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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 $\left | z \right |=2$
Res$\left ( -1 \right )=\frac{sinz}{2z}$
evaluate $\frac{sinz}{2z}$ at $z=-1$
Res$\left ( 1 \right )=\frac{sinz}{2z}$
evaluate $\frac{sinz}{2z}$ at $z=1$
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 $2\pi i[sum \; of \; both \; residues \; found]$
$2\pi i[ \frac{-1}{8}+\frac{-e^2}{8}]$ = $\frac{-e^2+(-1) \pi i}{4}$