Editing Cauchy's integral theorem (section)

==== 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.