A course in complex analysis in one variable.

*(English)*Zbl 0994.30001
Singapore: World Scientific. ix, 149 p. (2002).

This book discusses a variety of problems which are usually treated in a year-long graduate course on the theory of functions of one complex variable. In particular it covers: holomorphic functions, Cauchy-Riemann equations, conformality, power series, integration along a contour, Cauchy theorem in simply connected domains and its main consequences, Cauchy theorem in multiply connected domains and the pre-residue theorem, Cauchy integral formula and its consequences, analyticity, Taylor’s theorem, identity theorem, spaces of holomorphic functions and Montel’s theorem, maximum modulus theorem and Schwarz’ lemma, singularities, theorems of Riemann and Casorati-Weierstrass, principle of the argument, Rouché’s theorem, transcendental equation, Laurent expansion, calculation of residues, residue theorem, application to calculation of real integrals, a more general removable singularities theorem, Schwarz reflection principle, conformal mappings, linear fractional transformations, equivalence of the unit disk and the upper half plane, automorphism groups of the disk, upper half plane and entire plane, annuli, Riemann mapping theorem for planar domains. An unusual feature of this book is a short final chapter containing applications of complex analysis to Lie theory: complete reducibility of representations according to H. Weyl and the functional equation for the exponential map of a real Lie group, surjectivity of the exponential map for \(U(p,q)\), Zariski density of cofinite volume subgroups of complex Lie groups, differential topology and Lie groups. Although the material is classical, the exposition is organized in an especially efficient manner, presenting basic complex analysis in around 130 pages, with about 50 exercises.

Reviewer: Eleonora Storozhenko (Odessa)

##### MSC:

30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |