# Plemelj formulas

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

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

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

 $\displaystyle\left\{\begin{array}[]{ll}\Psi^{+}(t)\equiv\frac{1}{2}\psi(t)+% \frac{1}{2\pi i}\int_{C}\frac{\psi(\zeta)}{\zeta-t}d\zeta,\\ \Psi^{-}(t)\equiv-\frac{1}{2}\psi(t)+\frac{1}{2\pi i}\int_{C}\frac{\psi(\zeta)% }{\zeta-t}d\zeta.\end{array}\right.$ (2)

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.

## References

• 1 J. Plemelj, Monatshefte für Mathematik und Physik, vol. 19, pp. 205- 210, 1908.
• 2 N. I. Muskhelishvili, Singular Integral Equations, Groningen: Noordhoff (based on the second Russian edition published in 1946), 1953.
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