Global analytic function: Difference between revisions

Revision as of 03:50, 8 May 2012

Revision as of 11:57, 5 May 2014

Line 2:

==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''₁, ''U''₁) and (''f''₂, ''U''₂) are said to be [[analytic continuation]]s 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'').

Line 16:

[[Category:Complex analysis]]

[[Category:Types of functions]]

{{mathanalysis-stub}}