The history of three wrong definitions
The topic is the history of the concepts of equivalence relation, Cauchy sequence, and metric space. The thesis is that disused definitions of these notions could profitably be revived.
math.HO↗
arXiv subjects
Harold P. Boas
Publications and source records attributed to Harold P. Boas.
Cauchy's residue sore thumb
Cauchy's method from two centuries ago for computing integrals along the real axis by passing into the complex plane is not rigorous by present-day standards. Yet when properly formulated, his original approach is simpler than modern presentations of the residue calculus.
math.HO↗
Mocposite functions
Traditional mathematical notation can lead to confusion. Expressions that appear to define composite functions sometimes do not. A particular example with engineering applications is studied in detail.
math.HO↗
Julius and Julia: Mastering the art of the Schwarz lemma
This article discusses classical versions of the Schwarz lemma at the boundary of the unit disk in the complex plane. The exposition includes commentary on the history, the mathematics, and the applications.
math.CV↗
Lu Qi-Keng's Problem
This expository article, intended to be accessible to students, surveys results about the presence or absence of zeroes of the Bergman kernel function of a bounded domain in C^n. Six open problems are stated. The article is based on a lecture at the third Korean several complex variables symposium held at the Global Analysis Research Center at Seoul National University in December 1998.
math.CV↗
Majorant Series
This article discusses questions in one and several complex variables about the size of the sum of the moduli of the terms of the series expansion of a bounded holomorphic function. Although the article is partly expository, it also includes some previously unpublished results with complete proofs. The article is based on a lecture at the third Korean several complex variables symposium held at the Global Analysis Research Center at Seoul National University in December 1998.
math.CV↗
Vita: Friedrich Wilhelm Wiener
We report on the life and work of F. Wiener, whom we confused with N. Wiener in a previous article.
math.CV↗
The football player and the infinite series
This is the text of an expository talk given at the May 1997 Detroit meeting of the American Mathematical Society. It is a tale of a famous football player and a subtle problem he posed about the uniform convergence of Dirichlet series. Hiding in the background is the theory of analytic functions of an infinite number of variables.
math.CV↗
The Bergman kernel function: explicit formulas and zeroes
We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in C^3 defined by the inequality |z_1|+|z_2|+|z_3|<1, have zeroes.
math.CV↗
Global regularity of the dbar-Neumann problem: a survey of the L^2-Sobolev theory
This is a survey article written for the proceedings of the special year in several complex variables, 1995-1996, at the Mathematical Sciences Research Institute in Berkeley.
math.CV↗
Bohr's power series theorem in several variables
Generalizing a classical one-variable theorem of Harald Bohr, we show that if an n-variable power series has modulus less than 1 in the unit polydisc, then the sum of the moduli of the terms is less than 1 in the polydisc of radius 1/(3*n^{1/2}).
math.CV↗
The Lu Qi-Keng Conjecture Fails Generically
The bounded domains of holomorphy in~$\mathbf{C}^n$ whose Bergman kernel functions are zero-free form a nowhere dense subset (with respect to a variant of the Hausdorff distance) of all bounded domains of holomorphy.
math.CV↗