I’ll spare the specifics for brevity’s sake, but in essence the problem I’m posed is finding

$ $ \int_C \frac{(z-1)^3 \cdot e^z \cdot cos(z)}{z}dz$ $

along two different closed loops $ C$ . Each is a rectangle, oriented clockwise. One of them encloses the discontinuity of this function (i.e. $ z=0$ ) and another doesn’t.

I’m mostly just wanting to double-check my approach to this since it’s explicitly specified that “this shouldn’t take much computation,” and want to double-check I’m on the right path.

My thoughts on the matter:

• For $ C$ being the closed loop not enclosing the discontinuity, the integral would be $ 0$ per Cauchy’s integral theorem.

• For $ C$ being the closed loop that encloses the discontinuity, the integral would be $ 2\pi i f(0)$ , from Cauchy’s integral formula, where $ f(z)$ is the numerator of the integrand (sans the $ dz$ of course), and “$ 0$ ” coming from being the point of discontinuity.

I have a rough intuition for why this might be – it’s fairly heuristic and informal though – so I just wanted to make sure I was on the right track.

Thanks in advance.