Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 1 / 13A / b / iii

In the open strip $-c<\mathrm{Im}\lambda <c$, neither pole of $g$ is present. The two evaluations therefore giveBoth summands are holomorphic on this strip. This is a half-plane decomposition by Cauchy integrals; the pole subtraction in $H$ is what makes its extension holomorphic even at the upper pole of $g$.