Residues.

Let $f$ be a function analytic on a simple Jordan curve $C$ and at all the interior points, excepted at $z_0$ . The residue of $f$ at $z_0$ , denoted Res[$f(z),z_0$ ] is the complex number;

$$[f(z),z_0] = \frac {1}{2 \pi i} \int_C f(z) \; dz$$

Theorem 9.1.1   The residue of $f$ at $z_0$ is the coefficient $a_{-1}$ in the Laurent series expansion of $f$ in an annulus $0<\vert z-z_0\vert<r$ .

Example 9.1.2   Compute the integral $\int_C \frac {1}{z^2+z} \; dz$ , where $C=\{ z \in \mathbb{C}; \; \vert z+1\vert=1 \}$ .

We compute a Laurent series expansion for $z$ which is convergent on an annulus centered at -1.

$$f(z) = \frac {1}{z^2+z} =\frac { 1}{z(z+1)}$$

$$\quad = \frac {1}{z+1} \cdot \frac {1}{(z+1)-1}$$

$$\quad = - (z+1)^{-1} \cdot \frac {1}{1-(z+1)}$$

$$\quad = - (z+1)^{-1} ( 1 + (z+1) + (z+1)^2 + \dots )$$

$$\quad = - (z+1)^{-1} -1 - (z+1) - (z+1)^2 - \dots$$

Therefore $\int_C \frac {1}{z^2+z} \; dz = 2 \pi i (-1) = -2 \pi i$ .

How to compute quickly residues?

1. If $f$ has a pole of order 1 at $z_0$ , then

   $$[f(z),z_0]= \underset{z \rightarrow z_0}{\text{lim}} (z-z_0)f(z).$$

   
   

2. If $f$ has a pole of order 2 at $z_0$ , then

   $$[f(z),z_0]= \underset{z \rightarrow z_0}{\text{lim}} \frac {d}{dz}((z-z_0)^2f(z)).$$

   
   

3. If $f$ has a pole of order n at $z_0$ , then

   $$[f(z),z_0]= \underset{z \rightarrow z_0}{\text{lim}} \frac {1}{(n-1)!} \frac {d^{n-1}}{dz^{n-1}}((z-z_0)^nf(z)).$$

   
   

4. If $f(z)=g(z)/h(z)$ has a simple pole at $z_0$ , then

   $$[f(z),z_0]= \frac {g(z_0)}{h'(z_0)}$$