Editing Cauchy's integral theorem (section)

==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]].