Using Cauchys integral formula, find the value of $$\oint_{|z-1|=2} \frac{e^{-z}}{z^2}$$
I just want to make sure I am understanding this correctly. Applying cauchys integral formula we have $2i\pi(\frac{d}{dz}e^{-z})$ evaluated at $0$. We should just get $-2i\pi$ for the answer right? I just have doubt that I understand the formula correctly $($also, I understand more work must be done about the singularity at $0)$.
Isn't it a pole of order two? I would apply the residue theorem, as I don't see CIF applying. You should get $-2\pi i$, I think.
I guess, on second thought, you could be using the consequence of CIF sometimes known as Cauchy's differentiation formula. In which case I guess you are correct.