Research proof index

Let $f$ and its logarithmic derivative $f'/f$ be meromorphic on a set $R$, and assume the meromorphic order of $f$ is finite at every point of $R$. Then the poles of $f'/f$ in $R$ are exactly the points where the divisor of $f$ is nonzero, namely the zeros and poles of $f$.

An argument-principle identity on an axis-parallel rectangle. If $f$ and its logarithmic derivative $f'/f$ are meromorphic on the rectangle, every point has finite meromorphic order, and no zero or pole of $f$ lies on the boundary, then $\frac{1}{2\pi i}\oint_{\partial R}\frac{f'(z)}{f(z)}\,dz=\sum_{p\in R}\operatorname{ord}_p(f).$ Thus the normalized boundary integral counts zeros positively and poles negatively, with multiplicity.