John A. Pfaltzgraff

Schwarz-Christoffel mapping of multiply connected domains

A Schwarz-Christoffel mapping formula is established for polygonal domains of finite connectivitym≥2 thereby extending the results of Christoffel (1867) and Schwarz (1869) form=1 and Komatu (1945),m=2. A formula forf, the conformal map of the exterior ofm bounded disks to the exterior ofm bounded disjoint polygons, is derived. The derivation characterizes the global preSchwarzianf″ (z)/f′ (z) on the Riemann sphere in terms of its singularities on the sphere and its values on them boundary circles via the reflection principle and then identifies a singularity function with the same boundary behavior. The singularity function is constructed by a “method of images” infinite sequence of iterations of reflecting prevertex singularities from them boundary circles to the whole sphere.

Radial and circular slit maps of unbounded multiply connected circle domains

Infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain, or to domains with both radial and circular slit boundary components are derived and implemented numerically and graphically. The formulae are generated by analytic continuation with the reflection principle. Convergence of the infinite products is proved for domains with sufficiently well-separated boundary components. Some recent progress in the numerical implementation of infinite product mapping formulae is presented.

Norm order and geometric properties of holomorphic mappings in\(\mathbb{C}^n \)

We introduce a new notion of the order of a linear invariant family of locally biholomorphic mappings on then-ball. This order, which we call the norm order, is defined in terms of the norm rather than the trace of the “second Taylor coefficient operator” of mappings in a family. Sharp bounds on ‖Df(z)‖ and ‖f(z)‖, a general covering theorem for arbitrary LIFs and results about convexity, starlikeness, injectivity and other geometric properties of mappings given in terms of the norm order illustrate the useful nature of this notion. The norm order has a much broader range of influence on the geometric properties of mappings than does the “trace” order that the present authors and many others have used in recent years.

Distortion of locally biholomorphic maps of the N-Ball

Distortion Bounds mappings, in linear invariant families of locally hiholomorphic mappings from the n-Ball into Cn are established. Coordinate-free methods yield a short and straightforward proof of the distortion theorem. The n-dimensional versions of linear-invariant families, the order of a family and the Koebe transform are introduced. A method for generating many examples of linear-invariant families is presented.

Numerical conformal mapping of multiply connected regions by Fornberg-like methods

Summary. We develop a new algorithm for computing conformal maps from regions exterior to non-overlapping disks to unbounded multiply connected regions exterior to non-overlapping, smoothly bounded Jordan regions. The method is an extension of Fornbergs original Newton-like method for mapping of the disk to simply connected regions. A Fortran program based on the algorithm has been developed and tested for the 2 and 3 disk case. Numerical examples are reported.

Linear Invariance, Order and Convex Maps in

We continue the study, begun by the first author, of linear-invariant families in . We obtain the unexpected result that the Cayley transform (the analogue of the half-plane mapping in the complex plane) does not give the maximum or minimum distortion among all mappings of the unit ball of n≥ 2, onto convex domains. In addition a result analogous to that of Pommerenke that a linear invariant family has order 1 (the smallest possible order) if and only if it consists of convex mappings of the disk does not hold for the ball in n≥ 2. Finally, we extend these ideas to the polydisk in . The theory for this case bears some similarity to the theory for the ball but there are some striking differences as well. For example it is true that the family of convex holomorphic mappings of the polydisk in has minimum order (which is nin this case) but it is not true that for n>1 every linear-invariant family of minimum order consists of convex mappings.

Linear spaces and linear-invariant families of locally univalent analytic functions

Let X be the set of functions f analytic in the unit disk D with derivatives f′(z)≠0 in D, that satisfy the growth restriction f″(z)/f′(z)=0((1−|z|)−1 (z∈D). The set X is equipped with a normed linear space structure. The topological structure of X and its relationship with linear-invariant families of locally univalent functions is investigated.

k-Quasiconformal extension criteria in the disk ∗

The method of Loewner chains is used to establish univalence and k-quasiconformal extension criteria for functions analytic in the unit disk. Results include univalence and k-quasiconformal extension criteria that generalize two of Neharis Schwarzian derivative univalence criteria. Various criteria involving z 2 f1;(z)/f 0(z) and its logarithmic derivative are developed.

Numerical Conformal Mapping Methods for Simply and Doubly Connected Regions

Methods are presented and analyzed for approximating the conformal map from the interior (exterior) of the disk to the interior (exterior) of a smooth, simple closed curve and from an annulus to a bounded, doubly connected region with smooth boundaries. The methods are Newton-like methods for computing the boundary correspondences and conformal moduli similar to Fornbergs method for the interior of the disk. We show that the linear systems are discretizations of the identity plus a compact operator and, hence, that the conjugate gradient method converges superlinearly.

Schwarz--Christoffel Mapping of the Annulus

A new derivation of the Schwarz--Christoffel (S--C) transformation for doubly connected domains is given. The work is based on constructing the pre-Schwarzian by reflection of singularities. A derivation of the simply connected S--C map without appeal to Liouvilles theorem is a byproduct of this work.