Defining a new variable \( u=t-x \), we can rewrite the principal value integral as $$ \begin{equation} I_{\Delta}(x)={\cal P}\int_{-\Delta}^{+\Delta}du\frac{f(u+x)}{u}. \tag{19} \end{equation} $$ One possibility is to Taylor expand \( f(u+x) \) around \( u=0 \), and compute derivatives to a certain order as we did for the Trapezoidal rule or Simpson's rule. Since all terms with even powers of \( u \) in the Taylor expansion dissapear, we have that $$ \begin{equation*} I_{\Delta}(x)\approx \sum_{n=0}^{N_{max}}f^{(2n+1)}(x) \frac{\Delta^{2n+1}}{(2n+1)(2n+1)!}. \end{equation*} $$