Morera's theorem

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In complex analysis, a branch of mathematics, 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

<math>\int_C f(z)\,dz=0</math>

for C any simple closed curve, then f is differentiable or "holomorphic" at every point in that open set.

Uses

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

<math>\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}</math>

or the Gamma function

<math>\Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx.</math>

It also leads to a quick proof of the general result that if a sequence f_n(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.ru:Теорема Мореры