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