Cauchy integral formula Can someone please help me answer this question as I cannot seem to get to the answer.
Please note that the Cauchy integral formula must be used in order to solve it.
Many thanks in advance!
\begin{equation*}
\int_{|z|=3}\frac{e^{zt}}{z^2+4}=\pi i\sin(2t).
\end{equation*}
Also $|z| = 3$ is given the counterclockwise direction.

Hint. The denominator $\displaystyle z^2+4=(z-2i)(z+2i)$ of the function
$$
z \longmapsto \frac{e^{zt}}{z^2+4}
$$ gives two poles inside $|z|=3$ and by the Cauchy integral formula we have
$$
\int_{|z|=3}\frac{e^{zt}}{z^2+4}dz=2i\pi\left({\rm{Res}}_{z=-2i}f(z)+{\rm{Res}}_{z=2i}f(z)\right).
$$ You conclude using for example
$$
{\rm{Res}}_{z=-2i}f(z)=\lim_{z\to -2i}\left((z+2i)\times\frac{e^{zt}}{z^2+4}\right)=\ldots
$$
