Let $\psi(\zeta)$ be a density function of a complex variable satisfying the Hölder condition (the Lipschitz condition of order $\alpha$)¹¹A function $f(\zeta)$ satisfies the Hölder condition on a smooth curve $C$ if for every $\zeta_{1},\zeta_{2}\in C$ $|f(\zeta_{2})-f(\zeta_{1})|\leq M|\zeta_{2}-\zeta_{1}|^{\alpha}$, $M>0$, $0<\alpha\leq 1$. It is clear that the Hölder condition is a weaker restriction than a bounded derivative for $f(\zeta)$. on a smooth closed contour $C$ in the integral

then the limits $\Psi^{+}(t)$ and $\Psi^{-}(t)$ as $z$ approaches an arbitrary point $t$ on $C$ from the interior and the exterior of $C$, respectively, are

These are the Plemelj[1] formulas ²²cf.[2], where restrictions that Plemelj made, were relaxed. and the improper integrals in (2) must be interpreted as Cauchy’s principal values.