Essentials of Mathematical Methods in Science and Engineering, 2nd Edition

CHAPTER 10COMPLEX ANALYSIS

Line integrals, power series, and residues constitute an important part of complex analysis. Complex integration theorems are usually concise but powerful. Many of the properties of analytic functions are quite difficult to prove without the use of these theorems. Complex contour integration also allows us to evaluate various difficult proper or improper integrals encountered in physical theories. Just as in real analysis, in complex integration we distinguish between definite and indefinite integrals. Since differentiation and integration are inverse operations of each other, indefinite integrals can be found by inverting the known differentiation formulas of analytic functions. Definite integrals evaluated over continuous, or at least piecewise continuous, paths are not just restricted to analytic functions and thus can be defined exactly by the same limiting procedure used to define real integrals. Most complex definite integrals can be written in terms of two real integrals. Hence, in their discussion, we heavily rely on the background established in Chapters 1 and 2 on real integrals. One of the most important places, where the theorems of complex integration is put to use is in power series representation of analytic functions. In this regard, Laurent series play an important part in applications, which also allows us to classify singular points.

10.1 CONTOUR INTEGRALS

Each point in the complex plane is represented by two parameters, hence, ...