So-called principal value (PV) integrals are often employed in physics, from Green's functions for scattering to dispersion relations. Dispersion relations are often related to measurable quantities and provide important consistency checks in atomic, nuclear and particle physics. A PV integral is defined as $$ \begin{equation*} I(x)={\cal P}\int_a^bdt\frac{f(t)}{t-x}=\lim_{\epsilon\rightarrow 0^+} \left[\int_a^{x-\epsilon}dt\frac{f(t)}{t-x}+\int_{x+\epsilon}^bdt\frac{f(t)}{t-x}\right], \end{equation*} $$ and arises in applications of Cauchy's residue theorem when the pole \( x \) lies on the real axis within the interval of integration \( [a,b] \). Here \( {\cal P} \) stands for the principal value. An important assumption is that the function \( f(t) \) is continuous on the interval of integration.