The argument principle, Rouché’s theorem, and Hurwitz’s theorem count the zeros of a holomorphic or meromorphic function in a region.
How many zeros in this region?, the hypotheses under which each theorem applies.
The argument principle, zeros minus poles as an integral of the logarithmic derivative.
Rouché's theorem, equal counts for functions related by a strict inequality on the boundary.
Hurwitz's theorem, zeros of a locally uniform limit.