In
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 branch of
mathematics
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 ...
, analytic continuation is a technique to extend the
domain of definition 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 ...
. Analytic continuation often succeeds in defining further values of a function, for example in a new region where the
infinite series representation which initially defined the function 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 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 ...
defined on a non-empty
open subset ''U'' of the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
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 of ''F'' to ''U'' is the function ''f'' we started with.
Analytic continuations are unique in the following sense: if ''V'' is the
connected 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 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 de ...
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 on the small domain and then using this equation to extend the domain. Examples are the
Riemann zeta function and the
gamma function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
.
The concept of a
universal cover 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 ...
. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of
Riemann surfaces.
Analytic continuation is used in
Riemannian manifolds, in the context of solutions of
Einstein's equations. For example,
Schwarzschild coordinates can be analytically continued into
Kruskal–Szekeres coordinates.
Worked example
Begin with a particular analytic function
. In this case, it is given by a
power series centered at
:
By the
Cauchy–Hadamard theorem, 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 to calculate the new coefficients, one has
The last summation results from the th derivation of the
geometric series, which gives the formula
Then,
which has radius of convergence
around
. 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 further. This particular
can be analytically continued to the whole punctured complex plane
In this particular case the obtained values of
are the same when the successive centers have a positive imaginary part or a negative imaginary part. This is not always the case; in particular this is not the case for the
complex logarithm, the
antiderivative of the above function.
Formal definition of a germ
The power series defined below is generalized by the idea of a ''
germ''. The general theory of analytic continuation and its generalizations is known as
sheaf theory. Let
:
be a
power series converging in the
disk ''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'' 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