Morera's theorem

In complex analysis, Morera's theorem states that if the integral of a continuous complex-valued function f of a complex variable along every simple closed curve within an open set is zero, i.e. if

if C is any simple closed curve at all, then f is differentiable at every point in that open set.

Morera's theorem can be used to show the analyticity of functions defined by sums or integrals, such as the Riemann zeta function

or the Gamma function