Complex Function Tutorial

Author: Fang Qiqin
Publisher:
Publish Date: 2001-03-01
Features: The book "A Course in Complex Analysis" fully incorporates interesting and inspiring content from foreign textbooks. For example, Chapter 3 is used as an application of derivative rules to prove the relationship between the zeros of a polynomial and the zeros of its derivative; the proof of the Cauchy formula employs mathematical induction; Chapter 6 introduces the properties of local mappings of analytic functions in the application of single-valued analytic functions. The main difference between this book and current foreign textbooks lies in the treatment of the Cauchy theorem. Foreign textbooks introduce the concept of the winding number of a curve around a point and the homology of closed curve families in a region being zero before proving the theorem, using methods such as Dixon's proof, Beurdon's proof, or Artin's proof. However, this book still adopts the traditional Pringsheim method. This approach does not affect the subsequent study of specialized courses such as Riemann surfaces, several complex variables, Hardy spaces, and univalent function theory. Moreover, the presentation in foreign textbooks cannot replace the approach in this book, such as the discussion that the integral path can be the boundary of the region and the case where the region can include the point at infinity.