In the mathematical field of

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

, 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 In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...

. They are one foundation for the theory of Riemann surfaces.

Definition

The following definition is in , 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''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.

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 In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...

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=978-0-07-000657-7 Complex analysis Types of functions