Description
Complex Analysis develops the fundamental theory of functions of a complex variable and connects it with broader areas of mathematical analysis and number theory. The book begins with complex numbers and the complex plane, convergence, holomorphic functions, power series, and integration along curves, followed by Cauchy’s theorem and its applications. It then examines meromorphic functions, singularities, residues, the complex logarithm, Fourier transforms, entire functions, and infinite products. Later chapters develop the Gamma and Zeta functions, the prime number theorem, conformal mappings, elliptic functions, and applications of theta functions. Appendices address asymptotic methods and the Jordan curve theorem. Through rigorous development, examples, exercises, and applications, the book provides a foundation in complex analysis while demonstrating connections between complex analysis, Fourier analysis, and number theory.
