Conformal mapping

Enter conformal mapping homework here.

Question 1

compute the residue at the poles of the functions

1. a)
$\int _c\frac{sinz}{z^2-1}dz$ c is the circle $\left | z \right |=2$

(1)
$$z^2-1=0$$
(2)
\begin{align} z=\pm 1 \end{align}

Res$\left ( -1 \right )=\frac{sinz}{2z}$
evaluate $\frac{sinz}{2z}$ at $z=-1$

(3)
\begin{align} \frac{sin\left ( -1 \right )}{2\left ( -1 \right )}=\frac{-.84}{-2}=.42074 \end{align}

Res$\left ( 1 \right )=\frac{sinz}{2z}$
evaluate $\frac{sinz}{2z}$ at $z=1$

(4)
\begin{align} \frac{sin\left ( 1 \right )}{2\left ( 1 \right )}=\frac{.84}{2}=.42074 \end{align}
(5)
\begin{align} 2\pi i\left ( .42074+.42074 \right )=1.68\pi i \end{align}
C) Compute the residue at the poles of each of the following functions.

$\int _{c} \frac{e^z}{z(z-2)^3} \, dz$ where C is the circle |z|=3

Find problem spots for denominator: z=0 and z=2

expand $(z-2)^3$ = $z^4-6z^3+12z^2-8z$

take derivative of denominator: $= \frac{e^z}{4z^3-18z^2+24z-8}$

Res(0)= $\frac{e^0}{4(0)^3-18(0)^2+24(0)-8}$

$\; \; \; \; \; \; \; \; \; \;$=$\frac{-1}{8}$

Res(2)= $\frac{1}{2!} \frac{d^2}{d^2 z}[(z-2)^3 \frac{e^z}{z(x-2)^3}]$
$\; \; \; \; \; \; \;$ =$\frac{1}{2} \frac{d^2}{d^2z}[\frac{e^z}{z}]$

Take derivative of function inside square brackets, then combine with remaining parts:

=$\frac{-e^z(z^2-2z+2}{2(z^3)}$

Plug in value (2) into function:

=$\frac{-e^2(4-4+2)}{2(8)}=\frac{-e^2(2)}{16}=\frac{-e^2}{8}$

Multiply $2\pi i[sum \; of \; both \; residues \; found]$

$2\pi i[ \frac{-1}{8}+\frac{-e^2}{8}]$ = $\frac{-e^2+(-1) \pi i}{4}$