This book is a masterful account of the traditional first course in complex variables given at Helsinki University, and reflects in its first seven chapters the influence of E. Lindelöf's introductory text [Johdatus funktioteoriaan, Mercatorin Kirjapaino, Helsinki, 1936]. The scope of the book is best described by the chapter headings: (1) The concept of analytic function; (2) General properties of rational functions; (3) Linear transformations; (4) Mappings by second-order rational functions; (5) The exponential function and its inverse; (6) Trigonometrical functions; (7) Series with complex terms; (8) Integration in the complex domain and the Cauchy integral formula; (9) The Cauchy integral formula and its application; (10) The residue theorem and its applications; (11) Harmonic functions; (12) Analytic continuation; (13) Entire functions; (14) Periodic functions (most of this chapter is devoted to doubly periodic functions); (15) The Euler gamma function; (16) The Riemann zeta function; (17) The theory of conformal mapping (this includes the Riemann mapping theorem, boundary correspondence, triangle functions, and the Picard theorem). At the end of each chapter there is an extensive set of exercises ranging from routine to challenging.
Reviewer: Lohwater, A. J. [form MathSciNet]