I am going over the solutions to previous problems in order to prepare for a test. I am having a hard time understanding even basic applications of Cauchy's Integral Formula.
For example, I have tried to solve for the solution to the integral $$\int_c \frac{e^{-z^2}}{z^2}dz$$ where C is any positively-oriented simple closed contour surrounding the origin.
The solutions simply states that $$\frac{1}{2\pi i}\int_C \frac{e^{-z^2}}{z^2}dz = (e^{-z^2})'|_{z=0} = 0$$ by Cauchy.
How is Cauchy applied here? What are the steps taken to get from one side of the = to the other?
Hint: By Cauchy Integral Formula we have
$$f'(z_0) = \frac{1}{2\pi i} \int_C \frac{f(z)}{(z-z_0)^2} dz$$
Let $z_0 = 0$ and $f(z) = e^{-z^2}$.