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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, that is, if

$\int_C f(z)\,dz=0$

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

$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$

$\Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx.$

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.

Last updated: 05-23-2005 19:27:24

10-26-2009 08:16:03
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