5 Analytic Continuation

Abstract

Our focus in the last chapter was on the construction of an analytic function from a knowledge of its singularities.More often than not, however, we are confronted with the inverse problem: given some knowledge of a function in a restricted region of its domain of holomorphy, determine its singularities.This will be the focus of the present chapter.We have seen repeatedly that one need not know all that much about an analytic function in order to determine its value everywhere in the complex plane or on its Riemann surface.Cauchy's Integral Representation can be viewed as the embodiment of this property and thus far it has provided the key to exploiting it.We are now going to nd out what constitutes a minimal set of information for the determination of an analytic function.The answer is one that is best exploited not by Cauchy's Integral but by one of its consequences, the Taylor series.In so doing, we shall also nd out how to use a representation of a function that is valid in one domain of the complex plane to determine its values at points outside the domain or indeed, at any points where it is holomorphic.Our starting point is the following theorem which, despite its innocuous appearance, is one of the most remarkable results of complex analysis.Theorem: Let f (z) and f (z) be holomorphic in a domain D of the complex plane.If the two functions coincide in any neighbourhood, however small, of a point z in D, or even on a point set with an accumulation point in D, then they coincide throughout D. Proof: The function f (z)f (z) is holomorphic throughout D and has a set of zeros consisting of the points where f (z) and f (z) coincide, with an accumulation point in D. We know that in any domain where it is holomorphic a function either has isolated zeros or it is identically zero.Thus,What this theorem establishes is that a holomorphic function is uniquely determined everywhere within its domain of holomorphy by its behaviour in the neighbourhood of an arbitrary point of that domain.But how can one exploit this remarkable property?Obviously not by means of a Cauchy Integral or dispersion representation or anything else of that ilk as we lack the necessary input information.However, what we do have is precisely the information needed to determine a Taylor series representation.Suppose that we know the value of the function f (z) throughout a neighbourhood of the point z = z which is a point lying within the function's domain of holomorphy, D. This is su cient to permit calculation of the coe cients c = f (z ), c = f (z ), . . ., cm = m! f (m) (z ), . . .