Complex Analysis in One Variable

• I Complex Analysis in One Variable.- 1 Elementary Theory of Holomorphic Functions.- 2 Covering Spaces and the Monodromy Theorem.- 3 The Winding Number and the Residue Theorem.- 4 Picard’s Theorem.- 5 Inhomogeneous Cauchy-Riemann Equation and Runge’s Theorem.- 6 Applications of Runge’s Theorem.- 7 Riemann Mapping Theorem and Simple Connectedness in the Plane.- 8 Functions of Several Complex Variables.- 9 Compact Riemann Surfaces.- 10 The Corona Theorem.- 11 Subharmonic Functions and the Dirichlet Problem.- II Exercises.- 0 Review of Complex Numbers.- 1 Elementary Theory of Holomorphic Functions.- 2 Covering Spaces and the Monodromy Theorem.- 3 The Winding Number and the Residue Theorem.- 4 Picard’s Theorem.- 5 The Inhomogeneous Cauchy—Riemann Equation and Runge’s Theorem.- 6 Applications of Runge’s Theorem.- 7 The Riemann Mapping Theorem and Simple Connectedness in the Plane.- 8 Functions of Several Complex Variables.- 9 Compact Riemann Surfaces.- 10 The Corona Theorem.- 11 Subharmonic Functions and the Dirichlet Problem.- Notes for the exercises.- References for the exercises.

"The first part of the book under review represents essentially the material of R. Narasimhan's 'Complex analysis in one variable' (first edition, 1985). The second part of the book, authored by Y. Nievergelt, consists of exercises and relevant references... There are notes at the end of each chapter which contain brief remarks on the history of the material presented as well as references to the literature. The exercises of part II give the reader the opportunity to consolidate his knowledge in complex analysis. At the end of this part there are notes for the exercises and references. The book can be highly recommended for a thorough study of complex analysis." -ZENTRALBLATT MATH (Review of the second edition) "The book introduces and makes use of concepts from many different areas of mathematics, especially ideas used in several complex variables and differential geometry. There is also a short...introductory chapter dealing with several complex variables... The exercises in Part 2 vary from basic to advanced, and provide good practice for the concepts and techniques of the subject... The choice of topics covered gives an excellent introduction to modern complex analysis. The exposition is well written. All in all, this book is a welcome addition to the list of books presenting a first course in complex analysis." -MATHEMATICAL REVIEWS (Review of the second edition) "Provides a smooth and unintimidating transition from classical complex analysis in the plane to modern abstract theory on manifolds... An excellent, carefully written and thematically rich book which does not overwhelm the reader... Well-suited as a textbook either for sophisticated beginners or as a sequel to a one-semester introductory course." -JAHRESBERICHT DER DMV (Review of the first edition)