Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-17T00:38:47.304Z Has data issue: false hasContentIssue false
  • English
  • Français

Some integral equations involving finite parts of divergent integrals

Published online by Cambridge University Press:  18 May 2009

A. Erdélyi
A. Erdélyi
Affiliation:
Mathematical Institute of the University, 20Chambers Street, Edinburgh, 1.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In recent years, a number of special integral equations of the first kind was discussed by several authors (see [l]–[4], [6], [7], [9]–[18]). The kernels of these integral equations are special functions of the hypergeometric family, and it was necessary to restrict the parameters appearing in these functions to secure convergence of the integrals. If these restrictions are removed, the integral fails to converge but it may possess a finite part (in Hadamard's sense), and the question arises whether the methods used in the restricted case will alsoapply in the new situation. Indeed, one could pose the moregeneral problem of Volterra integral equations involving finite parts of divergent integrals [19]

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 8 , Issue 1 , January 1967 , pp. 50 - 54
Copyright
Copyright © Glasgow Mathematical Journal Trust 1967

References

1.Buschman, R. G., An inversion integral for a Legendre transformation, Amer. Math. Monthly 69 (1962), 288–289.CrossRefGoogle Scholar
2.Buschman, R. G., An inversion integral, Proc. Amer. Math. Soc. 13 (1962), 675–677.CrossRefGoogle Scholar
3.Buschman, R. G., An inversion integral for a general Legendretransformation, SIAM Rev. 5 (1963), 232–233.CrossRefGoogle Scholar
4.Buschman, R. G., Convolution equations with generalized Laguerre polynomial kernels, SIAM Rev. 6 (1964), 166–167.CrossRefGoogle Scholar
5.Butzer, P. L., Singular integral equations of Volterra type and the finite part of divergent integrals, Arch. Rational Mech. Anal. 3 (1959), 194–205.CrossRefGoogle Scholar
6.Erdelyi, A., An integral equation involving Legendre's polynomial, Amer. Math. Monthly 70 (1963), 651–652.CrossRefGoogle Scholar
7.Erdélyi, A., An integral equation involving Legendre functions, J. Soc. Indust. Appl. Math. 12 (1964), 15–30.CrossRefGoogle Scholar
8.Gelfand, I. M. and Shilov, G. E., Generalized functions, vol. I (Academic Press, New York and London, 1964).Google Scholar
9.Higgins, T. P., An inversion integral for a Gegenbauer transformation, J. Soc. Indust. Appl. Math. 11 (1963), 886–893.CrossRefGoogle Scholar
10.Higgins, T. P., A hypergeometric function transform, J. Soc. Indust. Appl. Math. 12 (1964), 601–612.CrossRefGoogle Scholar
11.Li, Ta, A new class of integral transforms. Proc. Amer. Math. Soc. 11 (1960), 290–298.CrossRefGoogle Scholar
12.Love, E. R., Some integral equations involving hypergeometric functions; to appear in Proc. Edinburgh Math. Soc.Google Scholar
13.Srivastava, K. N., On some integral transforms, Math. Japon. 6 (1961–1962), 65–72.Google Scholar
14.Srivastava, K. N., A class of integral equations involving ultraspherical polynomials as kernel, Proc. Amer. Math. Soc. 14 (1963), 932–940.CrossRefGoogle Scholar
15.Srivastava, K. N., Inversion integrals involving Jacobi's polynomials, Proc. Amer. Math. Soc. 15 (1964), 635–638.Google Scholar
16.Srivastava, K. N., On some integral transforms involving Jacobi functions, Ann. Polon. Math. 16 (1965), 195–199.CrossRefGoogle Scholar
17.Srivastava, K. N., On integral equations involving Whittaker's function, Proc. Glasgow Math. Assoc. 7 (1966), 125–127.CrossRefGoogle Scholar
18Wimp, Jet, Two integral transform pairs involving hypergeometric functions, Proc. Glasgow Math. Assoc. 7 (1965), 42–4.CrossRefGoogle Scholar
19.Wiener, Klaus, Lineare Integralgleichungen mit Hadamard-Integralen, Wiss. Z. Martin- Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe 11 (1962), 567–580.Google Scholar