Let $$f(z)=\frac{e^{az}}{(a+z)^2} , \,\,\ a,z \in \mathbb{C}$$
Compute the integral $\int_\gamma f(z) \,dz$ where $\gamma$ is the unit circle centered at the origin and $|a| \neq 1$.
I don't really know where to start with this question. My first check is usually to see whether $f$ is holomorhpic on some simply connected region containing $\gamma$ in order to see if the integral is just zero. However, how can I even do this here without knowing what $a$ is?
If $|a|<1$, then we have
From Cauchy's Integral Formula we have
$$\left.\left(\frac{de^{az}}{dz}\right)\right|_{z=-a}=\frac{1}{2\pi i}\oint_\gamma \frac{e^{az}}{(z-(-a))^2}\,dz$$
from which we see that
$$\oint_\gamma \frac{e^{az}}{(z+a)^2}\,dz=2\pi i ae^{-a^2}$$