Analytic Continuation - Initial Discussion

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.