Toronto Math Forum
MAT334--2020S => MAT334--Tests and Quizzes => Quiz 5 => Topic started by: Huyi Xiong on March 04, 2020, 08:08:09 PM
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Find the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point.
\begin{align*}
\frac{e^z-1}{z^2}, z_0=0
\end{align*}
\begin{align}
\frac{e^z-1}{z^2} &= \frac{\sum_{n=0}^{\infty}{\frac{z^n}{n!}}-1}{z^2} \\
&= \frac{\sum_{n=1}^{\infty}{\frac{z^n}{n!}}}{z^2} \\
&= \sum_{n=1}^{\infty}{\frac{z^{n-2}}{n!}} \\
&= \frac{1}{z}+\frac{1}{2}+\frac{1}{6}z +...\\
residue = 1
\end{align}