In
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 branch of
mathematics
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 ...
, analytic continuation is a technique to extend the
domain of definition
In mathematics, a partial function from a Set (mathematics), set to a set is a function from a subset of (possibly itself) to . The subset , that is, the Domain of a function, domain of viewed as a function, is called the domain of defini ...
of a given
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 an ...
. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
representation in terms of which it is initially defined becomes
divergent.
The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of
singularities. The case of
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of
sheaf cohomology.
Initial discussion
Suppose ''f'' is 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 an ...
defined on a non-empty
open subset ''U'' of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
If ''V'' is a larger open subset of containing ''U'', and ''F'' is an analytic function defined on ''V'' such that
:
then ''F'' is called an analytic continuation of ''f''. In other words, the
restriction
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
of ''F'' to ''U'' is the function ''f'' we started with.
Analytic continuations are unique in the following sense: if ''V'' is the
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
domain of two analytic functions ''F''
1 and ''F''
2 such that ''U'' is contained in ''V'' and for all ''z'' in ''U''
:
then
:
on all of ''V''. This is because ''F''
1 − ''F''
2 is an analytic function which vanishes on the open, connected domain ''U'' of ''f'' and hence must vanish on its entire domain. This follows directly from the
identity theorem
In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions ''f'' and ''g'' analytic on a domain ''D'' (open and connected subset of \mathbb or \mathbb), if ''f'' = ''g'' on so ...
for
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s.
Applications
A common way to define functions in complex analysis proceeds by first specifying the function on a small domain only, and then extending it by analytic continuation.
In practice, this continuation is often done by first establishing some
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
on the small domain and then using this equation to extend the domain. Examples are the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
and the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
.
The concept of a
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
was first developed to define a natural domain for the analytic continuation 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 an ...
. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea 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.
Analytic continuation is used in
Riemannian manifolds
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
, solutions of
Einstein's equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
. For example, the analytic continuation of
Schwarzschild coordinates In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coord ...
into
Kruskal–Szekeres coordinates
In general relativity, Kruskal–Szekeres coordinates, named after Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole. These coordinates have the advantage that they cover the entire space ...
.
Worked example
Begin with a particular analytic function
. In this case, it is given by a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
centered at
:
:
By the
Cauchy–Hadamard theorem
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by C ...
, its radius of convergence is 1. That is,
is defined and analytic on the open set
which has boundary
. Indeed, the series diverges at
.
Pretend we don't know that
, and focus on recentering the power series at a different point
:
:
We'll calculate the
's and determine whether this new power series converges in an open set
which is not contained in
. If so, we will have analytically continued
to the region
which is strictly larger than
.
The distance from
to
is
. Take
; let
be the disk of radius
around
; and let
be its boundary.
Then
. Using
Cauchy's differentiation formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundar ...
to calculate the new coefficients,
:
That is,
:
which has radius of convergence
, and
If we choose
with
, then
is not a subset of
and is actually larger in area than
. The plot shows the result for
We can continue the process: select
, recenter the power series at
, and determine where the new power series converges. If the region contains points not in
, then we will have analytically continued
even farther. This particular
can be analytically continued to the punctured complex plane
Formal definition of a germ
The power series defined below is generalized by the idea of a ''
germ
Germ or germs may refer to:
Science
* Germ (microorganism), an informal word for a pathogen
* Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually
* Germ layer, a primary layer of cells that forms during embry ...
''. The general theory of analytic continuation and its generalizations is known as
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 ...
. Let
:
be a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
converging in the
disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
''D''
''r''(''z''
0), ''r'' > 0, defined by
:
.
Note that without loss of generality, here and below, we will always assume that a maximal such ''r'' was chosen, even if that ''r'' is ∞. Also note that it would be equivalent to begin with an analytic function defined on some small open set. We say that the vector
:
is a ''
germ
Germ or germs may refer to:
Science
* Germ (microorganism), an informal word for a pathogen
* Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually
* Germ layer, a primary layer of cells that forms during embry ...
'' of ''f''. The ''base'' ''g''
0 of ''g'' is ''z''
0, the ''stem'' of ''g'' is (α
0, α
1, α
2, ...) and the ''top'' ''g''
1 of ''g'' is α
0. The top of ''g'' is the value of ''f'' at ''z''
0.
Any vector ''g'' = (''z''
0, α
0, α
1, ...) is a germ if it represents a power series of an analytic function around ''z''
0 with some radius of convergence ''r'' > 0. Therefore, we can safely speak of the set of germs
.
The topology of the set of germs
Let ''g'' and ''h'' be
germs. If