PlanetMath: Plemelj formulas

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Plemelj formulas

(Definition)

Let $\psi(\zeta)$ be a density function of a complex variable satisfying the Hölder condition (the Lipschitz condition of order $\alpha$ )¹ on a smooth closed contour 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

approaches an arbitrary point

from the interior and the exterior of

, respectively, are

$\displaystyle \left\{ \begin{array}{ll} \Psi^+(t) \equiv \frac{1}{2}\psi(t)+ \f... ...)+ \frac{1}{2\pi i}\int_C\frac{\psi(\zeta)}{\zeta-t}d\zeta. \end{array} \right.$

(2)

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

Bibliography

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.

Footnotes

...¹: A function $f(\zeta)$ satisfies the Hölder condition on a smooth curve if for every $\zeta_1,\zeta_2\in C$ $\vert f(\zeta_2)-f(\zeta_1)\vert\leq M\vert\zeta_2-\zeta_1\vert^\alpha$ , , $0<\alpha\leq 1$ . It is clear that the Hölder condition is a weaker restriction than a bounded derivative for $f(\zeta)$ .
...²: cf.[2], where restrictions that Plemelj made, were relaxed.

"Plemelj formulas" is owned by perucho.

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Cross-references: Cauchy principal values, improper integrals, exterior, interior, point, limits, integral, contour, closed, derivative, bounded, restriction, clear, curve, smooth, function, order, Lipschitz condition, variable, complex, density function
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This is version 2 of Plemelj formulas, born on 2006-06-24, modified 2006-08-24.
Object id is 8079, canonical name is PlemeljFormulas.
Accessed 1323 times total.

Classification:

AMS MSC:

30D10 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Representations of entire functions by series and integrals)

Pending Errata and Addenda

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