The Argument Principle concerns a loop integral, and is used to relate the number of zeros and poles in the interior of the region enclosed by the loop. Does this principle have an area-integral analog, by Stokes' Theorem, for instance?
Is the solution as simple as taking the derivative of $\frac{f'}{f}$ to make it into an area integral? Will the subtleties arising from complex analysis still hold? I could use any references on this matter. Apologies for these naive-sounding questions, but I am only getting started on understanding the complex nature of these theorems.
This question was motivated by an attempt to confirm some numerical winding number computations (using logarithms), against a discretized area integral.