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 nonzero 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 \equiv \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}

Proof

Proof. From the definition of a meromorphic function, we know that we can decompose $f(z)$ like so, where $h(z)$ is holomorphic and $z_p$ are all its poles:

\begin{aligned} f(z) = h(z) + \sum_{z_p} \frac{R_p}{z - z_p} \end{aligned}

We integrate this over a contour $C$ which contains all poles, and apply both Cauchy’s integral theorem and Cauchy’s integral formula to get:

\begin{aligned} \oint_C f(z) \dd{z} &= \oint_C h(z) \dd{z} + \sum_{p} R_p \oint_C \frac{1}{z - z_p} \dd{z} = \sum_{p} R_p \: 2 \pi i \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.