Overview
This very detailed treatise is devoted to the theory of analytic functions in one variable. In addition to the topics customary in an introductory course in the subject, it presents a number of more advanced topics often treated only in more specialized textbooks. The first volume covers the elementary theory (through Cauchy's theorem and power series) in great detail. The second volume covers isolated singular points and the theory of residues, inverse and implicit functions, univalent functions and the Schwarz-Christoffel transformation, harmonic and subharmonic functions, applications to fluid dynamics, the Poisson-Jensen formula and functions of bounded characteristic, and entire functions of finite order. Finally, the third volume covers conformal mapping, including boundary behavior and prime ends; approximation by rational functions and polynomials; elliptic functions; Riemann surfaces, analytic continuation, and the symmetry principle (culminating with Picard's theorem and Julia directions). An important feature of the book is the inclusion of problems (more than 700 overall) to all chapters of the book.