Categories: Complex analysis, Mathematics.

Residue theorem

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

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

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

Rplimzzp(zzp)f(z)\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 CC purely depends on the simple poles zpz_p enclosed by CC:

Cf(z)dz=i2πzpRp\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)f(z) like so, where h(z)h(z) is holomorphic and zpz_p are all its poles:

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

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

Cf(z)dz=Ch(z)dz+pRpC1zzpdz=pRp2πi\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 CC, 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.