# Plemelj formulas

Let $\psi (\zeta )$ be a density function of a complex variable satisfying the Hölder condition (the Lipschitz condition^{} of order $\alpha $)^{1}^{1}A function $f(\zeta )$ satisfies the Hölder condition on a smooth curve $C$ if for every ${\zeta}_{1},{\zeta}_{2}\in C$ $|f({\zeta}_{2})-f({\zeta}_{1})|\le M{|{\zeta}_{2}-{\zeta}_{1}|}^{\alpha}$, $M>0$, $$. 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

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

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

These are the Plemelj[1] formulas^{} ^{2}^{2}cf.[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.

Title | Plemelj formulas |
---|---|

Canonical name | PlemeljFormulas |

Date of creation | 2013-03-22 16:02:02 |

Last modified on | 2013-03-22 16:02:02 |

Owner | perucho (2192) |

Last modified by | perucho (2192) |

Numerical id | 5 |

Author | perucho (2192) |

Entry type | Definition |

Classification | msc 30D10 |