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Plemelj formulas
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(Definition)
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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
$C$ in the integral
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(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
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(2) |
These are the Plemelj[1] formulas 2 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.
Footnotes
- 1
- A function $f(\zeta)$ satisfies the Hölder condition on a smooth curve $C$ if for every $\zeta_1,\zeta_2\in C$ $|f(\zeta_2)-f(\zeta_1)|\leq M|\zeta_2-\zeta_1|^\alpha$ , $M>0$ , $0<\alpha\leq 1$ . It is clear that the Hölder condition is a weaker restriction than a bounded derivative for $f(\zeta)$ .
- 2
- cf.[2], where restrictions that Plemelj made, were relaxed.
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"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
There is 1 reference to this entry.
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 2131 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) |
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Pending Errata and Addenda
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