Chapter 14.11, Problem 11MP

Find the Laurent series of $f(z)=e^z/(1-z)$ for $\left|z\right|<1$ and $\left|z\right|>1.$ Hints: For $\left|z\right|<1,$ multiply two power series; you should find $f(z)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{a}_n}{z}^n$ with $a_n={\displaystyle \underset{k=0}{\overset{n}{\sum }}1}/k!$.

For $\left|z\right|>1,$ use (4.3) where $C$ is a circle $\left|z\right|=a$ with $a>1$ Evaluate the integrals by finding residues at 1 and 0. You should find $f(z)=$ ${\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{a}_n}{z}^n+{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{b}_n}{z}^{-n}$ where all $b_n=-e$ and $a_n=-e+{\displaystyle \underset{k=0}{\overset{n}{\sum }}1}/k!.$