I am given a complex integral: $\oint_c \frac{e^{iz}}{z(z-\pi)}dz$ where C is the boundary of the annulus between circles of radius 1 and radius 4. So we have singularities at $\ z=0$ and $\ z=\pi$
I know that this integral is equivalent to $\frac{1}{\pi}[\oint_c \frac{e^{iz}}{z-\pi}dz - \oint_c \frac{e^{iz}}{z}dz$] using partial fraction decomposition.
I think I know how to solve the second integral $\oint_c \frac{e^{iz}}{z}dz$ by expanding out the numerator into infinite series sum and the singularity point in the denominator "cancels out" and the integral becomes much easier to solve using Cauchy-Goursat theorem.
However, I cannot for the life of me evaluate $\oint_c \frac{e^{iz}}{z-\pi}dz$. Even if I expand out the numerator $\ e^{iz}$ into
$\ 1+iz-\frac{z^2}{2}-\frac{iz^3}{6}+...$ ,
the singularity does not cancel out, so I'm not sure how to go about doing this. Any help would be greatly appreciated!