Categories: Complex analysis, Mathematics.

# Residue theorem

A function $$f(z)$$ is meromorphic if it is holomorphic except in a finite number of simple poles, which are points $$z_p$$ where $$f(z_p)$$ diverges, but where the product $$(z - z_p) f(z)$$ is non-zero and still holomorphic close to $$z_p$$. In other words, $$f(z)$$ can be approximated close to $$z_p$$:

\begin{aligned} f(z) \approx \frac{R_p}{z - z_p} \end{aligned}

Where the residue $$R_p$$ of a simple pole $$z_p$$ is defined as follows, and represents the rate at which $$f(z)$$ diverges close to $$z_p$$:

\begin{aligned} \boxed{ R_p = \lim_{z \to z_p} (z - z_p) f(z) } \end{aligned}

Cauchy’s residue theorem for meromorphic functions is a generalization of Cauchy’s integral theorem for holomorphic functions, and states that the integral on a contour $$C$$ purely depends on the simple poles $$z_p$$ enclosed by $$C$$:

\begin{aligned} \boxed{ \oint_C f(z) \dd{z} = i 2 \pi \sum_{z_p} R_p } \end{aligned}

This theorem might not seem very useful, but in fact, by cleverly choosing the contour $$C$$, it lets us evaluate many integrals along the real axis, most notably Fourier transforms. It can also be used to derive the Kramers-Kronig relations.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.