We can then express a principal values integral as $$ \begin{equation} {\cal P}\int_{0}^{\infty} \frac{f(k)dk}{k^2-k_0^2} = \int_{0}^{\infty} \frac{(f(k)-f(k_0))dk}{k^2-k_0^2}, \tag{18} \end{equation} $$ where the right-hand side is no longer singular at \( k=k_0 \), it is proportional to the derivative \( df/dk \), and can be evaluated numerically as any other integral.