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 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 defined on a non-empty open subset U of the complex plane 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 F1 and F2 such that U is contained in V and for all z in U
then
on all of V. This is because F1 − F2 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 functions.
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.
The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. 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, solutions of Einstein's equations. For example, the analytic continuation of Schwarzschild coordinates 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,
That is,
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 punctured complex plane
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 Dr(z0), 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 g0 of g is z0, the stem of g is (α0, α1, α2, ...) and the top g1 of g is α0. The top of g is the value of f at z0.
Any vector g = (z0, α0, α1, ...) is a germ if it represents a power series of an analytic function around z0 with some radius of convergence r > 0. Therefore, we can safely speak of the set of germs .