Upper-half-plane indentation rule

Suppose a contour follows the real axis, closes in the upper half-plane, and avoids each simple real pole $a_j$ by a small semicircle above it. Each indentation is clockwise and tends toIf the large arc vanishes and there are no enclosed poles away from the real axis, the Cauchy principal value is therefore

By symmetry assume $a\ge b>0$ and integrate $f(z)=e^{iaz}\sin (bz)/z^2$ around an upper semicircle indented clockwise above zero. The residue at zero is $b$. The expansion $(e^{i(a+b)z}-e^{i(a-b)z})/(2i{z}^2)$ bounds the large arc by $\pi /R$, including when $a=b$. The upper-half-plane indentation rule therefore gives $\mathrm{PV}{\int }_{\mathbb{R}}f(x)dx=i\pi b$. Its imaginary part is even and ordinarily integrable, yielding the displayed value. At zero the sine-product integrand tends to $ab$; at infinity it is bounded by $1/x^2$. The principal value is needed for the auxiliary complex integral, not for the real sine-product integral.