Analytic continuation
Analytic continuation. Let f1(z) be a function that is analytic in a region R1 of the complex plane and f2(z) a function that is analytic in a region R2 partly overlapping R1. If f1(z) = f2(z) in the overlapping part, then f2(z) is called the analytic continuation of f1(z). This means that there is a function f(z) that is analytic in the combined regions R1 and R2 such that f(z) = f1(z) in R1 and f(z) = f2(z) in R2. It is sufficient for R1 and R2 to have only a small arc in common such as the arc ABC shown in Fig. 2.
By analytic continuation to regions R3, R4, etc. we can extend the original region of definition to other parts of the complex plane. The functions f1(z), f2(z), f3(z), ... defined in R1, R2, R3, ... respectively, are called function elements or briefly elements. It is sometimes impossible to extend a function analytically beyond the boundary of a region. In such a case the boundary is called a natural boundary.
Uniqueness theorem for analytic continuation. Let a function f1(z) defined in R1 be continued analytically to region Rn along two different paths. See Fig. 3. Then the two analytic continuations will be identical providing there is no singularity between the paths.
If we do get different results when using two different paths we can show that there is a singularity (specifically a branch point) between the paths.
One can illustrate analytic continuation with Taylor series expansions. Suppose we do not know the exact form of an analytic function f(z) but only know that inside some circle of convergence C1 with center at a f(z) is represented by a Taylor series
1) a0 + a1(z - a) + a2(z - a)2 + ...
If we then choose a point b inside C1 we can find the value of f(z) and its derivatives at b from 1) and thus arrive at a new series
2) b0 + b1(z - b) + b2(z - b)2 + ...
having a circle of convergence C2. See Fig. 4. We can then choose a point c inside C2 and repeat the process. The process can be repeated indefinitely.
The collection of all such power series representations, i.e. all possible analytic continuations, is defined as the analytic function f(z).
References
Spiegel. Complex Variables. (Schaum)