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.
Suppose f is an analytic function defined on the open subset U of the complex plane C. If V is an open subset of C containing U, and F is an analytic function defined on V such that F(z) = f(z) for all z in U, then F is called an analytic continuation of f.
Analytic continuations are unique in the following sense: if V is connected and F_{1} and F_{2} are two analytic continuations of f defined on V, then F_{1} = F_{2}. That is because the difference is an analytic function vanishing on a nonempty open set.
For example, if a power series with radius of convergence r is given, one can consider analytic continuations of the power series, i.e. analytic functions F which are defined on larger sets than
 {z : z − a < r},
and agree with the given power series on that set. The number r is maximal in the following sense: there always exists a complex number z with
 z − a = r
such that no analytic continuation of the series can be defined at z. Therefore there is a limitation to analytic continuation to bigger discs with the same centre a. On the other hand there may well be analytic continuations to some larger sets. If not, there is a natural boundary on the bounding circle.
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.
The power series defined above is generalized by the idea of a germ. The general theory of analytic continuation and its generalization is known as sheaf theory.
Formal definition
Let
 <math>f(z)=\sum_{k=1}^\infty \alpha_k (zz_0)^k<math>
be a power series converging in the disk D_{r}(z_{0}) := {z in C : z  z_{0} < r} for r > 0. (Note, without loss of generality, here and in the sequel, we will always assume that a maximal such r was chosen, even if it 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
 g = (z_{0}, α_{0}, α_{1}, α_{2}, ...)
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}, the bottom of g.
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 <math>\mathcal G<math>.
The topology of the set of germs
If g and h are germs, if h_{0}  g_{0} < r where r is the radius of convergence of g and if the power series that g and h represent define 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. If we extend the relation by transitivity, we obtain symmetric relation, which is therefore also an equivalence relation on germs (and not an ordering.) This extension by transitivity is one definition of analytic continuation. The equivalence relation will be denoted <math>\cong<math>.
We can define a topology on <math>\mathcal G<math>. Let r > 0, and let
 <math>U_r(g) = \{h \in \mathcal G : g \ge h, g_0  h_0 < r\}.<math>
The sets U_{r}(g), for all r > 0 and g ∈ <math>\mathcal G<math> define a basis of open sets for the topology on <math>\mathcal G<math>.
A connected component of <math>\mathcal G<math> (i.e., an equivalence class) is called a sheaf. We also note that the map φ_{g}(h) = h_{0} from U_{r}(g) to C where r is the radius of convergence of g, is a chart. The set of such charts forms an atlas for <math>\mathcal G<math>, hence <math>\mathcal G<math> is a Riemann surface. <math>\mathcal G<math> is sometimes called the universal analytic function.
Examples of analytic continuation
 <math>L(z1) = \sum_{k=1}^\infin \frac{(1)^{k+1}}{k}(z1)^k<math>
is a power series corresponding to the natural logarithm near z = 1. This power series can be turned into a germ
 g = (1, 0, 1, 1, 1, 1, 1, 1, ...)
This germ has a radius of convergence of 1, and so there is a sheaf S corresponding to it. This is the sheaf of the logarithm function.
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.
In older literature, sheaves of analytic functions were called multivalued functions. See sheaf for the general concept.sl:analitično nadaljevanje