Analogue Sochocky formulae for integral operators with additional logarithmic singularity

V. V. Kashevski

Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 4-10.

Résumé

In this paper we prove the limit formulas for singular integrals of the form $\Phi_{n}(z)=\int\limits_{0}^{1}\frac{\varphi(\tau)ln^{n}(\tau-z)}{\tau-z}\mathbb{d}\tau, n=1,2,\dots$ Limit values of such integrals are expressed in terms of singular integral operators $\Psi_{n}(t)=\int\limits_{0}^{1}\frac{\varphi(\tau)ln^{n}|\tau-t|}{\tau-t}\mathbb{d}\tau, n=1,2,\dots, t\in (0,1),$ and also integral operators $\int\limits_{0}^{t}\frac{\varphi(\tau)-\varphi(t)}{\tau-t}ln^{k}|t-\tau|\mathbb{d}\tau.$ As an application of these formulas derived additive representation for singular integrals $\int\limits_{0}^{1}\frac{\varphi(\tau)ln|\tau-t|}{\tau-t}\mathbb{d}\tau.$ The formulas are derived in the article can be used for research and operators singular integral equations solutions.

Détail

Keywords: integral operators; singular integrals.

Bibliographie

[1] F. D. Gakhov, “Kraevye zadachi”, Nauka, 1977 | MR

[2] N. I. Muskhelishvili, “Singulyarnye integralnye uravneniya”, Nauka, 1968