Find the Laurent series for $f(z)=\frac{e^z}{(z-i)^4}$ at $z=i$.

Question

Find the Laurent series for $f(z)=\frac{e^z}{(z-i)^4}$ at $z=i$.

What I was thinking of using $e^z=\sum_{i=0}^\infty \frac{x^i}{i!}$. But from there I am not sure what to do?

Answer 1

You have $$e^z = e^{z-i} e^i = e^i\sum_{n\ge 0} \frac{(z-i)^n}{n!}.$$ Now divide by $(z-i)^4$.