Complex Analysis: A General Higher Education National Planning Textbook for the 15th National Plan.

Author: Zheng Jianhua
Publisher:
Publish Date: 2005-01-01
Features: This book uses about half of its pages to introduce the fundamental conclusions of classical complex analysis established in the mid-19th century, including the complex number field, analytic functions, Cauchy's theorem, Cauchy's integral formula, Laurent series expansions, the argument principle, the residue theorem, and their applications in real integral calculations, etc. The other half primarily focuses on the fundamental conclusions unique to complex analytic functions, while also involving some newly developed conclusions and related disciplines. The main content includes: after the Great Modulus Theorem, it introduces Nevanlinna theory; after the basic conclusions of normal families, it briefly discusses normal families using Zalcman's new method and derives the equivalence between Picard's great and small theorems and the Montel theorem; it introduces the basic conclusions of conformal mappings and univalent functions; after the elementary Riemann surfaces, it further introduces the ideas, concepts, and fundamental conclusions of Riemann surfaces; through the Dirichlet boundary value problem on a disk, it introduces the basic knowledge of harmonic functions, and through general Dirichlet boundary value problems, it introduces harmonic measures, Green functions, etc.; finally, it introduces hyperbolic geometry and its applications from the perspective of hyperbolic metrics, using a geometric viewpoint to understand complex analytic functions.
This book is rich in content, rigorous in logic, and progressive in structure. It can serve as a textbook for undergraduate students in mathematics and applied mathematics departments, as well as a reference book for graduate students and teachers in related fields, and can also be read by relevant scientific and technological workers.