Editing Morera's theorem

{{Short description|Integral criterion for holomorphy}}
{{Complex_analysis_sidebar}}
[[File:Morera's Theorem.png|thumb|right|If the integral along every ''C'' is zero, then ''f'' is [[Holomorphic function|holomorphic]] on ''D''.]]
In [[complex analysis]], a branch of [[mathematics]], '''Morera's theorem''', named after [[Giacinto Morera]], gives a 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 ''f'' defined on an  [[open set]] ''D'' in the [[complex plane]] that satisfies
<math display="block">\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''.

The assumption of Morera's theorem is equivalent to ''f'' having an [[antiderivative (complex analysis)|antiderivative]] on&nbsp;''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.  

The standard counterexample is the function {{math|1=''f''(''z'') = 1/''z''}}, which is holomorphic on '''C'''&nbsp;−&nbsp;{0}.  On any simply connected neighborhood U in '''C'''&nbsp;−&nbsp;{0}, 1/''z'' has an antiderivative defined by {{math|1=''L''(''z'') = ln(''r'') + ''iθ''}}, where {{math|1=''z'' = ''re''<sup>''iθ''</sup>}}. Because of the ambiguity of ''θ'' up to the addition of any integer multiple of 2{{pi}}, any continuous choice of ''θ'' on ''U'' will suffice to define an antiderivative of 1/''z'' on ''U''.  (It is the fact that ''θ'' cannot be defined continuously on a simple closed curve containing the origin in its interior that is the root of why 1/''z'' has no antiderivative on its entire domain '''C'''&nbsp;−&nbsp;{0}.) And because the derivative of an additive constant is 0, any constant may be added to the antiderivative and the result will still be an antiderivative of 1/''z''.

In a certain sense, the 1/''z'' counterexample is universal: For every analytic function that has no antiderivative on its domain, the reason for this is that 1/''z'' itself does not have an antiderivative on '''C'''&nbsp;−&nbsp;{0}.

==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 ''f'' explicitly.

Without loss of generality, it can be assumed that ''D'' is [[connected space|connected]]. Fix a point ''z''<sub>0</sub> in ''D'', and for any <math>z\in D</math>, let <math>\gamma: [0,1]\to D</math> be a piecewise ''C''<sup>1</sup> curve such that <math>\gamma(0)=z_0</math> and <math>\gamma(1)=z</math>. Then define the function ''F'' to be
<math display="block">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 ''C''<sup>1</sup> 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''<sup>1</sup> curve in ''D''. Then,
<math display="block">\int_{\gamma} f(\zeta)\,d\zeta + \int_{\tau^{-1}} f(\zeta) \, d\zeta =\oint_{\gamma \tau^{-1}} f(\zeta)\,d\zeta = 0.</math>

And it follows that
<math display="block">\int_\gamma f(\zeta)\,d\zeta = \int_\tau f(\zeta)\,d\zeta.</math>

Then using the continuity of ''f'' to estimate difference quotients, we get that ''F''′(''z'')&nbsp;=&nbsp;''f''(''z'').  Had we chosen a different ''z''<sub>0</sub> in ''D'', ''F'' would change by a constant: namely, the result of integrating ''f'' along ''any'' piecewise regular curve between the new  ''z''<sub>0</sub> and the old, and this does not change the derivative.

Since ''f'' is the derivative of the holomorphic function ''F'', it is holomorphic. The fact that derivatives of holomorphic functions are holomorphic can be proved by using the fact that [[analyticity of holomorphic functions|holomorphic functions are analytic]], i.e. can be represented by a convergent [[power series]], and the fact that power series may be differentiated term by term.  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 ''f''<sub>1</sub>,&nbsp;''f''<sub>2</sub>,&nbsp;... is a sequence of holomorphic functions, [[uniform convergence|converging uniformly]] to a continuous function ''f'' on an open disc.  By [[Cauchy's integral theorem|Cauchy's theorem]], we know that
<math display="block">\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 display="block">\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 ''f'' must be holomorphic.  This fact can be used to show that, for any [[open set]] {{math|Ω ⊆ '''C'''}}, the set {{math|''A''(Ω)}} of all [[bounded function|bounded]], analytic functions {{math|''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 display="block">\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}</math>
or the [[Gamma function]]
<math display="block">\Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x}\,dx.</math>

Specifically one shows that
<math display="block"> \oint_C \Gamma(\alpha)\,d\alpha = 0 </math>
for a suitable closed curve ''C'', by writing
<math display="block"> \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 display="block"> \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 {{math|''α'' ↦ ''x''<sup>''α''−1</sup>}} to conclude that
<math display="block"> \oint_C x^{\alpha-1} \, d\alpha = 0, </math>
and hence the double integral above is&nbsp;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 display="block">\oint_{\partial T} f(z)\, dz</math>
to be zero for every closed (solid) triangle ''T'' contained in the region ''D''. This in fact [[characterization (mathematics)|characterizes]] holomorphy, i.e. ''f'' is holomorphic on ''D'' if and only if the above conditions hold. It also implies the following generalisation of the aforementioned fact about uniform limits of holomorphic functions: if ''f''<sub>1</sub>,&nbsp;''f''<sub>2</sub>,&nbsp;... is a sequence of holomorphic functions defined on an open set {{math|Ω ⊆ '''C'''}} that converges to a function ''f'' uniformly on compact subsets of Ω, then ''f'' is holomorphic.

==See also==
*[[Cauchy–Riemann equations]]
*[[Methods of contour integration]]
*[[Residue (complex analysis)]]
*[[Mittag-Leffler's theorem]]

==References==
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| date = January 1, 1979
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| publisher = McGraw-Hill
| isbn = 978-0-07-000657-7
| zbl = 0395.30001
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*{{Citation
| last = Morera
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| author-link = Giacinto Morera
| title = Un teorema fondamentale nella teorica delle funzioni di una variabile complessa
| journal = Rendiconti del Reale Instituto Lombardo di Scienze e Lettere
| volume = 19
| issue = 2
| pages = 304–307
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| url = https://archive.org/stream/rendiconti00unkngoog#page/n312/mode/2up
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* {{Citation
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| edition = 3rd
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==External links==
* {{springer|title=Morera theorem|id=p/m064920}}
* {{MathWorld | urlname= MorerasTheorem | title= Morera’s Theorem }}

[[Category:Theorems in complex analysis]]