Editing Analytic continuation (section)

{{Short description|Extension of the domain of an analytic function (mathematics)}}

In [[complex analysis]], a branch of [[mathematics]], '''analytic continuation''' is a technique to extend the [[domain of a function|domain of definition]] of a given [[analytic function]]. Analytic continuation often succeeds in defining further values of a function, for example in a new region where the [[infinite series]] representation which initially defined the function becomes [[Divergent series|divergent]].

The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of [[mathematical singularities|singularities]]. The case of [[Function of several complex variables|several complex variables]] is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of [[sheaf cohomology]].