PlemeljFormulas 2006-06-24 01:41:25 2006-08-24 00:18:57 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $\psi(\zeta)$ be a density function of a complex variable satisfying the H\"older condition (the Lipschitz condition of order $\alpha$){\footnote{A function $f(\zeta)$ satisfies the H\"older 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 H\"older condition is a weaker restriction than a bounded derivative for $f(\zeta)$.}} on a smooth closed contour $C$ in the integral \begin{align} \Psi(z)=\frac{1}{2\pi i}\int_C\frac{\psi(\zeta)}{\zeta-z}d\zeta, \end{align} 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 \begin{align} \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. \end{align} These are the Plemelj\cite{cite:Plemelj} formulas {\footnote{cf.\cite{cite:Musk}, where restrictions that Plemelj made, were relaxed.}} and the improper integrals in (2) must be interpreted as Cauchy's principal values. \begin{thebibliography}{99} \bibitem{cite:Plemelj} J. Plemelj, {\em Monatshefte f\"ur Mathematik und Physik,} vol. 19, pp. 205- 210, 1908. \bibitem{cite:Musk} N. I. Muskhelishvili, {\em Singular Integral Equations,} Groningen: Noordhoff (based on the second Russian edition published in 1946), 1953. \end{thebibliography}