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Homework 10 (MATH 1199 Fall 2019)
1. Find a Laurent series, centered at $0$, for the function $f(z)=\dfrac{1+5z^3}{z^4+z^7}$. Use that Laurent series with the Laurent series theorem to compute $\displaystyle\int_C \dfrac{1+5z^3}{z^4+z^7} \mathrm{d}z$ where $C$ is the circle $|z|=\frac{1}{2}$, oriented positively.
2. Find a Laurent series, centered at $0$, for $e^{\frac{1}{z^4}}$. Use that Laurent series with the Laurent series theorem to compute $\displaystyle\int_C e^{\frac{1}{z^4}} \mathrm{d}z$ where $C$ is the unit circle, oriented positively.
3. Find the three series for the function $f(z)=\dfrac{-1}{(z-2)(z-3)}$ in the relevant disks and annuli.
4. Find the residue at $z=0$ of...
(a) $\dfrac{1}{z+z^2}$
(b) $z \cos \left( \dfrac{1}{z} \right)$ (recall: $\cos(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k z^{2k}}{(2k)!}$ )
(c) $\dfrac{z-\sin(z)}{z}$ (recall: $\sin(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k z^{2k+1}}{(2k+1)!}$)
