Plemelj formulas

Let ψ(ζ) be a density function of a complex variable satisfying the Hölder condition (the Lipschitz conditionMathworldPlanetmath of order α)11A function f(ζ) satisfies the Hölder condition on a smooth curve C if for every ζ1,ζ2C |f(ζ2)-f(ζ1)|M|ζ2-ζ1|α, M>0, 0<α1. It is clear that the Hölder condition is a weaker restrictionPlanetmathPlanetmath than a bounded derivativePlanetmathPlanetmath for f(ζ). on a smooth closed contour C in the integral

Ψ(z)=12πiCψ(ζ)ζ-z𝑑ζ, (1)

then the limits Ψ+(t) and Ψ-(t) as z approaches an arbitrary point t on C from the interior and the exterior of C, respectively, are

{Ψ+(t)12ψ(t)+12πiCψ(ζ)ζ-t𝑑ζ,Ψ-(t)-12ψ(t)+12πiCψ(ζ)ζ-t𝑑ζ. (2)

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


  • 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