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In [[mathematics]], the '''Cauchy integral theorem''' (also known as the '''Cauchy–Goursat theorem''') in [[complex analysis]], named after [[Augustin-Louis Cauchy]], is an important statement about [[line integral]]s for [[holomorphic function]]s in the [[complex number|complex plane]]. Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere "in between" the two paths, then the two path integrals of the function will be the same.
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The theorem is usually formulated for closed paths as follows: let ''U'' be an [[open subset]] of '''C''' which is [[simply connected]], let ''f'' : ''U'' → '''C''' be a holomorphic function, and let <math>\!\,\gamma</math> be a [[rectifiable path]] in ''U'' whose start point is equal to its end point. Then
 
:<math>\oint_\gamma f(z)\,dz = 0. </math>
 
A precise ([[homology (mathematics)|homology]]) version can be stated using [[winding number]]s. The winding number of a closed curve around a point ''a'' not on the curve is the integral of ''f''(''z'')/[2 ''i''{{pi}}], where ''f''(''z'') = 1/(''z''&nbsp;&minus;&nbsp;''a'') around the curve. It is an [[integer]].
Briefly, the path integral along a [[Jordan curve]] of a function holomorphic in the interior of the curve, is zero. Instead of a single closed path we can consider a linear combination of closed paths, where the scalars are integers. Such a combination is called a closed chain, and one defines an integral along the chain as a linear combination of integrals over individual paths. A closed chain is called a cycle in a region if it is homologous to zero in the region; that is, the winding number, expressed by the integral of 1/(''z''&nbsp;&minus;&nbsp;''a'') over the closed chain, is zero for each point 'a' not in the region. This means that the closed chain does not wind around points outside the region. Then Cauchy's theorem can be stated as the integral of a function holomorphic in an open set taken around any cycle in the open set is zero. An example is furnished by the ring-shaped region. This version is crucial for rigorous derivation  of [[Laurent series]] and Cauchy's residue formula without involving any physical notions such as cross cuts or deformations. The version enables the extension of Cauchy's theorem to multiply-connected regions analytically.
 
==Discussion==
As was shown by [[Goursat]], Cauchy's integral theorem can be proven assuming only that the complex derivative ''f''&nbsp;'(''z'') exists everywhere in ''U''. This is significant, because one can then prove [[Cauchy's integral formula]] for these functions, and from that deduce these functions are in fact [[infinitely differentiable]].
 
The condition that ''U'' be [[simply connected]] means that ''U'' has no "holes" or, in [[homotopy]] terms, that the [[fundamental group]] of ''U'' is trivial; for instance, every open disk <math>U=\{ z: |z-z_{0}| < r\}</math> qualifies. The condition is crucial; consider
 
:<math>\gamma(t) = e^{it} \quad t \in \left[0,2\pi\right]</math>
 
which traces out the unit circle, and then the path integral
 
:<math>\oint_\gamma \frac{1}{z}\,dz = \int_0^{2\pi} { ie^{it} \over e^{it} }\,dt= \int_0^{2\pi}i\,dt = 2\pi i </math>
 
is non-zero; the Cauchy integral theorem does not apply here since <math>f(z) = 1/z</math> is not defined (and certainly not holomorphic) at <math>z = 0</math>.
 
One important consequence of the theorem is that path integrals of holomorphic functions on simply connected domains can be computed in a
manner familiar from the [[fundamental theorem of calculus|fundamental theorem of real calculus]]: let ''U'' be a [[simply connected]] [[open subset]] of '''C''', let ''f'' : ''U'' → '''C''' be a holomorphic function, and let γ be a [[piecewise continuously differentiable path]] in ''U'' with start point ''a'' and end point ''b''. If ''F'' is a [[complex antiderivative]] of ''f'', then
 
:<math>\int_\gamma f(z)\,dz=F(b)-F(a).</math>
 
The Cauchy integral theorem is valid in slightly stronger forms than given above. e.g. Let ''U'' be a simply connected open subset of  '''C''' and ''f'' a function which is holomorphic on ''U'' and continuous on [[closure (topology)|<math>\textstyle\overline{U}</math>]]. Let <math>\gamma</math> be a loop in <math>\textstyle\overline{U}</math> which is uniform limit of a sequence <math>\gamma_k</math> of rectifiable loops  in ''U'' with bounded length. Then, applying the Cauchy theorem to the  <math>\gamma_k</math>, and passing to the limit one has
 
:<math>\oint_\gamma f(z)\,dz=0.</math>
 
See e.g. {{harv|Kodaira|2007|loc=Theorem 2.3}} for a more general result.
 
The Cauchy integral theorem leads to [[Cauchy's integral formula]] and the [[residue theorem]].
 
==Proof==
If one assumes that the partial derivatives of a holomorphic function are continuous, the Cauchy integral theorem can be proved as a direct consequence of [[Green's theorem]] and the fact that the real and imaginary parts of <math>f=u+iv</math> must satisfy the [[Cauchy–Riemann equations]] in the region bounded by <math>\gamma</math>, and moreover in the open neighborhood ''U'' of this region. Cauchy provided this proof, but it was later proved by Goursat without requiring techniques from vector calculus, or the continuity of partial derivatives.
 
We can break the integrand <math>f</math>, as well as the differential <math>dz</math> into their real and imaginary components:
 
:<math> \displaystyle f=u+iv </math>
:<math> \displaystyle dz=dx+i\,dy </math>
 
In this case we have
 
:<math>\oint_\gamma f(z)\,dz = \oint_\gamma (u+iv)(dx+i\,dy) = \oint_\gamma (u\,dx-v\,dy) +i\oint_\gamma (v\,dx+u\,dy)</math>
 
By Green's theorem, we may then replace the integrals around the closed contour <math>\gamma</math> with an area integral throughout the domain <math>D</math> that is enclosed by <math>\gamma</math> as follows:
 
:<math>\oint_\gamma (u\,dx-v\,dy) =  \iint_D \left( -\frac{\partial v}{\partial x} -\frac{\partial u}{\partial y} \right )\,dx\,dy </math>
:<math>\oint_\gamma (v\,dx+u\,dy) =  \iint_D \left(  \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y} \right )\,dx\,dy</math>
 
However, being the real and imaginary parts of a function analytic in the domain <math>D</math>, <math>u</math> and <math>v</math> must satisfy the [[Cauchy–Riemann equations]] there:
 
:<math>{ \partial u \over \partial x } = { \partial v \over \partial y } </math>
:<math>{ \partial u \over \partial y } = -{ \partial v \over \partial x } </math>
 
We therefore find that both integrands (and hence their integrals) are zero
 
:<math>\iint_D \left(  -\frac{\partial v}{\partial x} -\frac{\partial u}{\partial y} \right )\,dx\,dy = \iint_D \left(  \frac{\partial u}{\partial y} -\frac{\partial u}{\partial y} \right )\,dx\,dy =0</math>
:<math>\iint_D \left(  \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y} \right )\,dx\,dy = \iint_D \left(  \frac{\partial u}{\partial x}-\frac{\partial u}{\partial x} \right ) \, dx \, dy = 0</math>
 
This gives the desired result
 
:<math>\oint_\gamma f(z)\,dz =0</math>
 
==See also==
*[[Cauchy&ndash;Riemann equations]]
*[[Morera's theorem]]
*[[Methods of contour integration]]
*[[Residue (complex analysis)]]
*[[Green's Theorem]]
 
==References==
* {{citation
|first = Kunihiko
|last = Kodaira
| authorlink = Kunihiko Kodaira
| year = 2007
| title =  Complex Analysis
| series= Cambridge Stud. Adv. Math., 107
| publisher = [[Cambridge University Press|CUP]]
| isbn = 978-0-521-80937-5}}
* {{citation
|first = Lars
|last = Ahlfors
| authorlink = Lars Ahlfors
| year = 2000
| title =  Complex Analysis
| series= McGraw-Hill series in Mathematics
| publisher = [[McGraw-Hill]]
| isbn = 0-07-000657-1
}}
* {{citation
|first = Serge
|last = Lang
| authorlink = Serge Lang
| year = 2003
| title =  Complex Analysis
| series= Springer Verlag GTM
| publisher = [[Springer Verlag]]
}}
* {{citation
|first = Walter
|last = Rudin
| authorlink = Walter Rudin
| year = 2000
| title = Real and Complex Analysis
| series= McGraw-Hill series in mathematics
| publisher = [[McGraw-Hill]]
}}
 
== External links ==
* {{springer|title=Cauchy integral theorem|id=p/c020900}}
* {{MathWorld | urlname= CauchyIntegralTheorem | title= Cauchy Integral Theorem}}
*[https://sites.google.com/site/anilsmathcomputernotes/ The site Anil's maths computer notes contains a simple treatment of the global (homology version of Cauchy's Theorem assuming Cauchy's theorem for a triangle and a convex region] offering simplified Dixon's proof and deduces Cauchy's theorem for a simply connected region
 
[[Category:Theorems in complex analysis]]

Latest revision as of 17:37, 3 November 2014


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