Function theory (Q5919255)

The book is based on the author's lectures to undergraduates at the ETH Zuerich. Besides presenting the unavoidable basic topics of complex analysis in one variable it contains some additional material: in particular, the boundary behaviour of the Riemann mapping function at regular analytic pieces of the boundary (studied via the Schwarz reflection principle), the Schwarz-Christoffel formulas and mapping theorems of \(n\)-connected domains onto slit annuli. Questions of plane topology around the concepts of connectedness, winding number, homotopy and homology are especially carefully discussed. The non-elementary functions presented are the Gamma-Function, Riemann's Zeta-Function, and -- very briefly mentioned -- elliptic functions. Quite a few interesting topics turn up as exercises. The book can be recommended as a first step into the field of complex analysis: it might encourage the reader to learn more, and it could also provide a lecturer in the field with useful hints.