With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The book begins with the main properties of holomorphic functions: the Cauchy theorems, residues, analytic continuation, and the argument principle. Then, the reader is introduced to a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory.

Elias M. Stein & Rami Shakarchi