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Home > Analytic continuation

Suppose f is an analytic function defined on the open subset U of the complex plane C. If V is an open subset of C containing U, and F is an analytic function defined on V such that F(z) = f(z) for all z in U, then F is called an analytic continuation of f.

Analytic continuations are unique in the following sense: if V is connected and F₁ and F₂ are two analytic continuations of f defined on V, then F₁ = F₂. That is because the difference is an analytic function vanishing on a non-empty open set.

For example, if a power series with radius of convergence r is given, one can consider analytic continuations of the power series, i.e. analytic functions F which are defined on larger sets than

{z : |z − a| < r},

and agree with the given power series on that set. The number r is maximal in the following sense: there always exists a complex number z with

|z − a| = r

such that no analytic continuation of the series can be defined at z. Therefore there is a limitation to analytic continuation to bigger discs with the same centre a. On the other hand there may well be analytic continuations to some larger sets. If not, there a natural boundary on the bounding circle.

A common way to define functions in complex analysis proceeds by first specifying the function on a small domain only, and then extending it by analytic continuation. In practice, this continuation is often done by first establishing some functional equation on the small domain and then using this equation to extend the domain. Examples are the Riemann zeta function and the gamma function.

The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function.

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