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Homework 9 (MATH 1199 Fall 2019)
1. Draw the contour, identify any "blow-up points", and compute the following integrals:
(a) $\displaystyle\int_C \dfrac{\sin(z)\cos(z)}{z^2+1} \mathrm{d}z$, where $C$ is the circle $|z-5|=1$
(b) $\displaystyle\int_C \dfrac{\sin(z)\cos(z)}{z^2+1} \mathrm{d}z$, where $C$ is the circle $|z-1.5i|=1$
(c) $\displaystyle\int_C \dfrac{e^z \sin(z)}{z+1} \mathrm{d}z$, where $C$ is the circle $|z-1|=1$
(d) $\displaystyle\int_C \dfrac{z^3+5z^2-2z+1}{z^2-4} \mathrm{d}z$, where $C$ is the unit circle
(e) $\displaystyle\int_C \dfrac{z^3+5z^2-2z+1}{z^2-4} \mathrm{d}z$, where $C$ is the circle $|z-2|=1$
(f) $\displaystyle\int_C \dfrac{z^3+5z^2-2z+1}{z^2-4} \mathrm{d}z$, where $C$ is the circle $|z+2|=1$
(g) $\displaystyle\int_C \dfrac{z\exp(e^z)}{z-\frac{1}{2}} \mathrm{d}z$, where $C$ is the unit circle
(h) $\displaystyle\int_C \dfrac{z\exp(e^z)}{z-\frac{1}{2}i} \mathrm{d}z$, where $C$ is the unit circle