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In the [[mathematics|mathematical]] field of [[complex analysis]], a '''global analytic function''' (or '''complete analytic function''') is a generalization of the notion of an [[analytic function]] which allows for functions to have multiple [[branch cut|branches]]. Global analytic functions arise naturally in considering the possible [[analytic continuation]]s of an analytic function, since analytic continuations may have a non-trivial [[monodromy]]. They are one foundation for the theory of [[Riemann surface]]s.
The definition of a global analytic function goes back to [[Karl Weierstrass]].
==Definition==
The following definition ismay duebe tofound in {{harvtxt|Ahlfors|1979}}. An analytic function in an [[open set]] ''U'' is called a '''function element'''. Two function elements (''f''<sub>1</sub>, ''U''<sub>1</sub>) and (''f''<sub>2</sub>, ''U''<sub>2</sub>) are said to be [[analytic continuation]]s of one another if ''U''<sub>1</sub> ∩∩ ''U''<sub>2</sub> ≠ ∅∅ and ''f''<sub>1</sub> = ''f''<sub>2</sub> on this intersection. A '''chain of analytic continuations''' is a finite sequence of function elements (''f''<sub>1</sub>, ''U''<sub>1</sub>), ……, (''f''<sub>''n''</sub>,''U''<sub>''n''</sub>) such that each consecutive pair are analytic continuations of one another; i.e., (''f''<sub>''i''+1</sub>, ''U''<sub>''i''+1</sub>) is an analytic continuation of (''f''<sub>''i''</sub>, ''U''<sub>''i''</sub>) 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.
===Sheaf-theoretic definition===
Using ideas from [[sheaf theory]], the definition can be streamlined. In these terms, a '''complete global analytic function''' is a [[Connected_space#Path_connectedness|path -connected]] sheaf of germs of analytic functions which is ''maximal'' in the sense that it is not contained (as an [[etale space]]) within any other path connected sheaf of germs of analytic functions.
==References==
* {{citation|first=Lars|last=Ahlfors|authorlink=Lars Ahlfors|title=Complex analysis|publisher=McGraw Hill|edition=3rd|year=1979|isbn=978-00700065770-07-000657-7}}
* {{cite book |last=Markushevich |first=A. I. |title=Theory of Functions of a Complex Variable, Volume 3 |year=1977 |publisher=Chelsea Publishing Company}}
* {{SpringerEOM|title=Complete analytic function|author=E. D. Solomentsev}}
[[Category:Complex analysis]]
[[Category:Types of functions]]
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