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[[Image:Morera's Theorem.png|thumb|right|If the integral along every ''C'' is zero, then ''&fnof;'' 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 (mathematics)|function]] is [[holomorphic function|holomorphic]].

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>

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

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 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 ==
[[Image:Morera's Theorem Proof.png|thumb|right|The integrals along two paths from ''a'' to ''b'' are equal, since their difference is the integral along a closed loop.]]

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 [[Proof that holomorphic functions are analytic|holomorphic functions are analytic]].

Without loss of generality, it can be assumed that ''D'' is [[connected space|connected]]. Fix a point ''z''0 in ''D'', and for any <math>z\in D</math>, let <math>\gamma: [0,1]\to D</math> be a piecewise ''C''1 curve such that <math>\gamma(0)=z_0</math> and <math>\gamma(1)=z</math>. Then define the function ''F'' to be

:<math>F(z) = \int_\gamma f(\zeta)\,d\zeta.\,</math>

To see that the function is well-defined, suppose <math>\tau: [0,1]\to D</math> is another piecewise ''C1'' curve such that <math>\tau(0)=z_0</math> and <math>\tau(1)=z</math>. The curve <math>\gamma \tau^{-1}</math> (i.e. the curve combining <math>\gamma</math> with <math>\tau</math> in reverse) is a closed piecewise ''C''1 curve in ''D''. Then,

:<math>\oint_{\gamma} f(\zeta)\,d\zeta\, + \oint_{\tau^{-1}} f(\zeta)\,d\zeta\,=\oint_{\gamma \tau^{-1}} f(\zeta)\,d\zeta\,=0</math>

And it follows that

:<math>\oint_{\gamma} f(\zeta)\,d\zeta\, = \oint_\tau f(\zeta)\,d\zeta.\,</math>

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, [[uniform convergence|converging uniformly]] to a continuous function ''ƒ'' on an open disc. By [[Cauchy's integral theorem|Cauchy's theorem]], we know that

:<math>\oint_C f_n(z)\,dz = 0</math>

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

:<math>\oint_C f(z)\,dz = \oint_C \lim_{n\to \infty} f_n(z)\,dz =\lim_{n\to \infty} \oint_C f_n(z)\,dz = 0 </math>

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 function|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]]

:<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>

Specifically one shows that

: <math> \oint_C \Gamma(\alpha)\,d\alpha = 0 </math>

for a suitable closed curve ''C'', by writing

: <math> \oint_C \Gamma(\alpha)\,d\alpha = \oint_C \int_0^\infty x^{\alpha-1} e^{-x}\,dx \,d\alpha </math>

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

: <math> \int_0^\infty \oint_C x^{\alpha-1} e^{-x} \,d\alpha \,dx = \int_0^\infty e^{-x} \oint_C x^{\alpha-1} \, d\alpha \,dx. </math>

Then one uses the analyticity of ''x'' ↦ ''x''''α''−1 to conclude that

: <math> \oint_C x^{\alpha-1} \, d\alpha = 0, </math>

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

:<math>\oint_{\partial T} f(z)\, dz</math>

to be zero for every closed triangle ''T'' contained in the region ''D''. This in fact [[characterization (mathematics)|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 ==
* {{Citation
| authorlink = Lars Ahlfors
| last = Ahlfors
| first = Lars
| date = January 1, 1979
| title = Complex Analysis
|series = International Series in Pure and Applied Mathematics
| publisher = McGraw-Hill
| isbn = 978-0-07-000657-7
| zbl = 0395.30001
}}.
* {{Citation
| last = Conway
| first = John B.
| year = 1973
| title = Functions of One Complex Variable I
| series = Graduate Texts in Mathematics
| volume = 11
| publisher = [[Springer Verlag]]
| isbn = 978-3-540-90328-4
| zbl = 0277.30001
}}.
*{{Citation
| last1 = Greene | first1 = Robert E.
| last2 = Krantz | first2 = Steven G.
| year = 2006
| title = Function Theory of One Complex Variable
| series = Graduate Studies in Mathematics
| volume = 40
| publisher = American Mathematical Society
| isbn = 0-8218-3962-4 }}
*{{Citation
| last = Morera
| first = Giacinto
| author-link = Giacinto Morera
| title = Un teorema fondamentale nella teorica delle funzioni di una variabile complessa
| journal = [http://www.istitutolombardo.it/pubblicazioni.html Rendiconti del Reale Instituto Lombardo di Scienze e Lettere]
| volume = 19
| issue = 2
| pages = 304–307
| language = Italian
| date =
| year = 1886
| url = http://www.archive.org/stream/rendiconti00unkngoog#page/n312/mode/2up
| archiveurl =
| archivedate =
| doi =
| jfm = 18.0338.02
}}.
* {{Citation
| last = Rudin
| first = Walter
| year = 1987
| origyear = 1966
| title = Real and Complex Analysis
| edition = 3rd
| publisher = [[McGraw-Hill]]
| pages = xiv+416
| isbn = 978-0-07-054234-1
| zbl = 0925.00005
}}.

== External links ==
* {{springer|title=Morera theorem|id=p/m064920}}
* {{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://eom.springer.de/M/m064920.htm EoM article]

[[Category:Theorems in complex analysis]]

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en>Yobot: /* Use in Astrophysics */WP:CHECKWIKI error fixes using AWB (10093)

131.107.0.89: typos