In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
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 functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, 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. 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 ne ...
s 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 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 vers ...
s.
Definition
The following definition is in , but also found in Weyl or perhaps Weierstrass. An analytic function in an
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
''U'' is called a function element. Two function elements (''f''
1, ''U''
1) and (''f''
2, ''U''
2) are said to be
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 ne ...
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 (: sheaves) 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 d ...
, 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 t ...
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 (: sheaves) 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 da ...
) 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