Conformal Mapping Problems Research Articles

Conformal Mapping of a Z-Shaped Domain

For the problem of conformal mapping of a half-plane onto a Z-shaped domain with arbitrary geometry, an efficient method is developed for finding parameters of the Schwarz–Christoffel integral, i.e., the preimages of the vertices (prevertices) and the pre-integral multiplier. Special attention is given to the situation of crowding prevertices, in which case conventional integration methods face significant difficulties. For this purpose, the concept of a cluster is introduced, its center is determined, and all integrand binomials with prevertices from this cluster are expanded into a fast-convergent series by applying a unified scheme. Next, the arising integrals are reduced to single or double series in terms of Gauss hypergeometric functions F(a, b, c, q). The fast convergence of the resulting expansions is ensured by applying formulas for analytic continuation of F(a, b, c, q) to a neighborhood of the point q = 1 and using numerically stable recurrence relations. The constructed expansions are also fairly efficient for choosing initial approximations for prevertices in Newton’s iteration method. By using the leading terms of these expansions, the approximations for the prevertices are expressed in explicit form in terms of elementary functions, and the subsequent iterations ensure the fast convergence of the algorithm. After finding the parameters in the integral, the desired mapping is constructed as a combination of power series expansions at prevertices, regular expansions at the preimage of the center of symmetry, a Laurent series in a semi-annulus, and special series near the preimages of the vertical segments. Numerical results demonstrate the high efficiency of the developed method, especially in the case of strong crowding of prevertices.

On the Solution of a Conformal Mapping Problem by Means of Weierstrass Functions

The conformal mapping problem for the section of a channel filled with porous material under a rectangular dam onto the upper half-plane is considered. Similar problems arise in computing of fluid flow in hydraulic structures. As a solution method, the representation of Christoffel–Schwartz elliptic integral in terms of Weierstrass functions is used. The calculation is based on Taylor series for the sigma function, the coefficients of which are determined recursively. A simple formula for a conformal mapping is obtained, which depends on four parameters and uses the sigma function. A numerical experiment was carried out for a specific area. The degeneration of the region, which consists in the dam width tending to zero, is considered, and it is shown that the resulting formula has a limit that implements the solution of the limiting problem. A refined proof of Weierstrass recursive formula for the coefficients of Taylor series of the sigma function is presented.

Infinite Product Representation for the Szegö Kernel for an Annulus

The Szegö kernel has many applications to problems in conformal mapping and satisfies the Kerzman-Stein integral equation. The Szegö kernel for an annulus can be expressed as a bilateral series and has a unique zero. In this paper, we show how to represent the Szegö kernel for an annulus as a basic bilateral series (also known as q -bilateral series). This leads to an infinite product representation through the application of Ramanujan’s sum. The infinite product clearly exhibits the unique zero of the Szegö kernel for an annulus. Its connection with the basic gamma function and modified Jacobi theta function is also presented. The results are extended to the Szegö kernel for general annulus and weighted Szegö kernel. Numerical comparisons on computing the Szegö kernel for an annulus based on the Kerzman-Stein integral equation, the bilateral series, and the infinite product are also presented.

Approximate method for solving the problem of conformal mapping of an arbitrary polygon to a unit circle

Approximate method for solving the problem of conformal mapping of an arbitrary polygon to a unit circle

Schwarz–Christoffel accessory parameter for quadrilaterals via isomonodromy

We develop the recent proposal by the authors to exploit the isomonodromic tau function defined by Jimbo, Miwa and Ueno (JMU) to solve the accessory parameter problem in conformal mapping theory. We focus here on mappings of Schwarz–Christoffel type: in particular, the mapping from the upper half plane to a four-sided polygon where the sides are all straight lines. We show that one can obtain the relevant accessory parameters—the pre-image of the polygonal vertices—via a special ‘zero curvature limit’ in which the radius of curvature of some of the edges tends to zero. We apply the procedure to rectangular domains where the JMU tau function is given by a ratio of Riemann theta functions, known as the Picard solution, and take the zero curvature limit to recover the accessory parameter obtained by Nehari using quite different methods. We then turn to trapezoids, deriving new asymptotic formulas for the accessory parameters in the limit of large and small aspect ratios. Our work lends a new geometrical perspective to problems of isomonodromy that we believe provides theoretical insight, while also showing how classical problems in conformal mapping can benefit from new ideas emerging from isomonodromic deformation theory.

The problem of extreme decomposition of a complex plane with fixed poles on a circle

The problem of extreme decomposition of a complex plane with fixed poles on a circle. Investigation on geometric function theory has been conducted by several researchers, however, few studies have reported on the problem considering extremal configurations the product of inner radii of non-overlapping domains with respect to fixed poles. The paper describes the problem of finding the maximum of the product of inner radii of mutually non-overlapping symmetric domains with respect to points on a unit circle multiply by a certain positive degree \(\gamma\) of the inner radius of the domain with respect to the zero. The problem was studied using the method of separating transformation. Proving the theorem shows that the maximum is obtained if \(\gamma\in(1,n^2]\) and for all \(n\geqslant 2\). Its results and the method for the obtaining of these results can be used in the theory of potential, approximations, holomorphic dynamics, estimation of the distortion problems in conformal mapping, and complex analysis.

Combinatorial and modular solutions of some sequences with links to a certain conformal map

If fn is a free parameter, we give a combinatorial closed form solution of the recursion (n + 1)2un+1 − fnun − n 2un−1 = 0, n ≥ 1, and a related generating function. This is used to give a solution to the Apery type sequence rnn3 + rn−1nαn3 −3α2n+α+ 2θon − θo+ rn−2(n − 1)3 = 0, n ≥ 2, for certain parameters α, θ. We show from another viewpoint two independent solutions of the last recursion related to certain modular forms associated with a problem of conformal mapping: Let f(τ) be a conformal map of a zero-angle hyperbolic quadrangle to an open half plane with values 0, ρ, 1, ∞ (0 < ρ < 1) at the cusps and define t = t(τ) := 1ρf(τ)f(τ)−ρ(τ)−1. Then the function E(τ) = 12πif0(τ)f(τ)11 −f(τ)ρ is a solution, as a generating function in the variable t, of the above recurrence. In other words, E(τ) = r0 +r1t+r2t 2 +. . . , where r0 = 1, r1 = −θ, α = 2− 4ρ.

Boundary First Flattening

A conformal flattening maps a curved surface to the plane without distorting angles—such maps have become a fundamental building block for problems in geometry processing, numerical simulation, and computational design. Yet existing methods provide little direct control over the shape of the flattened domain, or else demand expensive nonlinear optimization. Boundary first flattening (BFF) is a linear method for conformal parameterization that is faster than traditional linear methods, yet provides control and quality comparable to sophisticated nonlinear schemes. The key insight is that the boundary data for many conformal mapping problems can be efficiently constructed via the Cherrier formula together with a pair of Poincaré-Steklov operators ; once the boundary is known, the map can be easily extended over the rest of the domain. Since computation demands only a single factorization of the real Laplace matrix, the amortized cost is about 50× less than any previously published technique for boundary-controlled conformal flattening. As a result, BFF opens the door to real-time editing or fast optimization of high-resolution maps, with direct control over boundary length or angle. We show how this method can be used to construct maps with sharp corners, cone singularities, minimal area distortion, and uniformization over the unit disk; we also demonstrate for the first time how a surface can be conformally flattened directly onto any given target shape.

Abelian Higgs vortices and discrete conformal maps

We establish a connection between recent developments in the study of vortices in the abelian Higgs models, and in the theory of structure-preserving discrete conformal maps. We explain how both are related via conformal mapping problems involving prescribed linear combinations of the curvature and volume form, and show how the discrete conformal theory can be used to construct discrete vortex solutions.

Zolotarev's conformal mapping and Chebotarev's problem

Consider the Chebotarev problem of finding a continuum S in the complex plane including some given points such that the logarithmic capacity of S is minimal. In this paper, we give a complete solution of this problem for the case of three given points with the help of Zolotarev's conformal mapping using Jacobian elliptic and theta functions. Moreover, for four given points, some special cases can be treated.

Transformations of sections upon the conformal mapping of certain circular polygons appearing in flow problems with free surfaces

The linear differential equation of the Fuchsian class with six regular singular points, which corresponds to the problem of conformal mapping of circular hexagons in polar networks with two sections, is considered. It is shown that when fixing the parameter, which characterizes the deviation of the radii of the circles constituting the opposite sides of the polygon, on which the sections can appear, the configuration and mutual arrangement of the latter depend substantially not only and not even mainly on the known properties of θ functions, based on which the partial solutions of the equation under consideration are constructed, but mainly on the ranges of varying the unknown constants of the conformal mapping, which enter the expressions for imaging functions. The results of numerical calculations are presented.

Conformal mapping and inverse conductivity problem with one measurement

This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series of numerical examples illustrating the performance of the method.

A 3-dimensional Bergman kernel method with applications to rectangular domains

In this paper we revisit the so-called Bergman kernel method—BKM—for solving conformal mapping problems and propose a generalized BKM-approach to extend the theory to three-dimensional mapping problems. A special software package for quaternions was developed for the numerical experiments.

The Riemann–Hilbert problem and the generalized Neumann kernel

This paper presents two new Fredholm integral equations associated to the interior and the exterior Riemann–Hilbert problems in simply connected regions with smooth boundaries. The kernel of these integral equations is the generalized Neumann kernel. The solvability of the integral equations depends on whether λ = ± 1 are eigenvalues of the kernel which in turn depends on the index of the Riemann–Hilbert problem. The complete discussion of the solvability of the integral equations with the generalized Neumann kernel is presented. The integral equations can be used effectively to solve numerically the Riemann–Hilbert problems. The case of non-uniquely solvable Riemann–Hilbert problems is treated by imposing additional constraints to get a uniquely solvable problem. Fredholm integral equations with generalized Neumann kernels are also derived for the problem of the interior and the exterior harmonic conjugate functions. As applications, we study the problem of conformal mapping to a nearby region and extend Wegmann's iterative method to general regions. Numerical examples reveal that the present method offers an effective solution technique for the Riemann–Hilbert problems when the boundaries are sufficiently smooth.

Constructive solution of a certain class of Riemann–Hilbert problems on multiply connected circular regions

A class of Riemann–Hilbert (RH) problems on multiply connected circular regions is treated iteratively by a method of successive conjugation (SC), which solves the problem on a single boundary circle each time, running through all circles in each iterative step. It is shown that the method converges. Convergence is linear with a rate which can be calculated with an eigenvalue problem. The RH problems considered have negative index. In order to fulfill the solvability conditions, free parameters are introduced in form of polynomials. The proof of convergence of the SC method shows at the same time that the RH problems with this specification of the free parameters have unique solutions. RH problems of this kind are of practical use in conformal mapping problems of multiply connected regions to circular regions. They play a role in the mapping of nearly circular regions (Lavrentev's principle). There is also a close relationship to Koebe's iterative method for the calculation of conformal mappings. The properties of RH problems are different for unbounded and bounded regions. The SC iteration must be adapted accordingly. The performance of the SC iteration is illustrated with several examples.

Dispersionless limit of hirota equations in some problems of complex analysis

We study the integrable structure recently revealed in some classical problems in the theory of functions in one complex variable. Given a simply connected domain bounded by a simple analytic curve in the complex plane, we consider the conformal mapping problem, the Dirichlet boundary problem, and the 2D inverse potential problem associated with the domain. A remarkable family of real-valued functionals on the space of such domains is constructed. Regarded as a function of infinitely many variables, which are properly defined moments of the domain, any functional in the family gives a formal solution of the above problems. These functions satisfy an infinite set of dispersionless Hirota equations and are therefore tau-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda chain. In addition to our previous studies, we show that within a more general definition of the moments, this connection pertains not to a particular solution of the Hirota equations but to the hierarchy itself.

Applications of Fuchsian differential equations to free boundary problems

In a couple of recent papers free boundary problems and a class of conformal mappings involving curvilinear quadrilaterals were analysed primarily using transform methods. In both the free boundary and conformal mapping problems there is an underlying relation with Fuchsian differential equations and an alternative, conceptually simpler, solution technique for these problems would use solutions of these Fuchsian equations. Thus it was conjectured in those papers that a class of Fuchsian differential equations, commonly known as Heun9s equation, has in some special, but relatively important, cases degenerate solutions; these involve hypergeometric functions. This complementary solution method is, in many cases, more convenient than the transform approach. The purpose of this paper is to explore the connections with the Fuchsian equations directly, and use the solutions to solve free boundary problems. The specific examples treated here come from a quasi–steady approximation to solidification problems and are not without interest in their own right. There are few analytical solutions for solidification problems in geometries of practical interest, and the solutions found here should be of use in that regard.

From Potential Theory to Matrix Iterations in Six Steps

The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, and so on) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor $\rho \le 1$ can be derived by solving a problem of potential theory or conformal mapping. Six approximations are involved in relating the actual computation to this scalar estimate. These six approximations are discussed in asystematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.

Nonlinear Riemann ‐ Hilbert Problems without Transversality

AbstractNonlinear Riemann ‐ Hilbert problems (RHP) generalize two fundamental classical problems for complex analytic functions, namely: 1. the conformal mapping problem, and 2. the linear Riemann ‐ Hilbert problem. This paper presents new results on global existence for the nonlinear (RHP) in doubly connected domains with nonclosed restriction curves for the boundary data. More precisely, our nonlinear (RHP) is required to become „at infinity”︁, i.e., for solutions having large moduli, a linear (RHP) with variable coefficients. Global existence for q‐connected domains was already obtained in [9] for the special case that the restriction curves for the boundary data „at infinity”︁ coincide with straight lines corresponding to linear (RHP)‐s with special so‐called constant ‐ coefficient transversality boundary conditions. In this paper, the boundary conditions are much more general including highly nonlinear conditions for bounded solutions in the context of nontransversality. In order to prove global existence, we reduce the problem to nonlinear singular integral equations which can be treated by a degree theory of Fredholm ‐ quasiruled mappings specifically constructed for mappings defined by nonlinar pseudodifferential operators.

Some solutions of the problem of plane steady flow of a gas condensate mixture through a porous medium

The plane problem of steady two-phase flow of a multicomponent mixture through a porous medium with phase transitions is considered. It is shown that the system of equations for the two-phase multicomponent flow process, together with the equations of phase equilibrium, can be solved in quadratures if the solution of two auxiliary problems is known. These are the problem of conformal mapping of the neighborhood of a well onto a rectangle and the purely physicochemical problem of the description of the mechanical and thermodynamic properties of a mixture. The solutions for a vertical well with a barrier and for a horizontal well in a finite productive stratum are found under certain assumptions concerning the properties of the mixture.