Complex Analysis

Written by a master of the subject, this text will be appreciated by students and experts for the way it develops the classical theory of functions of a complex variable in a clear and straightforward manner. In general, the approach taken here emphasises geometrical aspects of the theory in order to avoid some of the topological pitfalls associated with this subject. Thus, Cauchy's integral formula is first proved in a topologically simple case from which the author deduces the basic properties of holomorphic functions. Starting from the basics, students are led on to the study of conformal mappings, Riemann's mapping theorem, analytic functions on a Riemann surface, and ultimately the Riemann-Roch and Abel theorems. Profusely illustrated, and with plenty of examples, and problems (solutions to many of which are included), this book should be a stimulating text for advanced courses in complex analysis.

The Riemann mapping theorem has been said by some to be the greatest theorem of the nineteenth century. The entire concept of the theorem is profoundly original, and its proof introduced many new ideas. Certainly normal families and the use of extremal problems in complex analysis are just two of the important techniques that have grown out of studies of the Riemann mapping theorem.

In this note, we give an explicit construction of iterative approximation of Riemann mappings. Our method is motivated by Koebe's proof of the Riemann mapping theorem.

(2007). Riemann Mapping Theorem by Steepest Descent. The American Mathematical Monthly: Vol. 114, No. 3, pp. 246-251.

INTRODUCTION. This note concerns Riemann mappings of rectangles onto the unit disk in the complex plane. Let's fix some notation for the discussion. For the unit circle and unit disk we'll use, respectively, S1 and D. For the rectangle parallel to the coordinate axes, centered at the origin 0 E C with height 1 and base length a > 0, so that the corners are at (?a/2, ? 1/2), we'll use R(a). In particular, it will be convenient to have R(b) c R(a) if b < a. Riemann's mapping theorem says that every simply connected plane domain Ql not equal to the whole plane can be mapped conformally one-to-one onto D. Thus this certainly holds for the interior of rectangles. These maps are thoroughly well understood from several points of view, primarily using the Schwarz-Christoffel formula (elliptic functions in a not too subtle disguise). The purpose of this note is to show that rectangles with different modulus (different values of a) can be seen as conformally different and completely characterized by sets of four distinct points on S with very little machinery. In fact, our only tools will be the Schwarz reflection principle, the Schwarz lemma (in other words, Schwarz seemed to have a pretty good sense about this sort of thing), some elementary conformal mappings, and the concept of normal families of analytic functions. Perhaps then the real purpose of this note is to serve as a reminder of the power of the profound, beautiful, and essentially elementary maximum modulus principle. The only reference necessary for this discussion is [A]. The theorem of Lowner may also be found in [C]. The proof given there is of a somewhat different flavor, and Lowner's proof [L] was roughly the same as the one given here.

In my talk I will attempt to draw some connections between complex analysis and computational geometry, particularly between conformal mappings, hyperbolic geometry, the medial axis and optimal meshing. The Riemann mapping theorem says that there is a conformal (angle preserving) map of the unit disk, DISK, to the interior Ω of any simple n-gon. How much work is needed to compute this map? [MR2671015] We can compute the conformal map f: DISK -> Ω to within error e in time O(n ⋅ log 1/e log log 1/e).The proof uses an iteration that converges quadratically to f, but it needs a good initial guess that is close to the correct answer. The medial axis allows us to quickly construct a map Ω -> DISK that is close to Riemann's map with precise estimates. This gives our starting point, and the time estimates in the theorem depend on time needed to compute the medial axis (linear by work of Chin, Snoeyink and Wang).Conversely, the proof of Theorem 1 uses ideas from analysis that may be of interest in CG. For example, the dome of a planar domain Ω is the surface in R+3 given by the upper envelope of all hemispheres whose base on R2 is a medial axis disk of Ω. Domes arise in the study of 3-manifolds, and a fundamental result of Dennis Sullivan about convex sets in hyperbolic 3-space lies at the heart of Theorem 1. The proof of Theorem 1 also introduces the thick-thin decomposition of a polygon, inspired by the thick-thin decomposition of a hyperbolic manifold. The thin parts correspond to long, narrow pieces of the polygon (but not in the obvious way; the precise definition uses a conformal invariant called extremal length). This decomposition plays in important role in proving: [MR2671014] Any simple n-gon has a O(n) quadrilateral mesh with all angles ≤ 120O and all new angles ≥ 60O.The sharp upper bound is due to Bern and Eppstein; the lower bound is the novel part here. The result is also true for PSLGs with at most n edges and vertices, although simple examples show the O(n) must be replaced by O(n2): [Bishop-Quad] Every PSLG has an O(n2) quadrilateral mesh with all angles ≤ 120O and all new angles ≥ 60O. Only O(n/e) angles satisfy |θ-90O| > e, for any e >0.

Normal families and the Riemann mapping theorem

Introduction. From the point of view of conformal mapping, it is unsatisfactory to consider the individual points of the boundary of a simply connected region as the primitive constituents of this boundarv. When such a region is mapped conformally onto the unit disk, in accordance with the Riemann mapping theorem, the points of the unit circumference correspond, indeed, to the prime ends of the region. These prime ends are here considered as equivalence classes of sequences of points of the region under a relation P*. The principal tool we employ to introduce this relation is extremal length, a conformally invariant quantity associated with a family of curves. Its definition was first given by Ahlfors and Beurling ([2], [3]). The term prinle end originated with Caratheodory [4], who initiated the systematic study of the structure of the boundary of a simply connected region. His approach was topological, and it dealt with conceptssubregions, crosscuts, etc.-which are defined with reference to the given region. It is easily seen that with Caratheodory's definition the prime ends of the unit disk correspond to points on the circumference in the sense that an equivalence class is associated with each boundary point and that two sequences belong to the same class if and only if they converge to the same point of the circumference. The problem that arises under his approach, however, and this is solved by one of Caratheodory's fundamental theorems, is that one must show that prime ends are preserved under conformal mapping. Lindelif [6] circumvented this difficulty by defining prime ends by reference to the conformal map of the disk onto the region; namely in terms of the set of indetermination or cluster set. However, his method does not obviate an explicit analysis of the topological situation in the region itself. Lindelof also obtained the following result, which is related to the classification of prime ends. Caratheodory had distinguished two classes of points, principal and subsidiary (see Section 2. 7, below), on the point set of a prime end, i. e. on the

This is an informal talk for non-specialists dealing on an elementary level with some aspects of the theory of partial differential equations. In recent years progress in the theory has been tremendous, often in unexpected directions, while also solving classical problems in more general settings. New fields have been added, like the study of variational inequalities, of solitons, of wave front sets, of pseudo-differential operators, of differential forms on manifolds, etc. Much of the progress has been made possible by the use of functional analysis. However, in the process much of the original simplicity of the theory has been lost. This is perhaps connected with the emphasis on solving problems, which often requires the piling up mountains of a priori inequalities and the skillful juggling of function spaces to make ends meet. It is good to remember that mathematics is not only concerned with solving problems, but with studying the structure and behavior of the objects it creates. One of the best examples is the classical theory of functions of a complex variable. It, incidentally, does solve problems as in the Riemann mapping theorem. But much of its beauty lies in statements that can hardly be considered as “solving” anything, like the calculus of residues, or Picard’s theorem, or Cauchy’s formula

History of the Riemann Mapping Theorem

Geometric Aspects and the Riemann Mapping Theorem

Exercise 293. Determine the types of fractional linear transformations that map the upper half plane onto the unit disc, with the real axis onto the unit circle.

The Riemann mapping theorem, that an arbitrary simply connected region of the plane can be mapped one-to-one and conformally onto a circle, first appeared in the Inaugural dissertation of Riemann (1826-1866) in 1851. The theorem is important, for by it a result proved for the circle can often be transformed from the circle to a more general region. The proof is difficult, as involving both behavior of a function in the small (conformal mapping) and behavior in the large (one-toone mapping). Riemann's proof was open to criticism and in the following decades numerous mathematicians sought for a proof, e.g., H. A. Schwarz (1843-1921), A. Harnack (1851-1888), H. Poincare (1854-1912), etc., until the first rigorous proof was given in 1900 by W. F. Osgood. The proof of Osgood represented, in my opinion, the coming of age of mathematics in America. Until then, numerous American mathematicians had gone to Europe for their doctorates, or for other advanced study, as indeed did Osgood. But the mathematical productivity in this country in quality lagged behind that of Europe, and no American before 1900 had reached the heights that Osgood then reached. William Fogg Osgood (1864-1943) was born in Boston in 1864, graduated from Harvard College in 1886, stayed in Cambridge for a year of graduate work, and then went to Gottingen with a Harvard fellowship for further study, especially with Felix Klein (1849-1925). According to gossip, Osgood became so enamored of a Gottingen lady that his work suffered and Klein sent him to Erlangen for his doctorate. In any case, he was accorded the degree from Erlangen in 1890 for a thesis on Abelian integrals, and one or two days later he married the girl in G6ttingen, and one or two days still later they sailed for the United States of America. His

Before proving the Riemann Mapping Theorem, we examine the relation between conformal mapping and the theory of fluid flow. Our main goal is to motivate some of the results of the next section and the treatment here will be less formal than that of the remainder of the book.

This research is motivated by the program of reverse mathematics and non‐standard arguments in second‐order arithmetic. Within a weak subsystem of second‐order arithmetic ACA0, we investigate some aspects of non‐standard analysis related to sequential compactness. Then, using arguments of non‐standard analysis, we show the equivalence of the Riemann mapping theorem and ACA0 over WKL0. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)