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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c0209001.png" /> is a regular analytic function of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c0209002.png" /> in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c0209003.png" /> in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c0209004.png" />, then the integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c0209005.png" /> along any closed rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c0209006.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c0209007.png" /> vanishes:
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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/c/c020/c020900/c0209008.png" /></td> </tr></table>
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A fundamental theorem in complex analysis which states the following.
  
An equivalent version of Cauchy's integral theorem states that the integral
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'''Theorem 1'''
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If $D\subset \mathbb C$ is a simply connected open set and $f:D\to \mathbb C$ a [[Holomorphic function|holomorphic funcion]], then the integral of $f(z)\, dz$ along any closed [[Rectifiable curve|rectifiable curve]] $\gamma\subset D$ vanishes:
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\begin{equation}\label{e:integral_vanishes}
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\int_\gamma f(z)\, dz = 0\, .
 +
\end{equation}
  
<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/c/c020/c020900/c0209009.png" /></td> </tr></table>
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The left hand side of \eqref{e:integral_vanishes} is the integral of the (complex) [[Differential form|differential form]] $f(z)\, dz$ (see also [[Integration on manifolds]]). More precisely, if $\alpha: \mathbb S^1 \to \mathbb C$ is a Lipschitz parametrization of the curve $\gamma$, then
 +
\begin{equation}\label{e:formula_integral}
 +
\int_\gamma f(z)\, dz = \int_0^{2\pi} f (\alpha (t))\, \dot{\alpha} (t)\, dt\,
 +
\end{equation}
 +
(observe that in order for \eqref{e:formula_integral} to be well defined, i.e. independent of the chosen parametrization, we must in general decide an [[Orientation|orientation]] for the curve $\gamma$; however since \eqref{e:integral_vanishes} stipulates that the integral vanishes, the choice of the orientation is not important in the present context).
  
is independent of the choice of the path of integration between the fixed points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090012.png" />. This, essentially, was the original formulation of the theorem as proposed by A.L. Cauchy (1825) (see [[#References|[1]]]); similar formulations may be found in the letters of C.F. Gauss (1811). Cauchy's proof involved the additional assumption that the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090013.png" /> is continuous; the first complete proof was given by E. Goursat [[#References|[2]]]. The property of analytic functions expressed by the Cauchy integral theorem fully characterizes them (see [[Morera theorem|Morera theorem]]), and therefore all the fundamental properties of analytic functions may be inferred from the Cauchy integral theorem.
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An equivalent version of Cauchy's integral theorem states that (under the same assuptions of Theorem 1), given any (rectifiable) path $\eta:[0,1]\to D$ the integral
 +
\[
 +
\int_\eta f(z)\, dz
 +
\]
 +
depends only upon the two endpoints $\eta (0)$ and $\eta(1)$, and hence it is  
 +
independent of the choice of the path of integration $\eta$. This, essentially, was the original formulation of the theorem as proposed by A.L. Cauchy (1825) (see {{Cite|Ca}}); similar formulations may be found in the letters of C.F. Gauss (1811). Cauchy's proof involved the additional assumption that the (complex) derivative $f'$ is continuous; the first complete proof was given by E. Goursat {{Cite|Go2}}. The property of analytic functions expressed by the Cauchy integral theorem fully characterizes them (see [[Morera theorem]]), and therefore all the fundamental properties of analytic functions may be inferred from the Cauchy integral theorem.
  
For an arbitrary domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090014.png" /> in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090015.png" /> or on a Riemann surface, the Cauchy integral theorem may be stated as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090016.png" /> is a regular analytic function in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090017.png" />, then the integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090018.png" /> along any rectifiable closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090019.png" /> which is homotopic to zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090020.png" /> vanishes.
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For an arbitrary open set $D\subset \mathbb C$ or on a [[Riemann surface]], the Cauchy integral theorem may be stated as follows: if $f:D\to \mathbb C$ is holomorphic and $\gamma \subset D$ a closed rectifiable curve [[Homotopy group|homotopic]] to $0$, then \eqref{e:integral_vanishes} holds.
  
A generalization of the Cauchy integral theorem to analytic functions of several complex variables is the Cauchy–Poincaré theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090022.png" />, is a regular analytic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090023.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090025.png" />, then, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090026.png" />-dimensional surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090027.png" /> with smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090028.png" />,
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A generalization of the Cauchy integral theorem to holomorphic functions of several complex variables (see [[Analytic function]] for the definition) is the Cauchy-Poincaré theorem.
  
<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/c/c020/c020900/c02090029.png" /></td> </tr></table>
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'''Theorem 2'''
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If $D\subset \mathbb C^n$ is an open set and $f:D \to \mathbb C$ a holomorphic function, then for any smooth oriented $n+1$-dimensional (real) surface $\Sigma$ with smooth boundary $\partial \Sigma$ we have
 +
\[
 +
\int_{\partial \Sigma} f(z)\, dz = 0\, ,
 +
\]
 +
where $dz$ denotes the differential form $dz_1\wedge dz_2 \wedge \ldots \wedge dz_n$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090030.png" /> is an abbreviation for the holomorphic differential 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/c/c020/c020900/c02090031.png" /></td> </tr></table>
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When $n=1$ the surface $\Sigma$ and the domain $D$ have the same (real) dimension (the case of the classical Cauchy integral theorem); when $n>1$, $\Sigma$ has strictly lower dimension than $D$. See also [[Residue of an analytic function|Residue of an analytic function]]; [[Cauchy integral|Cauchy integral]].
 
 
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090032.png" /> the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090033.png" /> and the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090034.png" /> have the same dimension, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090035.png" /> (the case of the classical Cauchy integral theorem); when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090037.png" /> has lower dimension than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090039.png" />. See also [[Residue of an analytic function|Residue of an analytic function]]; [[Cauchy integral|Cauchy integral]].
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.L. Cauchy,  "Oeuvres complètes, Ser. 1" , '''4''' , Paris  (1890)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Goursat,  "Démonstration du théorème de Cauchy"  ''Acta Math.'' , '''4'''  (1884)  pp. 197–200</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1–2''' , Chelsea  (1977)  (Translated from 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><TR><TD valign="top">[5]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
In [[#References|[2]]] Goursat still assumed continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020900/c02090040.png" />. He soon saw how to remove this assumption, cf. [[#References|[a1]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Goursat,  "Sur la définition générale des fonctions analytiques, d'après Cauchy"  ''Trans. Amer. Math. Soc.'' , '''1'''  (1900)  pp. 14–16</TD></TR></table>
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{|
 +
|-
 +
|valign="top"|{{Ref|Al}}|| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) {{MR|0188405}} {{ZBL|0154.31904}}
 +
|-
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|valign="top"|{{Ref|Ca}}|| A.L. Cauchy,  "Oeuvres complètes, Ser. 1" , '''4''' , Paris  (1890)
 +
|-
 +
|valign="top"|{{Ref|Go}}|| E. Goursat,  "Démonstration du théorème de Cauchy"  ''Acta Math.'' , '''4'''  (1884)  pp. 197–200
 +
|-
 +
|valign="top"|{{Ref|Go2}}|| E. Goursat,  "Sur la définition générale des fonctions analytiques, d'après Cauchy"  ''Trans. Amer. Math. Soc.'' , '''1'''  (1900)  pp. 14–16
 +
|-
 +
|valign="top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of  functions of a complex variable" ,  '''1–3''' , Chelsea (1977)  (Translated from Russian) {{MR|0444912}}  {{ZBL|0357.30002}}
 +
|-
 +
|valign="top"|{{Ref|Sh}}|| B.V. Shabat, "Introduction of complex  analysis" , '''1–2''' , Moscow  (1976) (In Russian) {{MR|}}  {{ZBL|0799.32001}} {{ZBL|0732.32001}}  {{ZBL|0732.30001}}  {{ZBL|0578.32001}} {{ZBL|0574.30001}} 
 +
|-
 +
|valign="top"|{{Ref|Vl}}|| V.S. Vladimirov, "Methods of the theory of  functions of several complex variables" , M.I.T. (1966) (Translated  from Russian)
 +
|-
 +
|}

Latest revision as of 13:04, 3 January 2014

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

A fundamental theorem in complex analysis which states the following.

Theorem 1 If $D\subset \mathbb C$ is a simply connected open set and $f:D\to \mathbb C$ a holomorphic funcion, then the integral of $f(z)\, dz$ along any closed rectifiable curve $\gamma\subset D$ vanishes: \begin{equation}\label{e:integral_vanishes} \int_\gamma f(z)\, dz = 0\, . \end{equation}

The left hand side of \eqref{e:integral_vanishes} is the integral of the (complex) differential form $f(z)\, dz$ (see also Integration on manifolds). More precisely, if $\alpha: \mathbb S^1 \to \mathbb C$ is a Lipschitz parametrization of the curve $\gamma$, then \begin{equation}\label{e:formula_integral} \int_\gamma f(z)\, dz = \int_0^{2\pi} f (\alpha (t))\, \dot{\alpha} (t)\, dt\, \end{equation} (observe that in order for \eqref{e:formula_integral} to be well defined, i.e. independent of the chosen parametrization, we must in general decide an orientation for the curve $\gamma$; however since \eqref{e:integral_vanishes} stipulates that the integral vanishes, the choice of the orientation is not important in the present context).

An equivalent version of Cauchy's integral theorem states that (under the same assuptions of Theorem 1), given any (rectifiable) path $\eta:[0,1]\to D$ the integral \[ \int_\eta f(z)\, dz \] depends only upon the two endpoints $\eta (0)$ and $\eta(1)$, and hence it is independent of the choice of the path of integration $\eta$. This, essentially, was the original formulation of the theorem as proposed by A.L. Cauchy (1825) (see [Ca]); similar formulations may be found in the letters of C.F. Gauss (1811). Cauchy's proof involved the additional assumption that the (complex) derivative $f'$ is continuous; the first complete proof was given by E. Goursat [Go2]. The property of analytic functions expressed by the Cauchy integral theorem fully characterizes them (see Morera theorem), and therefore all the fundamental properties of analytic functions may be inferred from the Cauchy integral theorem.

For an arbitrary open set $D\subset \mathbb C$ or on a Riemann surface, the Cauchy integral theorem may be stated as follows: if $f:D\to \mathbb C$ is holomorphic and $\gamma \subset D$ a closed rectifiable curve homotopic to $0$, then \eqref{e:integral_vanishes} holds.

A generalization of the Cauchy integral theorem to holomorphic functions of several complex variables (see Analytic function for the definition) is the Cauchy-Poincaré theorem.

Theorem 2 If $D\subset \mathbb C^n$ is an open set and $f:D \to \mathbb C$ a holomorphic function, then for any smooth oriented $n+1$-dimensional (real) surface $\Sigma$ with smooth boundary $\partial \Sigma$ we have \[ \int_{\partial \Sigma} f(z)\, dz = 0\, , \] where $dz$ denotes the differential form $dz_1\wedge dz_2 \wedge \ldots \wedge dz_n$.


When $n=1$ the surface $\Sigma$ and the domain $D$ have the same (real) dimension (the case of the classical Cauchy integral theorem); when $n>1$, $\Sigma$ has strictly lower dimension than $D$. See also Residue of an analytic function; Cauchy integral.

References

[Al] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) MR0188405 Zbl 0154.31904
[Ca] A.L. Cauchy, "Oeuvres complètes, Ser. 1" , 4 , Paris (1890)
[Go] E. Goursat, "Démonstration du théorème de Cauchy" Acta Math. , 4 (1884) pp. 197–200
[Go2] E. Goursat, "Sur la définition générale des fonctions analytiques, d'après Cauchy" Trans. Amer. Math. Soc. , 1 (1900) pp. 14–16
[Ma] A.I. Markushevich, "Theory of functions of a complex variable" , 1–3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[Sh] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001
[Vl] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
How to Cite This Entry:
Cauchy integral theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_integral_theorem&oldid=19065
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article