Initial Discussion
Suppose f is an analytic function defined on an open subset U of the complex plane . If V is a larger open subset of, containing U, and F is an analytic function defined on V such that
then F is called an analytic continuation of f. In other words, the restriction of F to U is the function f we started with.
Analytic continuations are unique in the following sense: if V is the connected domain of two analytic functions F1 and F2 such that U is contained in V and for all z in U
- F1(z) = F2(z) = f(z),
then
- F1 = F2
on all of V. This is because F1 − F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain. This follows directly from the identity theorem for holomorphic functions.
Read more about this topic: Analytic Continuation
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