Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 7D (Further Complex Methods)

For singularities $a<c_1<\cdots <c_N<b$, the Cauchy principal value iswhere the assumed tail integrals make the two unbounded pieces meaningful. Equivalently, remove a symmetric interval of radius $\epsilon$ about each $c_j$ and take the limit of the remaining integral.

Using the convention adopted here, the Hilbert transform isThe convention with denominator $x-y$ differs by a minus sign.

For $f(x)=1/(x^2+1)$, contour integration or partial fractions givesunder the convention in part (b). Therefore