Analytic continuation
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.
These may have an essentially topological nature, leading to inconsistencies (defining more than one value).
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.
Suppose f is an analytic function defined on a non-empty open subset U of the complex plane
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.
Analytic continuation is used in Riemannian manifolds, in the context of solutions of Einstein's equations.
, and focus on recentering the power series at a different point
's and determine whether this new power series converges in an open set
The last summation results from the kth derivation of the geometric series, which gives the formula
The power series defined below is generalized by the idea of a germ.
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.
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.
where r is the radius of convergence of g and if the power series defined by g and h specify identical functions on the intersection of the two domains, then we say that h is generated by (or compatible with) g, and we write g ≥ h. This compatibility condition is neither transitive, symmetric nor antisymmetric.
The uniqueness theorem for analytic functions also extends to sheaves of analytic functions: if the sheaf of an analytic function contains the zero germ (i.e., the sheaf is uniformly zero in some neighborhood) then the entire sheaf is zero.
Armed with this result, we can see that if we take any germ g of the sheaf S of the logarithm function, as described above, and turn it into a power series f(z) then this function will have the property that exp(f(z)) = z.
If we had decided to use a version of the inverse function theorem for analytic functions, we could construct a wide variety of inverses for the exponential map, but we would discover that they are all represented by some germ in S. In that sense, S is the "one true inverse" of the exponential map.
Suppose that a power series has radius of convergence r and defines an analytic function f inside that disc.
A point for which there is a neighbourhood on which f has an analytic extension is regular, otherwise singular.
The prime zeta function has an analytic continuation to all complex s such that
As a remark, this fact can be problematic if we are performing a complex contour integral over an interval whose real parts are symmetric about zero, say
given by For any positive natural numbers c, the lacunary series function diverges at
Hence, since the set formed by all such roots is dense on the boundary of the unit circle, there is no analytic continuation of
The proof of this fact is generalized from a standard argument for the case where
denotes the open unit disk in the complex plane and
distinct complex numbers z that lie on or inside the unit circle such that
Hence, there is no analytic continuation for these functions beyond the interior of the unit circle.
is an open set and f an analytic function on D. If G is a simply connected domain containing D, such that f has an analytic continuation along every path in G, starting from some fixed point a in D, then f has a direct analytic continuation to G. In the above language this means that if G is a simply connected domain, and S is a sheaf whose set of base points contains G, then there exists an analytic function f on G whose germs belong to S. For a power series with the circle of convergence is a natural boundary.
Let be a power series, then there exist εk ∈ {−1, 1} such that has the convergence disc of f around z0 as a natural boundary.
