Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 7A / Solution

Apply the residue theorem to the upper-half-plane semicircle, with a small semicircular indentation above the pole at $z=0$. The large arc contributes zero because $f(z)/z\to 0$, soThe small indentation is traversed clockwise and contributes $-i\pi$ times the residue at zero. The only pole strictly inside the indented contour is $z=ia$. ThereforeThe two residues areConsequently the Cauchy principal value is