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.

Since all the poles of the integrand are enclosed by the contour, we have 
\begin{eqnarray}
\int_{|z|=3}\frac{e^{zt}}{z^2+4}dz=2\pi i\text{Res}_{z=0}\frac{1}{z^2}\frac{e^{\frac{t}{z}}}{\frac{1}{z^2}+4}=2\pi i\text{Res}_{z=0}\frac{e^{\frac{t}{z}}}{1+4z^2}
\end{eqnarray}
Note that 
$\displaystyle \frac{e^{\frac{t}{z}}}{1+4z^2}=\left(\sum_{n=0}^\infty (-1)^n4^nz^{2n}\right)\left(\sum_{n=0}^\infty \frac{t^n}{n! z^n}\right)$. So $\displaystyle\text{Res}_{z=0}=\sum_{n=0}^\infty \frac{(-1)^n4^nt^{2n+1}}{(2n+1)!}=\frac{1}{2}\sin 2t$
