Plemelj formulas

Let $\psi (\zeta )$ be a density function of a complex variable satisfying the Hölder condition (the Lipschitz condition of order $\alpha$) on a smooth closed contour $C$ in the integral

$\mathrm{\Psi }(z)={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\int }_C}{\displaystyle \frac{\psi (\zeta )}{\zeta -z}}𝑑\zeta ,$(1)

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

$\{\begin{array}{cc}{\mathrm{\Psi }}^{+}(t)\equiv {\displaystyle \frac{1}{2}}\psi (t)+{\displaystyle \frac{1}{2\pi i}}{\displaystyle {\int }_C}{\displaystyle \frac{\psi (\zeta )}{\zeta -t}}𝑑\zeta ,\hfill & \\ {\mathrm{\Psi }}^{-}(t)\equiv -{\displaystyle \frac{1}{2}}\psi (t)+{\displaystyle \frac{1}{2\pi i}}{\displaystyle {\int }_C}{\displaystyle \frac{\psi (\zeta )}{\zeta -t}}𝑑\zeta .\hfill & \end{array}$(2)

These are the Plemelj[1] formulas and the improper integrals in (2) must be interpreted as Cauchy’s principal values.