A quick question about complex integrals and Cauchy's integral formula

Question

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.

Post-Script (December 2018): I recognize that the "discontinuity" mentioned is in reality a singularity.

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.

Answer 1

Full disclosure: I was going back through some of the questions I asked over the past semester and noticed this one was left unanswered, at least in the "official answer" sense, i.e. no one gave me an answer to accept. Since this was basically an "am I right" sort of question, I'm just going to make this a community wiki answer and accept it for the sake of closing the question. (I chose community wiki because it was other people who verified my solution and I don't have much to add to the solution. Plus if others have discussion to offer, they should do so.)

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The thoughts posted in the OP are indeed correct, and follow directly from Cauchy's integral theorem/formula. The first contour yields an integral evaluating to $0$, the second to $2\pi i$.

Thanks to José Carlos Santos and Kavi Rama Murthy for their input on the post.