Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T02:57:49.946Z Has data issue: false hasContentIssue false
  • English
  • Français

Integral equations for Lamé functions

Published online by Cambridge University Press:  20 January 2009

A. Erdélyi
A. Erdélyi
Affiliation:
Mathematical Institute, The University, 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 the theory of ordinary linear differential equations with three regular singularities and in the theory of their special and limiting cases, integral representations of the solutions are known to be very important. It seems that there is no corresponding simple integral representation of the solutions of ordinary linear differential equations with four regular singularities (Heun's equation) or of particular (e.g. Lamé's equation) or limiting (e.g. Mathieu's equation) cases of such equations. It has been suggested (Whittaker 1915 c) that the theorems corresponding in these latter cases to integral representations of the hypergeometric functions involve integral equations of the second kind. Such integral equations have been discovered for Mathieu functions (Whittaker 1912, cf. also Whittaker and Watson 1927 pp. 407–409 and 426) as well as for Lame functions (Whittaker 1915 a and b, cf. also Whittaker and Watson 1927 pp. 564–567) and polynomial or “quasi-algebraic” solutions of Heun's equation (Lambe and Ward 1934). Ince (1921–22) investigated general integral equations connected with periodic solutions of linear differential equations.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 7 , Issue 1 , October 1942 , pp. 3 - 15
Copyright
Copyright © Edinburgh Mathematical Society 1942

References

REFERENCES TO LITERATURE

Erdélyi, A., 1938, Die Funksche Infcegralgleichung der Kugelflächenfunktionen und ihre Uebertragung auf die Ueberkugel. Math. Annalen, 115, 356–365.Google Scholar
Funk, P., 1916, Beiträge zur Theorieder Kugelfunktionen. Math. Annalen, 77, 136–152.Google Scholar
Hecke, E., 1918, Ueber orthogonalinvariante Integralgleichungen. Math. Annalen, 78, 398–404.Google Scholar
Heine, E., 1878, Handbuch der Kugelfunktionen. 2nd edition, vol. i. (Berlin.)Google Scholar
Hobson, E. W., 1931, Spherical and ellipsoidal harmonics. (Cambridge.)Google Scholar
Ince, E. L., 1921–1922, On the connection between linear differential systems and integral equations. Proc. Royal Soc. Edinburgh, 42, 43–53.Google Scholar
Ince, E. L., 1940 a, The periodic Lamé functions. Proc. Royal Soc. Edinburgh, 60, 47–63.Google Scholar
Ince, E. L., 1940 b, Further investigations into the periodic Lamé functions. Proc. Royal Soc. Edinburgh, 60, 83–99.CrossRefGoogle Scholar
Lambe, C. G. and Ward, D. R., 1934, Some differential equations and associated integral equations. Quart. J. of Math. (Oxford), 5, 81–97.Google Scholar
Sharma, J. L., 1937, An integral equation satisfied by the Lamé functions. Journal de Math., 16, 199–203.Google Scholar
Whittaker, E. T., 1912, On the functions associated with the elliptic cylinder in harmonic analysis. Intern. Congress of Math. Cambridge. Vol. 1, 366–371.Google Scholar
Whittaker, E. T., 1915 a, On an integral equation whose solutions are the functions of Lamé. Proc. Royal Soc. Edinburgh, 35, 70–77.CrossRefGoogle Scholar
Whittaker, E. T., 1915 b, On Lamé's differential equation and ellipsoidal harmonics. Proc. London Math. Soc. (2), 14, 260–268.CrossRefGoogle Scholar
Whittaker, E. T., 1915 c, On a class of differential equations whose solutions satisfy integral equations. Proc. Edinburgh Math. Soc., 33, 14–23.Google Scholar
Whittaker, E. T. and Watson, G. N., 1927, Modern Analysis. (Cambridge.)Google Scholar