Find the Laurent series expansion in powers of z

Question

Find the Laurent series expansion in powers of $z$ of

$$f(z)=\frac{e^{2z}} {z}$$

valid in the region $|z|>$0.

Any help appriciated. Thanks

Answer 1

The Maclaurin series of $e^{2z}$ is

$$ 1 + (2z) + \frac{(2z)^2}{2!} + \cdots = \sum_{k=0}^\infty \frac{2^k z^k}{k!} $$ so the Laurent series you're looking for is simply $1/z$ times this, i.e.: $$ \sum_{k=0}^\infty \frac{2^k z^{k-1}}{k!} = \sum_{k=-1}^\infty \frac{2^{k+1} z^{k}}{(k+1)!}. $$

The Maclaurin series of $e^{2z}$ converges on all of $\mathbb{C}$, since $e^{2z}$ is entire. Hence the Laurent series above converges for $z \neq 0$.