Laurent's expansion

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)

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