Im having a bit trouble getting started in complex integrals.
Im integrating along a circle the integral $\int \frac{e^z}{z^2 + 1}$ where
a) center O radius 1/2 b) center i radius 1 c) center i radius 3
I have set $\gamma(t) = \frac{1}{2}e^{it}$ and $\gamma' = \frac{i}{2}e^{it}$ and $\gamma = z$ for part a) but how do I get around to doing the integral?
If you know the residue theorem, then these integrals may be done easily. The integrand has poles at $z=i$ and $z=-i$. In part a), neither pole is inside the contour, so the integral is zero. For part b), only the pole at $z=i$ is inside, so the integral is $i 2 \pi$ times the residue at $z=i$, or
$$i 2 \pi \frac{e^i}{2 i} = \pi (\cos{1} + i \sin{1})$$
In part c), both poles are inside the contour, so you will need to sum the residues at both poles. I get that the integral is then $i 2 \pi \sin{1}$.