Morera's theorem
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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, that is, if
- <math>\int_C f(z)\,dz=0<math>
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
- <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
- 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.