How to integrate $e^z/z^2$?

Question

This may be a very basic question.

How to compute the integral $ \int_\gamma \frac{e^z}{z^2} \, dz$, where $\gamma$ is the unit circle? I did it with Cauchy's integral formula for $\int_\gamma \frac{e^z}{z}\,dz$, but how about this one?

Answer 1

The function $f(z) = e^z$ is analytic inside and on $\gamma$ (in fact, it's entire) and $0$ lies inside $\gamma$. So by Cauchy's differentiation formula,

$$\int_\gamma \frac{e^z}{z^2}\, dz = \int_\gamma \frac{f(z)}{z^2}\, dz = 2\pi i f'(0) = 2\pi i.$$