# Difference between revisions of "Morera's theorem"

en>Daniele.tampieri m (→References: Minor corrections) |
(The example has been removed as it is wrong. It's a basic result that if f(z) is analytic, then 1/f(z) is analytic anywhere f(z) is not zero. f(z) = z^2 is analytic and zero only at z = 0, thus g(z) = 1/f(z) = 1/z^2 is analytic in C not 0.) |
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[[Image:Morera's Theorem.png|thumb|right|If the integral along every ''C'' is zero, then ''ƒ'' is [[holomorphic]] on ''D''.]] | [[Image:Morera's Theorem.png|thumb|right|If the integral along every ''C'' is zero, then ''ƒ'' is [[holomorphic]] on ''D''.]] | ||

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 function|holomorphic]]. | In [[complex analysis]], a branch of [[mathematics]], '''Morera's theorem''', named after [[Giacinto Morera]], gives an important criterion for proving that a [[function (mathematics)|function]] is [[holomorphic function|holomorphic]]. | ||

Morera's theorem states that a [[continuous function|continuous]], [[complex number|complex]]-valued function ''ƒ'' defined on a [[ | Morera's theorem states that a [[continuous function|continuous]], [[complex number|complex]]-valued function ''ƒ'' defined on a [[Simply connected space|simply connected]] [[open set]] ''D'' in the [[complex plane]] that satisfies | ||

:<math>\oint_\gamma f(z)\,dz = 0</math> | :<math>\oint_\gamma f(z)\,dz = 0</math> | ||

for every closed piecewise ''C''<sup>1</sup> curve <math>\gamma</math> in ''D'' must be holomorphic on ''D''. | for every closed piecewise ''C''<sup>1</sup> curve <math>\gamma</math> in ''D'' must be holomorphic on ''D''. | ||

The assumption of Morera's theorem is equivalent to that ''ƒ'' has an | The assumption of Morera's theorem is equivalent to that ''ƒ'' has an [[antiderivative (complex analysis)|antiderivative]] on ''D''. | ||

The converse of the theorem is not true in general. A holomorphic function need not possess an | 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. The converse does hold e.g. if the domain is [[simply connected]]; this is [[Cauchy's integral theorem]], stating that the [[line integral]] of a holomorphic function along a [[closed curve]] is zero. | ||

== Proof == | == Proof == | ||

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*[[Methods of contour integration]] | *[[Methods of contour integration]] | ||

*[[Residue (complex analysis)]] | *[[Residue (complex analysis)]] | ||

*[[Mittag-Leffler's theorem]] | |||

== References == | == References == | ||

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== External links == | == External links == | ||

* {{springer|title=Morera theorem|id=p/m064920}} | |||

* {{MathWorld | urlname= MorerasTheorem | title= Morera’s Theorem }} | * {{MathWorld | urlname= MorerasTheorem | title= Morera’s Theorem }} | ||

* [http://math.fullerton.edu/mathews/c2003/LiouvilleMoreraGaussMod.html Module for Morera's Theorem by John H. Mathews] | * [http://math.fullerton.edu/mathews/c2003/LiouvilleMoreraGaussMod.html Module for Morera's Theorem by John H. Mathews] | ||

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[[Category:Theorems in complex analysis]] | [[Category:Theorems in complex analysis]] | ||

## Revision as of 20:45, 16 September 2013

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.

Morera's theorem states that a continuous, complex-valued function *ƒ* defined on a simply connected open set *D* in the complex plane that satisfies

for every closed piecewise *C*^{1} curve in *D* must be holomorphic on *D*.

The assumption of Morera's theorem is equivalent to that *ƒ* has an antiderivative 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. The converse does hold e.g. if the domain is simply connected; this is Cauchy's integral theorem, stating that the line integral of a holomorphic function along a closed curve is zero.

## Proof

There is a relatively elementary proof of the theorem. One constructs an anti-derivative for *ƒ* 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 *z*_{0} in *D*, and for any , let be a piecewise *C*^{1} curve such that and . Then define the function *F* to be

To see that the function is well-defined, suppose is another piecewise *C ^{1}* curve such that and . The curve (i.e. the curve combining with in reverse) is a closed piecewise

*C*

^{1}curve in

*D*. Then,

And it follows that

By continuity of f and the definition of the derivative, we get that F'(z) = f(z). Note that we can apply neither the Fundamental theorem of Calculus nor the mean value theorem since they are only true for real-valued functions.

Since *f* is the derivative of the holomorphic function *F*, it is holomorphic. This completes the 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 *ƒ*_{1}, *ƒ*_{2}, ... is a sequence of holomorphic functions, converging uniformly to a continuous function *ƒ* on an open disc. By Cauchy's theorem, we know that

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

for every closed curve *C*, and therefore by Morera's theorem *ƒ* 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.

### Infinite sums and integrals

Morera's theorem can also be used in conjunction with Fubini's theorem and the Weierstrass M-test to show the analyticity of functions defined by sums or integrals, such as the Riemann zeta function

or the Gamma function

Specifically one shows that

for a suitable closed curve *C*, by writing

and then using Fubini's theorem to justify changing the order of integration, getting

Then one uses the analyticity of *x* ↦ *x*^{α−1} to conclude that

and hence the double integral above is 0. Similarly, in the case of the zeta function, the M-test justifies interchanging the integral along the closed curve and the sum.

## Weakening of hypotheses

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

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

## See also

- Cauchy–Riemann equations
- Methods of contour integration
- Residue (complex analysis)
- Mittag-Leffler's theorem

## References

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## External links

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