This book of the well-known Japanese laureate and winner of the Fields medal appeared for the first time in Japanese (1977, 1978) in three volumes. The present version contains all three volumes and it appears in English for the first time in a such form (the first two volumes were published in English under the title “Introduction to Complex Analysis“ in 1984). The book starts on a basic level and many theorems (which are quite often considered as prerequisites of such texts) are included. The approach can be characterized as geometrical: it was the intention of the author to avoid a detailed study of topological properties of the complex plane.
The first part of the book contains classical material on holomorphic functions, the Cauchy theorem and conformal mapping, followed by chapters on analytic continuation and the Riemann mapping theorem. More than the last third of the book is devoted to Riemann surfaces. It is divided into three chapters: Riemann surfaces, The structure of Riemann surfaces and Analytic functions on a closed Riemann surface. The last chapter includes the Riemann-Roch and Abel theorems. While most of the material included in the first part could be used in a basic course on complex analysis, the whole book could serve as a text for an advanced course on Riemann surfaces. The book contains many pictures (helping to build geometric intuition) and problems (elementary and advanced). The book could be very helpful for students as well as for experts in the field.