Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 4 / d / Solution

For a double pole at $z_0\in \{i,-i\}$, differentiate the remaining holomorphic factor:Since $z_0^2=-1$ and $1+z_0+z_0^2=z_0$, this simplifies toOn the selected branch $\sqrt{i}=e^{i\pi /4}$ and $\sqrt{-i}=e^{3i\pi /4}$, givingThe residue theorem therefore givesThe equality $C=2I$ uses the limiting contour integrals established in (c).