CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well. Contributed papers should be written in English (exceptions in rare cases are tolerated), and in a lucid, expository style. Papers should not exceed 30 printed pages.

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Computational Methods and Function Theory

Chapter 9 – Methods for Numerical Conformal Mapping

We show that given a nonvanishing particular solution of the equation the corresponding differential operator can be factorized into a product of two first-order operators. The factorization allows us to reduce the above equation to a first-order equation which in a two-dimensional case is a Vekua equation of a special form. Under quite general conditions on the coefficients p and q, we obtain an algorithm which allows us to construct in explicit form positive formal powers (solutions of the Vekua equation generalizing the usual powers (z − z0)n, n = 0, 1, ...). This result means that under quite general conditions one can construct an infinite system of exact solutions of the above equation explicitly and, moreover, at least when p and q are real valued this system will be complete in ker(divp grad + q) in the sense that any solution of the above equation in a simply connected domain Ω can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of Ω. Finally, we give a similar factorization of the operator (divp grad + q) in a multidimensional case and obtain a natural generalization of the Vekua equation which is related to second-order operators in a similar way as its two-dimensional prototype does.

On a factorization of second-order elliptic operators and applications

We propose a method for solving boundary value and eigenvalue problems for the elliptic operator D = div p grad + qin the plane using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be constructed following a simple algorithm consisting in recursive integration. This system of solutions is used for solving boundary value and spectral problems for the operator D in bounded simply connected domains. We study theoretical and numerical aspects of the method. Copyright © 2010 John Wiley & Sons, Ltd.

Solution of boundary and eigenvalue problems for second-order elliptic operators in the plane using pseudoanalytic formal powers

We present recently obtained results in the theory of pseudoanalytic functions and its applications to elliptic second-order equations. The operator (divpgrad+q) with p and q being real valued functions is factorized with the aid of Vekua type operators of a special form and as a consequence the elliptic equation 
(divpgrad+q)u=0, (1) reduces to a homogeneous Vekua equation describing generalized analytic (or pseudoanalytic) functions. As a tool for solving the Vekua equation we use the theory of Taylor and Laurent series in formal powers for pseudoanalytic functions developed by L. Bers. The series possess many important properties of the usual analytic power series. Their applications until recently were limited mainly because of the impossibility of their explicit construction in a general situation. We obtain an algorithm which in a really broad range of practical applications allows us to construct the formal powers and hence the pseudoanalytic Taylor series in explicit form precisely for the Vekua equation related to equation (1). In other words, in a bounded domain this gives us a complete (in C-norm) system of exact solutions of (1).

Recent developments in applied pseudoanalytic function theory

In the theory of pseudoanalytic functions one can define (pseudoanalytic) rational functions, especially polynomials called “pseudopolynomials”. (See Bers [3], [4], Vekua [12]) Therefore it can be developed a theory of approximation and interpolation by rational functions. First results have been published by Bers [3] (Runge's theorem), Ismailov and Taglieva [8]. Let G be a domain of the complex plane bounded by a closed Jordan curve, let w(z) be pseudoanalytic in G. In this paper we deal with a relation between the behaviour of w(z) on C (Holder-continuity) and the degree of approximation of w(z) by pseudopolynomials. The results correspond to certain theorems of Curtiss, Sewell and Walsh in the theory of analytic functions.

Zur Approximation pseudoanalytischer Funktionen durch Pseudopolynome

Extremalpunkte und konforme Abbildung

Let and E be a compact subset of D.Tsuji [4] introduced a point system on E by which the hyperbolic capacity ρ(E) of E can be approximated. If E is a continuum, the point system can also be used to approximate the conformal mapping of D\E onto (see Pommerenke [1], Siciak [3]). Here some sharper estimates for the approximations are given under additional assumptions on E.

On tsuji points in a continuum

On the distribution of Tsuji points

LetC be a closed Jordan curve in the complex plane and letf(z)=dz+a0+a1z−1+… be the analytic function mapping |z|>1 schlicht onto the exterior ofC (d>0 is the transfinite diameter ofC). Similar to the Fekete points a point system will be defined calledextremal points. One can use the Fekete points or the extremal points to approximated. The author has proved [3] that in the case of an analytic closed Jordan curve the approximation ofd by means of extremal points is much better than the approximation ofd by the use of Fekete points. Here we show how to approximated by means of extremal points in the case of a piecewise analytic, closed Jordan curve possessing corners of openingαπ (0<α<2).

Klaus Menke

Papers

Zur Approximation pseudoanalytischer Funktionen durch Pseudopolynome

Extremalpunkte und konforme Abbildung

On tsuji points in a continuum

On the distribution of Tsuji points

Zur approximation des transfiniten durchmessers bei bis auf ecken analytischen geschlossenen jordankurven