Analytic continuation

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.

Example I: A function with a natural boundary at zero (the prime zeta function)

The monodromy theorem gives a sufficient condition for the existence of a direct analytic continuation (i.e., an extension of an analytic function to an analytic function on a bigger set).

A useful theorem: A sufficient condition for analytic continuation to the non-positive integersExample I: The connection of the Riemann zeta function to the Bernoulli numbers

Suppose that F is a smooth, sufficiently decreasing function on the positive reals satisfying the additional condition that