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Homework 11 (MATH 1199 Fall 2019)
1. Let $C$ be a contour that surrounds all of the poles. Use the residue theorem (and the "shortcut theorem" to find residues) to compute the integral of...
(a) $f(z)=\dfrac{e^z \cos(z)}{z^2+1}$
(b) $f(z)=\dfrac{z^2\sin(z)}{(z+1)^2}$
(c) $f(z)=\dfrac{\cos(z)\sin(z)}{z^2+2z+1}$
2. Recall that the inversion integral for the Laplace transform of a function $F$ with poles at $z_1, \ldots, z_n$ is
$$f(t) = \dfrac{1}{2\pi i} \displaystyle\int_C F(z)e^{zt} \mathrm{d}z,$$
where $C$ is a contour around the poles. Use the inversion integral to invert...
(a) $F(z)=\dfrac{1}{z^2+9}$
(b) $F(z)=\dfrac{z}{z^2+4}$
(c) $F(z)=\dfrac{1}{(z+a)^2(z+b)^2}$
