Analytic Continuation

Abstract

We have previously seen that an analytic function is determined by its behavior at a sequence of points having a limit point. This was precisely the content of the identity theorem (see Theorem 8.48) which is also referred to as the principle of analytic continuation. For example, as a consequence, there is precisely a unique entire function on ℂ which agrees with sin x on the real axis, namely sin z. But we have not yet explored the following question: If f(z) is analytic in a domain D₁, is there a function analytic in a different domain D₂ that agrees with f(z) in D₁ ∩ D₂? Analytic continuation deals with the problem of properly redefining an analytic function so as to extend its domain of analyticity. In the process, we come across functions for which no such extension exists. Finally, we apply our knowledge of analytic continuation to two of the most important functions in analysis, the gamma function and the Riemann-zeta function, defined originally by a definite integral and an infinite series, respectively.