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In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic. ## Proof

## Applications

Morera's theorem is a standard tool in complex analysis. It is used in almost any argument that involves a non-algebraic construction of a holomorphic function.
### Uniform limits

For example, suppose that f_{1}, f_{2}, ... is a sequence of holomorphic functions, converging uniformly to a continuous function f on an open disc. By Cauchy's theorem, we know that### Infinite sums and integrals

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

The hypotheses of Morera's theorem can be weakened considerably. In particular, it suffices for the integral## References

## External links

Morera's theorem states that if f is a continuous, complex-valued function defined on an open set D in the complex plane, satisfying

- $oint\_C\; f(z),dz\; =\; 0$

for every closed curve C in D, then f must be holomorphic on D.

The assumption of Morera's theorem is equivalent to that f has an anti-derivative on D.

The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. For instance, Cauchy's integral theorem states that the line integral of a holomorphic function along a closed curve is zero, provided that the domain of the function is simply connected.

There is a relatively elementary proof of the theorem. One constructs an anti-derivative for f explicitly. The theorem then follows from the fact that holomorphic functions are analytic.

Without loss of generality, it can be assumed that D is connected. Fix a point a in D, and define a complex-valued function F on D by

- $F(b)\; =\; int\_a^b\; f(z),dz.,$

The integral above may be taken over any path in D from a to b. The function F is well-defined because, by hypothesis, the integral of f along any two curves from a to b must be equal. It follows from the fundamental theorem of calculus that the derivative of F is f:

- $F\text{'}(z)\; =\; f(z).,$

In particular, the function F is holomorphic. Then f must be holomorphic as well, being the derivative of a holomorphic function.

- $oint\_C\; f\_n(z),dz\; =\; 0$

for every n, along any closed curve C in the disc. Then the uniform convergence implies that

- $oint\_C\; f(z),dz\; =\; lim\_\{nrightarrowinfty\}\; oint\_C\; f\_n(z),dz\; =\; 0$

for every closed curve C, and therefore by Morera's theorem f must be holomorphic. This fact can be used to show that, for any open set Ω ⊆ C, the set A(Ω) of all bounded, analytic functions u : Ω → C is a Banach space with respect to the supremum norm.

- $zeta(s)=sum\_\{n=1\}^infty\; frac\{1\}\{n^s\}$

or the Gamma function

- $Gamma(alpha)=int\_0^infty\; x^\{alpha-1\}\; e^\{-x\},dx.$

- $oint\_\{partial\; T\}\; f(z),\; dz$

to be zero for every closed triangle T contained in the region D. This in fact characterizes holomorphy, i.e. f is holomorphic on D if and only if the above conditions hold.

- G. Morera, "Un teorema fondamentale nella teoria delle funzioni di una variabile complessa", Rend. del R. Instituto Lombardo di Scienze e Lettere (2) 19 (1886) 304–307

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Last updated on Friday September 19, 2008 at 05:10:09 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday September 19, 2008 at 05:10:09 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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