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 D is zero, that is, if

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

[edit] Proof

We have

for any C. Therefore for any two simple curves γ₁ and γ₂ from within D starting at z₀ ∈ D and ending at z ∈ D we have

hence

exists and it is holomorphic function, and

is holomorphic as well.

[edit] Uses

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 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.

[edit] External links

• Eric W. Weisstein, Morera’s Theorem at MathWorld.
• Module for Morera’s Theorem by John H. Mathews