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If a (single-valued) function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m0649201.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m0649202.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m0649203.png" /> is continuous and if its integral over any closed rectifiable contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m0649204.png" /> is equal to zero, that is, if
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{{MSC|30-XX|32-XX}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m0649205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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A fundamental theorem in complex analysis first proved by G. Morera in {{Cite|Mo}}, which is an (incomplete) converse of the [[Cauchy integral theorem]]. The theorem states the following.  
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m0649206.png" /> is an [[Analytic function|analytic function]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m0649207.png" />. This theorem was obtained by G. Morera [[#References|[1]]].
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'''Theorem'''
 +
Let $D\subset \mathbb C$ be an open set and $f: D\to \mathbb C$ a continuous function. If
 +
the integral
 +
\begin{equation}\label{e:integral}
 +
\int_\gamma f(z)\, dz = 0
 +
\end{equation}
 +
vanishes for every [[Rectifiable curve|rectifiable]] contour $\gamma\subset D$, then the function $f$ is [[Holomorphic function|holomorphic]].
 +
 +
The integral in \eqref{e:integral} must be understood in the sense of the usual [[Integration on manifolds|integration]] of a $1$-[[Differential form|form]]. In particular, if $z: [0,T]\to D$ is a Lipschitz parametrization of the contour $\gamma$, then the right hand side of \eqref{e:integral} is given by
 +
\[
 +
\int_0^T f (z(t))\, \dot{z} (t)\, dt\, .
 +
\]
 +
Indeed the assumption of the theorem can be considerably weakened: to conclude that $f$ is holomorphic it suffices to know \eqref{e:integral} whenever $\gamma$ is the boundary of any triangle $\Delta\subset\subset D$.
  
The conditions of Morera's theorem can be weakened by restricting the requirement on vanishing integrals (*) to those taken over the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m0649208.png" /> of any triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m0649209.png" /> that is compactly contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492010.png" />, i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492011.png" />. Morera's theorem is an (incomplete) converse of the [[Cauchy integral theorem|Cauchy integral theorem]] and is one of the basic theorems in the theory of analytic functions.
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Morera's theorem can be generalized to functions of several complex variables.  
 
 
Morera's theorem can be generalized to functions of several complex variables. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492012.png" /> be a function of the complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492014.png" />, continuous in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492016.png" /> and such that its integral vanishes when taken over the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492017.png" /> of any prismatic domain compactly contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492018.png" /> of the form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492019.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492020.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492023.png" />, are line segments in the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492024.png" /> with end points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492027.png" /> is a triangle in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492028.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492029.png" /> is a holomorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064920/m06492030.png" />.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Morera,  "Un teorema fondamentale nella teorica delle funzioni di una variabili complessa"  ''Rend. R. Ist. Lomb. Sci. Lettere'' , '''19'''  (1886)  pp. 304–308</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
  
 +
'''Theorem'''
 +
Let $D\subset \mathbb C^n$ be an open set and $f: D \to \mathbb C$ a continuous function. Denote by $f (z)\, dz$ the (complex) differential form
 +
\[
 +
f (z)\, dz_1\wedge dz_2\wedge \ldots \wedge dz_n\, .
 +
\]
 +
Consider the class $\mathcal{P}$ of ''prismatic'' domains $\Gamma\subset\subset D$ of the form
 +
\[
 +
[a_1, b_1] \times \ldots \times [a_{i-1}, b_{i-1}]\times \partial \Delta \times [a_{i+1}, b_{i+1}] \times \ldots \times [a_n b_n]\, ,
 +
\]
 +
where $\Delta\subset \mathbb C$ is a arbitrary triangle, $a_k, b_k$ are complex numbers and $[a_k, b_k]$ denotes the segment $\sigma\subset \mathbb C$ given by $\{\lambda a_k + (1-\lambda b_k): \lambda \in [0,1]\}$. If
 +
\[
 +
\int_\Gamma f(z)\, dz = 0\,  \qquad\qquad \mbox{for any}\, \Gamma \in \mathcal{P}\, ,
 +
\]
 +
then $f$ is holomorphic.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"R. Remmert,  "Funktionentheorie" , '''1''' , Springer  (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.B. Conway,  "Functions of one complex variable" , Springer  (1978)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Co}}|| J.B. Conway, "Functions of one complex variable" , Springer (1973) {{MR|0447532}} {{ZBL|0277.30001}}
 +
|-
 +
|valign="top"|{{Ref|Re}}|| R. Remmert,  "Funktionentheorie" , '''1''' , Springer  (1984)
 +
|-
 +
|}

Latest revision as of 17:23, 12 January 2014

2020 Mathematics Subject Classification: Primary: 30-XX Secondary: 32-XX [MSN][ZBL]

A fundamental theorem in complex analysis first proved by G. Morera in [Mo], which is an (incomplete) converse of the Cauchy integral theorem. The theorem states the following.

Theorem Let $D\subset \mathbb C$ be an open set and $f: D\to \mathbb C$ a continuous function. If the integral \begin{equation}\label{e:integral} \int_\gamma f(z)\, dz = 0 \end{equation} vanishes for every rectifiable contour $\gamma\subset D$, then the function $f$ is holomorphic.

The integral in \eqref{e:integral} must be understood in the sense of the usual integration of a $1$-form. In particular, if $z: [0,T]\to D$ is a Lipschitz parametrization of the contour $\gamma$, then the right hand side of \eqref{e:integral} is given by \[ \int_0^T f (z(t))\, \dot{z} (t)\, dt\, . \] Indeed the assumption of the theorem can be considerably weakened: to conclude that $f$ is holomorphic it suffices to know \eqref{e:integral} whenever $\gamma$ is the boundary of any triangle $\Delta\subset\subset D$.

Morera's theorem can be generalized to functions of several complex variables.

Theorem Let $D\subset \mathbb C^n$ be an open set and $f: D \to \mathbb C$ a continuous function. Denote by $f (z)\, dz$ the (complex) differential form \[ f (z)\, dz_1\wedge dz_2\wedge \ldots \wedge dz_n\, . \] Consider the class $\mathcal{P}$ of prismatic domains $\Gamma\subset\subset D$ of the form \[ [a_1, b_1] \times \ldots \times [a_{i-1}, b_{i-1}]\times \partial \Delta \times [a_{i+1}, b_{i+1}] \times \ldots \times [a_n b_n]\, , \] where $\Delta\subset \mathbb C$ is a arbitrary triangle, $a_k, b_k$ are complex numbers and $[a_k, b_k]$ denotes the segment $\sigma\subset \mathbb C$ given by $\{\lambda a_k + (1-\lambda b_k): \lambda \in [0,1]\}$. If \[ \int_\Gamma f(z)\, dz = 0\, \qquad\qquad \mbox{for any}\, \Gamma \in \mathcal{P}\, , \] then $f$ is holomorphic.

References

[Co] J.B. Conway, "Functions of one complex variable" , Springer (1973) MR0447532 Zbl 0277.30001
[Re] R. Remmert, "Funktionentheorie" , 1 , Springer (1984)
How to Cite This Entry:
Morera theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morera_theorem&oldid=31239
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article