Complex Analysis

Abstract

As we have already seen, complex numbers crop up quite naturally in physics. Often, the solutions to problems are conveniently expressed in terms of complex functions [e.g., the angular momentum eigenfunctions Y _l ^m (θ, ø) of Eq. (5.26)]. Here, we employ the techniques of complex analysis to establish the equivalence of various definitions of each special function. Most physics and applied mathematics texts in which special functions appear make use of generating functions, asymptotic forms, and recursion formulas for these functions. The properties of analytic functions will be useful to us in deriving these relations. In the next two chapters we develop enough of the theory of complex variables to enable us to carry out these tasks.