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 \( s=0 \).