Editing Residue (complex analysis) (section)

== Contour integration<!-- Section needs introduction --> ==
{{See also|Contour integration}}

=== Contour integral of a monomial ===
Computing the residue of a [[monomial]]

:<math>\oint_C z^k \, dz</math>

makes most residue computations easy to do. Since path integral computations are [[homotopy]] invariant, we will let <math>C</math> be the circle with radius <math>1</math> going counter clockwise. Then, using the change of coordinates <math>z \to e^{i\theta}</math> we find that

: <math>dz \to d(e^{i\theta}) = ie^{i\theta} \, d\theta</math>

hence our integral now reads as

:<math>
\oint_C z^k dz = \int_0^{2\pi} i e^{i(k+1)\theta} \, d\theta
= \begin{cases}
2\pi i & \text{if } k = -1, \\
0 & \text{otherwise}.
\end{cases}
</math>
Thus, the residue of <math>z^k</math> is 1 if integer <math>k=-1</math> and 0 otherwise.

=== Generalization to Laurent series ===
If a function is expressed as a [[Laurent series]] expansion around c as follows:<math display="block">f(z) = \sum_{n=-\infty}^\infty a_n(z-c)^n.</math>Then, the residue at the point c is calculated as:<math display="block">\operatorname{Res}(f,c) = {1 \over 2\pi i} \oint_\gamma f(z)\,dz = {1 \over 2\pi i} \sum_{n=-\infty}^\infty \oint_\gamma a_n(z-c)^n \,dz = a_{-1} </math>using the results from contour integral of a monomial for counter clockwise contour integral <math>\gamma</math> around a point c. Hence, if a [[Laurent series]] representation of a function exists around c, then its residue around c is known by the coefficient of the <math>(z-c)^{-1}</math> term.

=== Application in residue theorem ===
{{Main|Residue theorem}}
For a [[meromorphic function]] <math>f</math>, with a finite set of singularities within a [[Curve orientation|positively oriented]] [[Jordan curve|simple closed curve]] <math>C</math> which does not pass through any singularity, the value of the contour integral is given according to [[residue theorem]], as:<math display="block">
\oint_C f(z)\, dz = 2\pi i \sum_{k=1}^n \operatorname{I}(C, a_k) \operatorname{Res}(f, a_k).
</math>where <math>\operatorname{I}(C, a_k)</math>, the winding number, is <math>1</math> if <math>a_k</math> is in the interior of <math>C</math> and <math>0</math> if not, simplifying to:<math display="block">
\oint_\gamma f(z)\, dz = 2\pi i \sum \operatorname{Res}(f, a_k)
</math>where <math>a_k</math> are all isolated singularities within the contour <math>C</math>.