Editing Cauchy's integral theorem

{{Short description|Theorem in complex analysis}}
{{Distinguish|Cauchy's integral formula|Cauchy formula for repeated integration}}

{{Complex analysis sidebar}}
In [[mathematics]], the '''Cauchy integral theorem''' (also known as the '''Cauchy–Goursat theorem''') in [[complex analysis]], named after [[Augustin-Louis Cauchy]] (and [[Édouard Goursat]]), is an important statement about [[line integral]]s for [[holomorphic function]]s in the [[complex number|complex plane]]. Essentially, it says that if <math>f(z)</math> is holomorphic in a [[simply connected]] [[Domain (mathematical analysis)|domain]] Ω, then for any simply closed contour <math>C</math> in Ω, that contour integral is zero. 

<math display="block">\int_C f(z)\,dz = 0. </math>

==Statement==
=== Fundamental theorem for complex line integrals ===
If {{math|''f''(''z'')}} is a holomorphic function on an open [[region (mathematical analysis)|region]] {{mvar|U}}, and <math>\gamma</math> is a curve in {{mvar|U}} from <math>z_0</math> to <math>z_1</math> then,
<math display="block">\int_{\gamma}f'(z) \, dz = f(z_1)-f(z_0).</math>

Also, when {{math|''f''(''z'')}} has a single-valued antiderivative in an open region {{mvar|U}}, then the path integral <math display="inline">\int_{\gamma}f(z) \, dz</math> is path independent for all paths in {{mvar|U}}.

==== Formulation on simply connected regions ====

Let <math>U \subseteq \Complex</math> be a [[Simply connected space|simply connected]] [[open subset|open]] set, and let <math>f: U \to \Complex</math> be a [[holomorphic function]]. Let <math>\gamma: [a,b] \to U</math> be a smooth closed curve. Then:
<math display="block">\int_\gamma f(z)\,dz = 0. </math>
(The condition that <math>U</math> be [[simply connected]] means that <math>U</math> has no "holes", or in other words, that the [[fundamental group]] of <math>U</math> is trivial.)

==== General formulation ====

Let <math>U \subseteq \Complex</math> be an [[open subset|open set]], and let <math>f: U \to \Complex</math> be a [[holomorphic function]]. Let <math>\gamma: [a,b] \to U</math> be a smooth closed curve. If  <math>\gamma</math> is [[Homotopy|homotopic]] to a constant curve, then:
<math display="block">\int_\gamma f(z)\,dz = 0. </math>where z є ''U''

(Recall that a curve is [[Homotopy|homotopic]] to a constant curve if there exists a smooth [[homotopy]] (within <math>U</math>) from the curve to the constant curve. Intuitively, this means that one can shrink the curve into a point without exiting the space.) The first version is a special case of this because on a [[Simply connected space|simply connected]] set, every closed curve is [[Homotopy|homotopic]] to a constant curve.

==== Main example ====

In both cases, it is important to remember that the curve <math>\gamma</math> does not surround any "holes" in the domain, or else the theorem does not apply. A famous example is the following curve: 
<math display="block">\gamma(t) = e^{it} \quad t \in \left[0, 2\pi\right] ,</math>
which traces out the [[unit circle]]. Here the following integral:
<math display="block">\int_{\gamma} \frac{1}{z}\,dz = 2\pi i \neq 0 , </math>
is nonzero. The Cauchy integral theorem does not apply here since <math>f(z) = 1/z</math> is not defined at <math>z = 0</math>. Intuitively, <math>\gamma</math> surrounds a "hole" in the domain of <math>f</math>, so <math>\gamma</math> cannot be shrunk to a point without exiting the space. Thus, the theorem does not apply.

==Discussion==
As [[Édouard Goursat]] showed, Cauchy's integral theorem can be proven assuming only that the complex derivative <math>f'(z)</math> exists everywhere in <math>U</math>. This is significant because one can then prove [[Cauchy's integral formula]] for these functions, and from that deduce these functions are [[infinitely differentiable]].

The condition that <math>U</math> be [[simply connected]] means that <math>U</math> has no "holes" or, in [[homotopy]] terms, that the [[fundamental group]] of <math>U</math> is trivial; for instance, every open disk <math>U_{z_0} = \{ z : \left|z-z_{0}\right| < r\}</math>, for <math>z_0 \in \Complex</math>, qualifies. The condition is crucial; consider
<math display="block">\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 display="block">\oint_\gamma \frac{1}{z}\,dz = \int_0^{2\pi} \frac{1}{e^{it}}(ie^{it} \,dt) = \int_0^{2\pi}i\,dt = 2\pi i </math>
is nonzero; the Cauchy integral theorem does not apply here since <math>f(z) = 1/z</math> is not defined (and is 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]]: let <math>U</math> be a [[simply connected]] [[open subset]] of <math>\Complex</math>, let <math>f: U \to \Complex</math> be a holomorphic function, and let <math>\gamma</math> be a [[piecewise continuously differentiable path]] in <math>U</math> with start point <math>a</math> and end point <math>b</math>. If <math>F</math> is a [[complex antiderivative]] of <math>f</math>, then
<math display="block">\int_\gamma f(z)\,dz=F(b)-F(a).</math>

The Cauchy integral theorem is valid with a weaker hypothesis than given above, e.g. given <math>U</math>'','' a simply connected open subset of <math>\Complex</math>, we can weaken the assumptions to <math>f</math> being holomorphic on <math>U</math> and continuous on [[closure (topology)|<math display="inline">\overline{U}</math>]] and <math>\gamma</math> a [[rectifiable curve|rectifiable]] [[Jordan curve theorem|simple loop]] in <math display="inline">\overline{U}</math>.<ref>{{Cite journal|last=Walsh|first=J. L.|date=1933-05-01|title=The Cauchy-Goursat Theorem for Rectifiable Jordan Curves|journal=Proceedings of the National Academy of Sciences|volume=19|issue=5|pages=540–541| doi=10.1073/pnas.19.5.540|pmid=16587781|pmc=1086062|issn=0027-8424|doi-access=free|bibcode=1933PNAS...19..540W }}</ref>

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 proven 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 {{nowrap|<math>\gamma</math>,}} and moreover in the open neighborhood {{mvar|U}} of this region. Cauchy provided this proof, but it was later proven by Goursat without requiring techniques from [[vector calculus]], or the continuity of partial derivatives.

We can break the integrand {{nowrap|<math>f</math>,}} as well as the differential <math>dz</math> into their real and imaginary components:

<math display="block"> f=u+iv </math>
<math display="block"> dz=dx+i\,dy </math>

In this case we have
<math display="block">\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 display="block">\oint_\gamma (u\,dx-v\,dy) = \iint_D \left( -\frac{\partial v}{\partial x} -\frac{\partial u}{\partial y} \right) \,dx\,dy </math>
<math display="block">\oint_\gamma (v\,dx+u\,dy) = \iint_D \left(  \frac{\partial u}{\partial x} -\frac{\partial v}{\partial y} \right) \,dx\,dy </math>

But as the real and imaginary parts of a function holomorphic in the domain {{nowrap|<math>D</math>,}} <math>u</math> and <math>v</math> must satisfy the [[Cauchy–Riemann equations]] there:
<math display="block">\frac{ \partial u }{ \partial x } = \frac{ \partial v }{ \partial y } </math>
<math display="block">\frac{ \partial u }{ \partial y } = -\frac{ \partial v }{ \partial x } </math>

We therefore find that both integrands (and hence their integrals) are zero

<math display="block">\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 display="block">\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 display="block">\oint_\gamma f(z)\,dz = 0</math>

==See also==
*[[Morera's theorem]]
*[[Methods of contour integration]]
*[[Star domain]]

==References==
{{Reflist}}
* {{citation
 |first = Kunihiko
 |last = Kodaira
 | author-link = 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
 | author-link = 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
 | author-link = Serge Lang
 | year = 2003
 | title =  Complex Analysis
 | series= Springer Verlag GTM
 | publisher = [[Springer Verlag]]
}}
* {{citation
 |first = Walter
 |last = Rudin
 | author-link = Walter Rudin
 | year = 2000
 | title = Real and Complex Analysis
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== External links ==
* {{springer|title=Cauchy integral theorem|id=p/c020900}}
* {{MathWorld | urlname= CauchyIntegralTheorem | title= Cauchy Integral Theorem}}
*Jeremy Orloff, 18.04 [https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-spring-2018/lecture-notes/ Complex Variables with Applications] Spring 2018 Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons.
[[Category:Augustin-Louis Cauchy]]
[[Category:Theorems in complex analysis]]