Past exam of the metric spaces and complex analysis course of the University of Oxford / 2023 / 4 / c / Solution

Use the local factorization for a pole of order $n$:Shrink the disc so that the holomorphic function $h$ has no zeros there. Differentiating and dividing gives the logarithmic derivativeThe second term is holomorphic across $a$. Since $n\ge 1$, the first term cannot cancel. Hence $f^{\prime }/f$ has a simple pole, and the coefficient of $(z-a{)}^{-1}$ in its Laurent series isThis is the pole case of the integer residue of a logarithmic derivative.