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==Definition==
The following definition is due to {{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'').
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