In the mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 field of complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, a global analytic function is a generalization of the notion of an analytic function

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, categories that are similar in some ways, but different in others...

 which allows for functions to have multiple branches. Global analytic functions arise naturally in considering the possible analytic continuation

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

s of an analytic function, since analytic continuations may have a non-trivial monodromy

Monodromy

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

. They are one foundation for the theory of Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

s.

Definition

The following definition is due to . An analytic function in an open set

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 U is called a function element. Two function elements (f₁, U₁) and (f₂, U₂) are said to be analytic continuation

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

s of one another if U₁ ∩ U₂ ≠ ∅ and f₁ = f₂ on this intersection. A chain of analytic continuations is a finite sequence of function elements (f₁, U₁), …, (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 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.