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Authors: MIHAI IANCU.
Published Online: 2022-12-02
Published Print: 2022-12-30
pp. 907-908
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The book under review is an excellent introduction to Complex Analysis. The author managed to put together in a harmonious way a large variety of classical results of the theory. Here is a list with the most important topics and results with complete self-contained proofs in the book: the Cauchy-Riemann equations, Cauchy’s theorems and their homology versions, Liouville’s theorem and its hyperbolic version, the identity theorem, the open mapping theorem, the maximum principle for holomorphic and harmonic functions, the residue theorem, the argument principle, Möbius maps and their dynamics, conformal metrics, the Schwarz-Pick lemma and Ahlfor’s generalization, Montel’s theorem and its generalization, the convergence results of Weierstrass, Hurwitz and Vitali, Marty’s theorem, the Riemann mapping theorem, Koebe’s distorsion bounds for the class of schlicht functions, the Carathéodory extension theorem, the solution of the Dirichlet problem on the disk with the Poisson kernel, the Fatou theorem, harmonic measures and Blaschke products, Weierstrass’ factorization theorem, Jensen’s formula, Mittag-Leffler’s theorem, elliptic functions, Runge’s theorem, Schönflies’ theorem, conformal models of finitely connected domains, natural boundaries, Ostrowski’s theorem, the monodromy theorem, the Schwarz reflection principle for analytic arcs, the Hausdorff measure and holomorphic removability, the Schwarz-Christoffel formula, Bloch’s theorem, Schottky’s theorem, Picard’s theorems, Zalcman’s rescaling theorem, branched coverings, the Riemann-Hurwitz formula, the modular group, the uniformization theorem for spherical domains, the characterization of hyperbolic domains, holomorphic covering maps of topological annuli.
