Editing Contour integration (section)

==Evaluation with residue theorem==
Using the [[residue theorem]], we can evaluate closed contour integrals. The following are examples on evaluating contour integrals with the residue theorem.

Using the residue theorem, let us evaluate this contour integral.
<math display=block>\oint_C \frac{e^z}{z^3}\,dz</math>

Recall that the residue theorem states
<math display=block>\oint_{C} f(z) dz=2\pi i\cdot \sum\operatorname{Res}(f,a_k)</math>

where <math>\operatorname{Res}</math> is the residue of <math>f(z)</math>, and the <math>a_k</math> are the singularities of <math>f(z)</math> lying inside the contour <math>C</math> (with none of them lying directly on <math>C</math>).

<math>f(z)</math> has only one pole, <math>0</math>. From that, we determine that the [[Residue (complex analysis)|residue]] of <math>f(z)</math> to be <math>\tfrac{1}{2}</math>
<math display=block>\begin{align} 
\oint_C f(z) dz&=\oint_C \frac{e^z}{z^3}dz\\ 
&=2\pi i \cdot \operatorname{Res}_{z=0} f(z)\\
&=2\pi i\operatorname{Res}_{z=0} \frac{e^z}{z^3}\\
&=2\pi i \cdot \frac{1}{2}\\
&=\pi i
\end{align}</math>

Thus, using the [[residue theorem]], we can determine:
<math display=block>\oint_C \frac{e^z}{z^3} dz = \pi i.</math>