Computational Methods and Function Theory

Bénéteau, Catherine; Khavinson, Dmitry

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321614

In this survey paper, we examine the isoperimetric inequality from an analytic point of view. We use as a point of departure the concept of analytic content in approximation theory: this approach reveals ties to overde-termined boundary problems and hydrodynamics. In particular, we look at problems connected to determining the shape of an electrified droplet or equivalently, that of an air bubble in fluid flow. We also discuss the connection with the Schwarz function and quadrature domains. Finally, we survey some known generalizations to higher dimensions and list many open problems that remain. This paper is an expanded version of the plenary talk given by the second author at the fifth CMFT Conference in Joensuu, Finland, in June 2005.

DeLillo, Thomas

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321615

A Schwarz-Christoffel formula for conformal maps from the exterior of a finite number of disks to the exterior of polygonal curves was derived by DeLillo, Elcrat, and Pfaltzgraff in [9] using the Reflection Principle. The derivative of the map is expressed as an infinite product. In this paper, the formula for the map from bounded circular domains to bounded polygonal domains is derived by the same method. Convergence of the resulting infinite product is proved for sufficiently well-separated domains. A formula for the bounded case was also derived by Crowdy in [5] using Schottky-Klein prime functions. We show that Crowdy’s formula can be reduced to ours. In addition, we discuss the relation of these formulae to the Poincaré theta series for functions automorphic under the Schottky group of Moebius transformations generated by reflections in circles. We also derive a formula for the map to circular slit domains.

DeLillo, Thomas; Driscoll, Tobin; Elcrat, Alan; Pfaltzgraff, John

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321616

We have recently derived a Schwarz-Christoffel formula for the conformal mapping of the exterior of a finite number of disks to the exterior of a set of polygonal curves [5]. In this work we show how to formulate a set of equations for determining the parameters of such a map. A number of examples are computed, including exteriors of multiple slits. We also recall the derivation of the mapping formulae and give a new formula for the doubly connected case.

Steinmetz, Norbert

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321617

In this note we prove that the so-called Sierpi\’nski holes in the parameter plane 0 < ¦λ¦ < ∞ of the McMullen family Fλ(z) = z m + λ/z ℓ, m ≥ 2 and ℓ ≥ 1 fixed, are simply connected, and determine the total number of these domains.

Clarkson, Peter

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321618

Rational solutions of the second, third and fourth Painlevé equations can be expressed in terms of special polynomials defined through second order bilinear differential-difference equations which are equivalent to the Toda equation. In this paper the structure of the roots of these special polynomials, as well as the special polynomials associated with algebraic solutions of the third and fifth Painlevé equations and equations in the PII hierarchy, are studied. It is shown that the roots of these polynomials have an intriguing, highly symmetric and regular structure in the complex plane. Further, using the Hamiltonian theory for the Painlevé equations, other properties of these special polynomials are studied. Soliton equations, which are solvable by the inverse scattering method, are known to have symmetry reductions which reduce them to Painlevé equations. Using this relationship, rational solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations and rational and rational-oscillatory solutions of the non-linear Schrödinger equation are expressed in terms of these special polynomials.

Poggi-Corradini, Pietro

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321619

We study the uniqueness properties of classical semiconjugations for analytic self-maps of the disk, by characterizing them as canonical solutions to certain functional equations. As a corollary, we obtain a complete description of all possible solutions to such equations.

Sola, Alan

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321620

In the theory of conformal mappings, one way of measuring how much a univalent function expands or contracts the unit disk is to study the integral means of its derivative along circles of increasing radii. Recently, Hedenmalm and Shimorin were able to find estimates on the universal means spectrum of conformal maps of the unit disk by using a combination of area-type methods and Bergman space techniques. Their paper included a numerical implementation based on the first two terms in a series expansion. We take three terms into account and show that it is possible to improve the previous results slightly.

Kariofillis, Christos; Nestoridis, Vassili

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321621

Let Ω be a simply connected domain in ℂ For a function f holomorphic in Ω let S n (f, ζ) denote the partial sum of the Taylor development of f with center ζ ∈ Ω. We show that generically overconvergence phenomena of S n(f, ζ) and their derivatives can occur on the boundary ∂Ω or in parts of it. In the rest of the boundary ∂Ω, the universal function f may be smooth. These parts of the boundary do not have to be connected.

Andrievskii, Vladimir; Carpenter, Amos; Varga, Richard

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321622

We continue the work of Szegő [18] on describing the angular distribution of the zeros of the normalized partial sum s n(nz) of e z, where $s _{n}(z):={\sum _{k=0} ^{n}}z ^k/k!$ . We imbed this problem in some inverse problem of potential theory and prove a so-called Erdős-Turán-type theorem, which is of interest in itself.

Maergoiz, Lev

2013 Computational Methods and Function Theory

doi: 10.1007/BF03321623

We obtain an analogue of the well-known Paley-Wiener Theorem on integral representation of entire functions of exponential type at most σ, σ > 0, which belong to the space L 2(ℝ). We choose 1 < p < 2 and σ > 0 and work in L p (ℝ). We find optimal estimates of the modulus on any line parallel to ℝ, and present applications to best analytic continuation from a finite set in ℂ for entire functions of this class. The main result was announced in [7].