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global analytic function
In the mathematical field of complex analysis, a global analytic function is a generalization of the notion of an analytic function which allows for functions to have multiple branches. Global analytic functions arise naturally in considering the possible analytic continuations of an analytic function, since analytic continuations may have a non-trivial monodromy. They are one foundation for the theory of Riemann surfaces.
==Definition==
The following definition is due to . An analytic function in an open set ''U'' is called a function element. Two function elements (''f''1, ''U''1) and (''f''2, ''U''2) are said to be analytic continuations of one another if ''U''1 ∩ ''U''2 ≠ ∅ and ''f''1 = ''f''2 on this intersection. A chain of analytic continuations is a finite sequence of function elements (''f''1, ''U''1), …, (''f''''n'',''U''''n'') such that each consecutive pair are analytic continuations of one another; i.e., (''f''''i''+1, ''U''''i''+1) is an analytic continuation of (''f''''i'', ''U''''i'') for ''i'' = 1, 2, …, ''n'' − 1.
A global analytic function is a family f of function elements such that, for any (''f'',''U'') and (''g'',''V'') belonging to f, there is a chain of analytic continuations in f beginning at (''f'',''U'') and finishing at (''g'',''V'').
A complete global analytic function is a global analytic function f which contains every analytic continuation of each of its elements.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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