Laurent's expansion

Theorem 5..1 (Laurent's expansion)   Concentric circles centered on $C_{1}, C_{2}$, their radii are $r_{1}, r_{2}$, and $C$ is any concentric circle between $C_{1}$ and $C_{2}$. The function $f(z)$ is a circular region surrounded by $C_{1}$ and $C_{2}$ with $r_{2} \leq \vert z –a\vert \leq r_{1}$. When monovalent and analytic, for any $z$ satisfying $r_{2} < \vert z -a\vert <r_{1}$,

$$f(z) = \sum_{n=-\infty}^{\infty}c_{n}(z-a)^{n}$$

$$= \cdots + \frac{c_{-m}}{(z-a)^{m}} + \cdots + \frac{c_{-2}}{(z-a)^2} + \frac{c_{-1}}{z-a}$$

$$+ c_{0} + c_{1}(z-a) + \cdots + c_{n}(z-a)^{n} + \cdots$$

Here coefficients of each term is

$$c_{n} = \frac{1}{2\pi i}\int_{C}\frac{f(\zeta)}{(\zeta - a)^{n+1}}\; d\zeta (n = 0, \pm 1, \pm 2, \cdots)$$

Exercise5.1

1. For the function $f(z) = \frac{1}{z^2 - 3z + 2}$, find the Laurent's expansion centered at the origin for the points in each of the following regions.

(a)

$\vert z\vert < 1$

(b)

$a < \vert z\vert < 2$

(c)

$\vert z\vert > 2$

2. Find the Laurent's expansion of the following functions at the center of the specified point. What kind of singularity is the center?

(a)

$\frac{1}{z^{3}(z+1)} [z=0]$

(b)

$\frac{z^3}{(z+1)} [z = -1]$

(c)

$\frac{e^{z^2}}{z^3} [z = 0]$

(d)

$\frac{\sin{z}}{z - \pi} [z = \pi]$