The integration limits are now from $-1$ to $1$, as for the Legendre polynomials. The principal value in Eq. (20) is however rather tricky to evaluate numerically, mainly since computers have limited precision. We will here use a subtraction trick often used when dealing with singular integrals in numerical calculations. We introduce first the calculus relation $$\begin{equation*} \int_{-1}^{+1} \frac{ds}{s} =0. \end{equation*}$$ It means that the curve $1/(s)$ has equal and opposite areas on both sides of the singular point $s=0$.