While closed contour integrals on holomorphic functions disappear, [meromorphic functions](Meromorphic%20functions.md), when integrated on such paths contain non-zero terms proportional to the sum of the [residues](Residues.md) at [[pole]]s $z_i$ such that the [closed contour integral](Closed%20contour%20integral.md) $\gamma$ is
$\oint_{\gamma}dz f(z) = 2 \pi i \sum_i \mbox{Res}_{z\rightarrow z_0}(f(z))$
where $\gamma$ is a _positively_ oriented (counterclockwise) contour. While
$\oint_{\gamma}dz f(z) = -2 \pi i \sum_i \mbox{Res}_{z\rightarrow z_0}(f(z))$
for a _negatively_ (clockwise) oriented contour.
# The residue theorem for infinitessimal circular arcs
The residue theorem also applies for contours in the shape of circular arcs as long as the radius of the circular arc approaches 0. Thus we may also state that given a simple [pole](Pole.md) at $z_0$ centered at the arc center,
$\lim_{r\rightarrow 0}\int_{\gamma}dz f(z) = \alpha i\, \mbox{Res}_{z\rightarrow z_0}(f(z)).$ ^6ef7fc
where $\gamma$ is a semicircle with a radius $r$ and $\alpha$ is the arc measure expressed in radians.
Image adapted from Orloff, J., Lecture Notes, MIT 18.04 _Complex Variables with Applications_
---
# Proofs and Examples
## Proof of the residue theorem
### Additional proof for [semicircular arcs](Residue%20theorem.md#The%20residue%20theorem%20for%20infinitessimal%20circular%20arcs)
---
# Recommended reading
The Residue theorem is given in many introductory level texts in Complex Analysis, complex variables, and mathematical methods for physicists. Thus one may find it introduced in the following texts:
* [Brown, J. W., Churchill R. V., _Complex Variables and Applications_, McGraw Hill, 8th edition, 2009.](Brown,%20J.%20W.,%20Churchill%20R.%20V.,%20Complex%20Variables%20and%20Applications,%20McGraw%20Hill,%208th%20edition,%202009..md) pg. 235. This text is aimed at undergraduates studying mathematics but is appropriate for anyone being introduced to complex analysis for the first time.
* [Altland, A. von Delft, J. _Mathematics for Physicists_, Cambridge University Press, 2019](Altland,%20A.%20von%20Delft,%20J.%20Mathematics%20for%20Physicists,%20Cambridge%20University%20Press,%202019.md) pg. 350.
For a discussion of the application of the residue theorem along a circular arc see:
* [Orloff, J., Lecture Notes, MIT 18.04 _Complex Variables with Applications_](MIT18_04S18_topic9.pdf) pgs. 15-17. These notes are also publicly available [here](https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-spring-2018/lecture-notes/MIT18_04S18_topic9.pdf) and part of a full course on Complex Analysis available [here](https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-spring-2018/index.htm) on MIT Opencourseware.
#MathematicalFoundations/Analysis/ComplexAnalysis/Integrals