In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of complex analysis, a global analytic function is a generalization of the notion of an

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...

which allows for functions to have multiple

branches A branch, sometimes called a ramus in botany, is a woody structural member connected to the central trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term ''twig'' usually r ...

. Global analytic functions arise naturally in considering the possible

analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...

s of an analytic function, since analytic continuations may have a non-trivial monodromy. They are one foundation for the theory of

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...

Definition

The following definition is in , but also found in Weyl or perhaps Weierstrass. An analytic function in an

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

''U'' is called a function element. Two function elements (''f''₁, ''U''₁) and (''f''₂, ''U''₂) are said to be

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''). 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 In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...

, 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 In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...

) 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