง)๐‘–||๐œƒ๐‘‡(๐‘ง)๎…ž๐‘(||๐‘ก)||๐œƒ๎…ž๐‘||๐‘“(๐‘ก)๎…ž(๐‘ง)||๐‘“๎…ž||(๐‘ง),๐‘งโˆˆฮ“.(6.25) The approximate interior value of the function ๐‘“(๐‘ง) is calculated by the Cauchy integral formula๐‘“(๐‘ง)=๐‘Žโˆ’๐‘ง1๐‘Ž๐‘ง๎€œ2๐œ‹๐‘–ฮ“๐‘Ž๐‘ค๐‘“(๐‘ค)1๐‘Žโˆ’๐‘ค๐‘คโˆ’๐‘ง๐‘‘๐‘ค,๐‘งโˆˆฮ“.(6.26) For points ๐‘ง which are not close to the boundary, the integral in (6.26) is approximated by the trapezoidal rule. However, for the points ๐‘ง closed to the boundary ฮ“, the numerical integration in (6.26) is nearly singular. This difficulty is overcome by using the fact that โˆซ(1/2๐œ‹๐‘–)ฮ“(1/(๐‘คโˆ’๐‘ง))๐‘‘๐‘ค=1, and rewrite ๐‘“(๐‘ง) as๐‘“โˆซ(๐‘ง)=((๐‘Žโˆ’๐‘ง)/๐‘Ž๐‘ง)(1/2๐œ‹๐‘–)ฮ“(๐‘Ž๐‘ค๐‘“(๐‘ค)/(๐‘Žโˆ’๐‘ค))(1/(๐‘คโˆ’๐‘ง))๐‘‘๐‘คโˆซฮ“(1/(๐‘คโˆ’๐‘ง))๐‘‘๐‘ค,๐‘งโˆˆฮฉ.(6.27) 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 function ๐‘ง๐‘(๐‘ก) is 2๐œ‹-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 Nystrรถmโ€™s method with the trapezoidal rule [24]. The trapezoidal rule is the most accurate method for integrating periodic functions numerically [25, page 134โ€“142]. Thus, solving the integral equations numerically reduces to solving linear systems of the form๐ด๐‘‹=๐ต.(7.1) 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, 11โ€“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, 27โ€“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 ฮ“๐‘— is ๐‘›.

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 of ๐‘“, ๐œ‡1 and approximate values ๐‘“๐‘›, ๐œ‡1๐‘› are shown in Table 1.


๐‘› โ€– ๐œ‡ โˆ’ ๐œ‡ ๐‘› โ€– โˆž โ€– ๐‘“ โˆ’ ๐‘“ ๐‘› ( ๐‘ก ) โ€– โˆž

8 1 . 8 ร— 1 0 โˆ’ 0 5 2 . 2 ร— 1 0 โˆ’ 0 2
16 3 . 7 ร— 1 0 โˆ’ 1 0 5 . 0 ร— 1 0 โˆ’ 0 6
32 8 . 8 ร— 1 0 โˆ’ 1 6 3 . 4 ร— 1 0 โˆ’ 1 4

Example 7.1. Consider a region ฮฉ bounded by the unit circle ๎€ฝ๐‘งฮ“โˆถ(๐‘ก)=๐‘’๐‘–๐‘ก๎€พ,๐‘Ž=โˆ’0.2+0.2๐‘–,(7.2) Then, the exact mapping function is given by [1, page 340] ๐‘”(๐‘ง)=(๐‘Žโˆ’๐‘ง)๎€ท๐‘Ž๐‘ง1โˆ’๎€ธ1๐‘Ž๐‘ง,๐‘Ÿ=|๐‘Ž|.(7.3) Figure 2 shows the region and its image based on our method. See Table 1 for results.

Example 7.2. Consider the elliptical region bounded by the ellipse ฮ“โˆถ{๐‘ง(๐‘ก)=4cos๐‘ก+2๐‘–sin๐‘ก},๐‘Ž=โˆ’0.2โˆ’0.2๐‘–.(7.4) Figure 3 shows the region and its image based on our method. See Table 2 for our computed value of ๐œ‡0.


๐‘› ๐œ‡ 0

16 3.5383174719052
32 3.5355590602433
64 3.5355585660566
128 โ€”

Example 7.3. Consider a region ฮฉ bounded by ๎€ฝ๐‘งฮ“โˆถ(๐‘ก)=(10+3cos3๐‘ก)๐‘’๐‘–๐‘ก๎€พ,๐‘Ž=0.1โˆ’0.6๐‘–.(7.5) Figure 4 shows the region and its image based on our method. See Table 3 for comparison between our computed values of ๐œ‡0 with those computed values ๐œ‡0๐‘› of Nasser [12, 13].


๐‘› โ€– ๐œ‡ 0 โˆ’ ๐œ‡ 0 ๐‘› โ€– โˆž

8 1 . 0 ร— 1 0 โˆ’ 0 2
16 7 . 2 ร— 1 0 โˆ’ 0 5
32 1 . 1 ร— 1 0 โˆ’ 0 8
64 4 . 6 ร— 1 0 โˆ’ 1 5

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. Figures 5โ€“7 show the region and its image based on our method, and approximate values of ๐œ‡0 and ๐œ‡1 are shown in Tables 4โ€“6.


2 ๐‘› โ€– ๐œ‡ 0 โˆ’ ๐œ‡ 0 ๐‘› โ€– โˆž โ€– ๐œ‡ 1 โˆ’ ๐œ‡ 1 ๐‘› โ€– โˆž

32 3 . 2 ร— 1 0 โˆ’ 0 3 5 . 8 ร— 1 0 โˆ’ 0 3
64 2 . 4 ร— 1 0 โˆ’ 0 6 5 . 1 ร— 1 0 โˆ’ 0 6
128 1 . 7 ร— 1 0 โˆ’ 1 2 3 . 5 ร— 1 0 โˆ’ 1 2
256 8 . 8 ร— 1 0 โˆ’ 1 6 2 . 2 ร— 1 0 โˆ’ 1 5


2 ๐‘› โ€– ๐œ‡ 0 โˆ’ ๐œ‡ 0 ๐‘› โ€– โˆž โ€– ๐œ‡ 1 โˆ’ ๐œ‡ 1 ๐‘› โ€– โˆž

64 1 . 5 ร— 1 0 โˆ’ 0 3 6 . 2 ร— 1 0 โˆ’ 0 4
128 4 . 9 ร— 1 0 โˆ’ 0 7 8 . 5 ร— 1 0 โˆ’ 1 0
256 7 . 1 ร— 1 0 โˆ’ 1 4 3 . 5 ร— 1 0 โˆ’ 1 4


2 ๐‘› โ€– ๐œ‡ 0 โˆ’ ๐œ‡ 0 ๐‘› โ€– โˆž โ€– ๐œ‡ 1 โˆ’ ๐œ‡ 1 ๐‘› โ€– โˆž

64 2 . 3 ร— 1 0 โˆ’ 0 3 2 . 4 ร— 1 0 โˆ’ 0 3
128 7 . 4 ร— 1 0 โˆ’ 0 7 9 . 5 ร— 1 0 โˆ’ 0 7
256 7 . 3 ร— 1 0 โˆ’ 1 4 9 . 9 ร— 1 0 โˆ’ 1 4

Example 7.4 (circular frame). Consider a pair of circles [28] ฮ“0โˆถ๎€ฝ๐‘ง(๐‘ก)=๐‘’๐‘–๐‘ก๎€พ,ฮ“1โˆถ๎€ฝ๐‘ง(๐‘ก)=โˆ’0.6+0.2๐‘’โˆ’๐‘–๐‘ก๎€พ,๐‘กโˆถ0โ‰ค๐‘กโ‰ค2๐œ‹,๐‘Ž=0.25+0.25๐‘–,(7.6) such that the region bounded by ฮ“0 and ฮ“1 is the region between a unit circle and a circle centered at โˆ’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 ๐œ‡0 and ๐œ‡1 with those computed values ๐œ‡0๐‘› and ๐œ‡1๐‘› of Nasser [12, 13].

Example 7.5 (ellipse with one circle). Consider a region ฮฉ bounded by an ellipse and a circle ฮ“0ฮ“โˆถ{๐‘ง(๐‘ก)=4cos๐‘ก+๐‘–sin๐‘ก},1โˆถ๎€ฝ๐‘ง(๐‘ก)=โˆ’1+0.25๐‘’โˆ’๐‘–๐‘ก๎€พ,๐‘กโˆถ0โ‰ค๐‘กโ‰ค2๐œ‹,๐‘Ž=โˆ’1.4,(7.7) such that the region bounded by ฮ“0 and ฮ“1 is the region between an ellipse and a circle centered at โˆ’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 ๐œ‡0 and ๐œ‡1 with those computed values ๐œ‡0๐‘› and ๐œ‡1๐‘› of Nasser [12, 13].

Example 7.6 (two ellipses). Consider a region ฮฉ bounded by pair of ellipses ฮ“0ฮ“โˆถ{๐‘ง(๐‘ก)=4cos๐‘ก+๐‘–sin๐‘ก},1โˆถ{๐‘ง(๐‘ก)=1+0.7cos๐‘กโˆ’0.3๐‘–sin๐‘ก},๐‘กโˆถ0โ‰ค๐‘กโ‰ค2๐œ‹,๐‘Ž=2.3.(7.8) Figure 7 shows the region and its image based on our method. See Table 6 for comparison between our computed values of ๐œ‡0 and ๐œ‡1 with those computed values ๐œ‡0๐‘› and ๐œ‡1๐‘› 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 of ๐œ‡0, ๐œ‡1, ๐œ‡2 and the computed values of ๐œ‡0๐‘›, ๐œ‡1๐‘›, ๐œ‡2๐‘› obtained from [12, 13] are shown in Tables 7โ€“9.


3 ๐‘› โ€– ๐œ‡ 0 โˆ’ ๐œ‡ 0 ๐‘› โ€– โˆž โ€– ๐œ‡ 1 โˆ’ ๐œ‡ 1 ๐‘› โ€– โˆž โ€– ๐œ‡ 2 โˆ’ ๐œ‡ 2 ๐‘› โ€– โˆž

48 5 . 1 ร— 1 0 โˆ’ 0 4 1 . 3 ร— 1 0 โˆ’ 0 3 4 . 7 ร— 1 0 โˆ’ 0 4
96 2 . 8 ร— 1 0 โˆ’ 0 6 7 . 5 ร— 1 0 โˆ’ 0 6 3 . 9 ร— 1 0 โˆ’ 0 6
192 2 . 4 ร— 1 0 โˆ’ 1 0 6 . 3 ร— 1 0 โˆ’ 1 0 3 . 1 ร— 1 0 โˆ’ 1 0
384 5 . 5 ร— 1 0 โˆ’ 1 7 2 . 7 ร— 1 0 โˆ’ 1 6 4 . 9 ร— 1 0 โˆ’ 1 6


3 ๐‘› โ€– ๐œ‡ 0 โˆ’ ๐œ‡ 0 ๐‘› โ€– โˆž โ€– ๐œ‡ 1 โˆ’ ๐œ‡ 1 ๐‘› โ€– โˆž โ€– ๐œ‡ 2 โˆ’ ๐œ‡ 2 ๐‘› โ€– โˆž

96 1 . 6 ร— 1 0 โˆ’ 0 5 1 . 0 ร— 1 0 โˆ’ 0 3 4 . 9 ร— 1 0 โˆ’ 0 3
192 2 . 7 ร— 1 0 โˆ’ 0 6 2 . 8 ร— 1 0 โˆ’ 0 6 8 . 6 ร— 1 0 โˆ’ 0 7
384 1 . 2 ร— 1 0 โˆ’ 1 1 1 . 4 ร— 1 0 โˆ’ 1 1 1 . 2 ร— 1 0 โˆ’ 1 1


3 ๐‘› ๐œ‡ 0 ๐œ‡ 1 ๐œ‡ 2

96 1.144844712112 1.333447560114 1.711779222648
192 1.144844080644 1.333446944282 1.711778670173
384โ€” 1.333446944281 โ€”

Example 7.7 (three ellipses). Let ฮฉ be the region bounded by ฮ“0ฮ“โˆถ{๐‘ง(๐‘ก)=10cos๐‘ก+6๐‘–sin๐‘ก},1โˆถฮ“{๐‘ง(๐‘ก)=โˆ’4โˆ’2๐‘–+3cos๐‘กโˆ’2๐‘–sin๐‘ก},2โˆถ{๐‘ง(๐‘ก)=4+2cos๐‘กโˆ’3๐‘–sin๐‘ก},0โ‰ค๐‘กโ‰ค2๐œ‹,๐‘Ž=7.(7.9) Figure 8 shows the region and its image based on our method. See Table 7 for comparison between our computed values of ๐œ‡0, ๐œ‡1, and ๐œ‡2 with those computed values of Nasser [12].

Example 7.8 (ellipse with two circles). Let ฮฉ be the region bounded by [7, 13, 15] ฮ“0ฮ“โˆถ{๐‘ง(๐‘ก)=4cos๐‘ก+๐‘–sin๐‘ก},1โˆถฮ“{๐‘ง(๐‘ก)=1.2+0.3(cos๐‘กโˆ’๐‘–sin๐‘ก)},2โˆถ{๐‘ง(๐‘ก)=โˆ’1+0.6(cos๐‘กโˆ’๐‘–sin๐‘ก)},0โ‰ค๐‘กโ‰ค2๐œ‹,๐‘Ž=โˆ’2.5โˆ’0.1๐‘–.(7.10) Figure 9 shows the region and its image based on our method. See Table 8 for comparison between our computed values of ๐œ‡0, ๐œ‡1, and ๐œ‡2 with those computed values of Nasser [13].

Example 7.9 (three circles). Let ฮฉ be the region bounded by ฮ“0โˆถ๎€ฝ๐‘ง(๐‘ก)=2๐‘’๐‘–๐‘ก๎€พ,ฮ“1โˆถ๎€ฝ๐‘ง(๐‘ก)=1.2+0.3๐‘’โˆ’๐‘–๐‘ก๎€พ,ฮ“2โˆถ๎€ฝ๐‘ง(๐‘ก)=โˆ’1+0.6๐‘’โˆ’๐‘–๐‘ก๎€พ,0โ‰ค๐‘กโ‰ค2๐œ‹,๐‘Ž=0.5โˆ’1.25๐‘–.(7.11) Figure 10 shows the region and its image based on our method. See Table 9 for our computed values of ๐œ‡0, ๐œ‡1, and ๐œ‡2.

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 of ๐œ‡0, ๐œ‡1, ๐œ‡2, ๐œ‡3, ๐œ‡4 and the computed values of ๐œ‡0๐‘›, ๐œ‡1๐‘›, ๐œ‡2๐‘›, ๐œ‡3๐‘›, ๐œ‡4๐‘› obtained from [12] are shown in Tables 10 and 12.


4 ๐‘› โ€– ๐œ‡ 0 โˆ’ ๐œ‡ 0 ๐‘› โ€– โˆž โ€– ๐œ‡ 1 โˆ’ ๐œ‡ 1 ๐‘› โ€– โˆž โ€– ๐œ‡ 2 โˆ’ ๐œ‡ 2 ๐‘› โ€– โˆž โ€– ๐œ‡ 3 โˆ’ ๐œ‡ 3 ๐‘› โ€– โˆž

64 6 . 7 ร— 1 0 โˆ’ 0 5 7 . 2 ร— 1 0 โˆ’ 0 5 9 . 9 ร— 1 0 โˆ’ 0 5 2 . 2 ร— 1 0 โˆ’ 0 5
128 6 . 4 ร— 1 0 โˆ’ 0 9 5 . 0 ร— 1 0 โˆ’ 0 8 1 . 8 ร— 1 0 โˆ’ 0 9 4 . 5 ร— 1 0 โˆ’ 0 8
256 6 . 8 ร— 1 0 โˆ’ 1 3 1 . 0 ร— 1 0 โˆ’ 1 2 9 . 8 ร— 1 0 โˆ’ 1 3 9 . 7 ร— 1 0 โˆ’ 1 3
512 1 . 3 ร— 1 0 โˆ’ 1 6 1 . 2 ร— 1 0 โˆ’ 1 5 3 . 0 ร— 1 0 โˆ’ 1 6 4 . 4 ร— 1 0 โˆ’ 1 6

Example 7.10. Let ฮฉ be the region bounded by [12] ฮ“0โˆถ๎€ฝ๐‘ง(๐‘ก)=(10+3cos3๐‘ก)๐‘’๐‘–๐‘ก๎€พ,ฮ“1โˆถ๎€ฝ๐‘ง(๐‘ก)=โˆ’3.5+6๐‘–+0.5๐‘’โˆ’๐‘–๐œ‹/4๎€ท๐‘’๐‘–๐‘ก+4๐‘’โˆ’๐‘–๐‘ก,ฮ“๎€ธ๎€พ2โˆถ๎€ฝ๐‘ง(๐‘ก)=5+0.5๐‘’๐‘–๐œ‹/4๎€ท๐‘’๐‘–๐‘ก+4๐‘’โˆ’๐‘–๐‘ก,ฮ“๎€ธ๎€พ3โˆถ๎€ฝ๐‘ง(๐‘ก)=โˆ’3.5โˆ’6๐‘–+0.5๐‘’๐‘–๐œ‹/4๎€ท๐‘’๐‘–๐‘ก+4๐‘’โˆ’๐‘–๐‘ก๎€ธ๎€พ,0โ‰ค๐‘กโ‰ค2๐œ‹,๐‘Ž=8.5+0.1๐‘–.(7.12) Figure 11 shows the region and its image based on our method. See Table 10 for comparison between our computed values of ๐œ‡0, ๐œ‡1, ๐œ‡2, and ๐œ‡3 with those computed values of Nasser [12].

Example 7.11 (ellipse with three circles). Let ฮฉ be the region bounded by ฮ“0ฮ“โˆถ{๐‘ง(๐‘ก)=2cos๐‘ก+1.5๐‘–sin๐‘ก},1โˆถฮ“{๐‘ง(๐‘ก)=1+0.25(cos๐‘กโˆ’๐‘–sin๐‘ก)},2ฮ“โˆถ{๐‘ง(๐‘ก)=โˆ’1+0.25(cos๐‘กโˆ’๐‘–sin๐‘ก)},3โˆถ{๐‘ง(๐‘ก)=0.75๐‘–+0.25(cos๐‘กโˆ’๐‘–sin๐‘ก)},0โ‰ค๐‘กโ‰ค2๐œ‹,๐‘Ž=0.25โˆ’0.25๐‘–.(7.13) Figure 12 shows the region and its image based on our method. See Table 11 for our computed values of ๐œ‡0, ๐œ‡1, ๐œ‡2, and ๐œ‡3.


4 ๐‘› ๐œ‡ 0 ๐œ‡ 1 ๐œ‡ 2 ๐œ‡ 3

64 2.97316998311 2.50170500411 3.45373711618 3.69125205510
128 2.96757277502 2.49923061605 3.45041067650 3.69904161729
256 2.96756361086 2.49922735100 3.45040617845 3.69905124306
512 2.96756361085 2.49922735099 3.45040617844 3.69905124308


5 ๐‘› โ€– ๐œ‡ 0 โˆ’ ๐œ‡ 0 ๐‘› โ€– โˆž โ€– ๐œ‡ 1 โˆ’ ๐œ‡ 1 ๐‘› โ€– โˆž โ€– ๐œ‡ 2 โˆ’ ๐œ‡ 2 ๐‘› โ€– โˆž โ€– ๐œ‡ 3 โˆ’ ๐œ‡ 3 ๐‘› โ€– โˆž โ€– ๐œ‡ 4 โˆ’ ๐œ‡ 4 ๐‘› โ€– โˆž

80 4 . 2 ร— 1 0 โˆ’ 0 5 4 . 5 ร— 1 0 โˆ’ 0 5 4 . 5 ร— 1 0 โˆ’ 0 5 4 . 4 ร— 1 0 โˆ’ 0 5 4 . 3 ร— 1 0 โˆ’ 0 5
160 1 . 1 ร— 1 0 โˆ’ 0 7 3 . 2 ร— 1 0 โˆ’ 0 8 3 . 2 ร— 1 0 โˆ’ 0 8 6 . 6 ร— 1 0 โˆ’ 0 8 6 . 6 ร— 1 0 โˆ’ 0 8
320 1 . 6 ร— 1 0 โˆ’ 1 3 5 . 7 ร— 1 0 โˆ’ 1 4 5 . 7 ร— 1 0 โˆ’ 1 4 1 . 2 ร— 1 0 โˆ’ 1 3 1 . 2 ร— 1 0 โˆ’ 1 3
400 9 . 9 ร— 1 0 โˆ’ 1 6 0 9 . 9 ร— 1 0 โˆ’ 1 6 0 0

Example 7.12 (ellipse with four circles). Let ฮฉ be the region bounded by ฮ“0ฮ“โˆถ{๐‘ง(๐‘ก)=0.2+8cos๐‘ก+6๐‘–sin๐‘ก},1โˆถฮ“{๐‘ง(๐‘ก)=3+2๐‘–+cos๐‘กโˆ’๐‘–sin๐‘ก},2ฮ“โˆถ{๐‘ง(๐‘ก)=โˆ’3+2๐‘–+cos๐‘กโˆ’๐‘–sin๐‘ก},3ฮ“โˆถ{๐‘ง(๐‘ก)=โˆ’3โˆ’2๐‘–+cos๐‘กโˆ’๐‘–sin๐‘ก},4โˆถ{๐‘ง(๐‘ก)=3โˆ’2๐‘–+cos๐‘กโˆ’๐‘–sin๐‘ก},0โ‰ค๐‘กโ‰ค2๐œ‹,๐‘Ž=4๐‘–.(7.14) Figure 13 shows the region and its image based on our method. See Table 12 for comparison between our computed values of ๐œ‡0, ๐œ‡1, ๐œ‡2, ๐œ‡3, and ๐œ‡4 with those computed values of Nasser [12].

Example 7.13 (five ellipses). Let ฮฉ be the region bounded by ฮ“0ฮ“โˆถ{๐‘ง(๐‘ก)=โˆ’1.5๐‘–+6cos๐‘ก+8๐‘–sin๐‘ก},1โˆถฮ“{๐‘ง(๐‘ก)=3+0.5๐‘–+1.5cos๐‘กโˆ’๐‘–sin๐‘ก},2ฮ“โˆถ{๐‘ง(๐‘ก)=โˆ’3+0.5๐‘–+1.5cos๐‘กโˆ’๐‘–sin๐‘ก},3ฮ“โˆถ{๐‘ง(๐‘ก)=โˆ’3๐‘–+0.7cos๐‘กโˆ’1.7๐‘–sin๐‘ก},4โˆถ{๐‘ง(๐‘ก)=โˆ’6๐‘–+1.7cos๐‘กโˆ’0.7๐‘–sin๐‘ก},0โ‰ค๐‘กโ‰ค2๐œ‹,๐‘Ž=0.4๐‘–.(7.15) Figure 14 shows the region and its image based on our method. See Table 13 for our computed values of ๐œ‡0, ๐œ‡1, ๐œ‡2, ๐œ‡3, and ๐œ‡4.


5 ๐‘› ๐œ‡ 0 ๐œ‡ 1 ๐œ‡ 2 ๐œ‡ 3 ๐œ‡ 4

160 0.4081769461 0.5470254751 0.5470254751 0.6850879289 0.5258641902
320 0.4081097591 0.5470505181 0.5470505181 0.6850466360 0.5258066821
400 0.4081097885 0.5470505071 0.5470505071 0.6850466537 0.5258067072

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 of ๐‘“(๐‘ง). 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.

Acknowledgments

This work was supported in part by the Malaysian Ministry of Higher Education (MOHE) through the Research Management Centre (RMC), Universiti Teknologi Malaysia (FRGS Vote 78479). This support is gratefully acknowledged. The authors would like to thank an anonymous referee for careful reading of the paper and constructive comments and suggestions that substantially improved the presentation of the paper.

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Copyright ยฉ 2012 Ali W. K. Sangawi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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