Categories: Complex analysis, Mathematics.

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 \(C\):

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

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.

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