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Morera's TheoremIn 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, that is, if -
for C any simple closed curve, 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 -
It also leads to a quick proof of the general result that if a sequence - fn(z),
of analytic functions on a given open set D of complex numbers, converges to a function - f(z)
uniformly on every compact subset K, then f is analytic. The condition can easily be reduced to K being a closed disk.
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