Given a polynomial $p(z)$, and a rectangle with vertices $2+iM, -3+iM, -3-iM, 2-iM$ what is the value of $f(z) = p(z) \log(z)$ around the contour? Or equivalently the change in argument?
In particular I'm looking at how the change in argument grows as $M$ grows. The particular polynomial I'm interested in is only quadratic, however I'd like to know if it's possible to solve this problem more generally.
Suppose we define $\log(z) = \log(|z|) + \arg(z)i$ where the branch cut is the positive real axis. Note that the only part of the integral that matters is the part with $\log$. Why? Then integrating we get:
$$2\pi i\cdot [\Sigma_{n=0}^{N} a_{n}\cdot \frac{z^{n+1}}{n+1}|_{z=2}]$$ where $p(z) = \sum_{n=0}^{N} a_{n}z^n$.
Clearly, this generalises easily to arbitrary polynomials. Also note that for clockwise integration, there is a change of sign.