The following definition is in {{harvtxt|Ahlfors|1979}}, but also found in Weyl or perhaps Weierstrass. 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 [[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.
