Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T09:40:14.025Z Has data issue: false hasContentIssue false
  • English
  • Français

A Determinantal Expansion for a Class of Definite Integral

Published online by Cambridge University Press:  20 January 2009

L. R. Shenton
L. R. Shenton
Affiliation:
College of Technology, Manchester, 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 a previous paper (Shenton, 1953) we have given an expansion for integrals of the form This expansion may be expressed as a determinantal quotient or Schweinsian series. In the present paper we state more general terms under which the expansion holds and consider the case when the limits of integration are infinite and the weight function of the form In particular we giye expansions for the Psi function, and where C (x) is a positive polynomial.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 10 , Issue 2 , December 1954 , pp. 78 - 91
Copyright
Copyright © Edinburgh Mathematical Society 1954

References

REFERENCES

Burgess, J. (1895). “The definite integral with extended tables of values,” Trans. Roy. Soc. Edinburgh, XXXIX, 257–321.Google Scholar
Laplace, P. S. (1805). Traité de Mécanique Céleste, Vol. 4 (Paris).Google Scholar
Rogers, L. J. (1905). “On the representation of certain asymptotic series as convergent continued fractions.” Proc. London Math. Soc. (2), 4, 72–89 and 393–395.Google Scholar
Romanovsky, V. (1927). “Note on orthogonalising series of functions and interpolation,” Biometrika, XIX, 93–99.CrossRefGoogle Scholar
Shenton, L. R. (1953). A determinantal expansion for a class of definite integral Part 1, Proc. Edinburgh Math. Soc. (2), 9 (1953), 44–52.CrossRefGoogle Scholar
Stieltjes, T. J. (1890). Sur quelques intégrales définies et leur développement en fractions continues.Google Scholar
Stieltjes, T. J. (1894) Recherches sur les fractions continues. Oeuvres complétes (Groningen), p. 387 and pp. 520, 521.Google Scholar
Tchebycheff, P. L. (1859). Sur le développement des fonctions á une seule variable. Oeuvres Tome 1, pp. 501–508. (St Pétersbourg, 1899).Google Scholar
Wall, H. S. (1945). “Note on the expansion of a power series into a continued fraction,” Bull. American Math. Soc., 51, 97–105.CrossRefGoogle Scholar