Residue

Exercise5.2

1. Find the residue at the singularity of the following function.

(a)

$\frac{1}{z(z-1)^2}$

(b)

$\frac{z}{(2z+1)(z-2)}$

(c)

$\frac{1}{\sin{z}}$

(d)

$\frac{e^{z}}{(z-1)(z+2)^2}$

2. Evaluate the following integrals.

(a)

$\int_{\vert z\vert=2}\frac{dz}{z(z-1)^2}$

(b)

$\int_{\vert z\vert=1}\frac{z}{(2z+1)(z-2)} dz$

(c)

$\int_{\vert z\vert=1}\frac{dz}{\sin{z}}$

(d)

$\int_{\vert z\vert=3}\frac{e^{z}}{(z-1)(z+2)^2} dz$

3. Evaluate $\frac{e^{2z}}{z^{2}(z^2 + 2z + 2)}$ along the following curves.

(a)

$\vert z\vert = 1$

(b)

$\vert z - i\vert = 2$

(c)

$\vert z\vert = 3$