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In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes 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. The case of 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. ==Initial discussion== Suppose ''f'' is an analytic function defined on a non-empty open subset ''U'' of the complex plane C. If ''V'' is a larger open subset of C, 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 ''F''1 and ''F''2 such that ''U'' is contained in ''V'' and for all ''z'' in ''U'' :''F''1(''z'') = ''F''2(''z'') = ''f''(''z''), then :''F''1 = ''F''2 on all of ''V''. This is because ''F''1 − ''F''2 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. ==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 surfaces. The power series defined below is generalized by the idea of a ''germ''. The general theory of analytic continuation and its generalizations are known as sheaf theory. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「analytic continuation」の詳細全文を読む スポンサード リンク
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