Editing Analytic continuation (section)

==Applications==
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]]. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of [[Riemann surface]]s.

Analytic continuation is used in [[Riemannian manifold|Riemannian manifolds]], in the context of solutions of [[Einstein field equations|Einstein's equations]]. For example, [[Schwarzschild coordinates]] can be analytically continued into [[Kruskal–Szekeres coordinates]].<ref>{{Cite journal |last=Kruskal |first=M. D. |date=1960-09-01 |title=Maximal Extension of Schwarzschild Metric |url=https://link.aps.org/doi/10.1103/PhysRev.119.1743 |journal=Physical Review |volume=119 |issue=5 |pages=1743–1745 |doi=10.1103/PhysRev.119.1743|bibcode=1960PhRv..119.1743K |url-access=subscription }}</ref>