Analytic continuation

Extending an analytic function's domain by patching together overlapping power-series pieces.

Analytic continuation is the process of extending the domain on which an analytic function is defined, by finding a second analytic function on an overlapping, larger region that agrees with the first wherever both are defined. Because two analytic functions agreeing on any set with a limit point inside a connected domain must agree everywhere on that domain, such an extension -- whenever it exists -- is completely determined by the original function, which is what makes it meaningful to speak of "the" analytic continuation rather than "an" analytic continuation.

Examples

The geometric series $\sum z^n$ converges only for $|z|<1$ , but it analytically continues to $1/(1-z)$ , defined everywhere except $z=1$ .
The Riemann zeta function is originally defined only for $\operatorname{Re}(z)>1$ ; its most famous properties concern its analytic continuation to the rest of the plane.