MAT334--2020S > Quiz 5

QUIZ5 TUT0101

(1/1)

Huyi Xiong:
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}

Navigation

[0] Message Index