Number of zeros equals number of poles The following is an old qualifying exam problem I cannot solve:
Let $f$ be a meromorphic function (quotient of two holomorphic functions) on an open neighborhood of the closed unit disk. Suppose that the imaginary part of $f$ does not have any zeros on the unit circle, then the number of zeros of $f$ in the unit disk equals the number of poles of $f$ in the unit disk.
It seems this problem begs Rouche's theorem, but I cannot seem to apply it correctly.

This is not quite Rouche though everything in the end boils down to the Cauchy integral formula.  To count zeros and poles you pass to the logarithmic derivative i.e. $\frac{f'}{f}$ and then integrate over the unit circle (which you are allowed to do since $f$ does not vanish there by hypothesis). Why you can take a continuous branch of the log is explained in the answer by @Sourav.
