Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T17:16:28.194Z Has data issue: false hasContentIssue false
  • English
  • Français

Two inequalities for parabolic cylinder functions

Published online by Cambridge University Press:  24 October 2008

F. W. J. Olver
F. W. J. Olver
Affiliation:
The National Physical Laboratory Teddington
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper upper bounds are established for the principal solution of the differential equation

and its derivative, for unrestricted values of the complex variable t and the complex parameter μ. The results may have little interest in their own right, but they are of great value in developing the asymptotic theory of linear second-order differential equations in a domain containing two turning points. Equation (1·1) is the simplest example of a differential equation of this type.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 4 , October 1961 , pp. 811 - 822
Copyright
Copyright © Cambridge Philosophical Society 1961

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Copson, E. T., Theory of functions of a complex variable (Oxford, 1935).Google Scholar
(2)Miller, J. C. P., Tables of Weber parabolic cylinder functions (London: H. M. Stationery Office, 1955).Google Scholar
(3)Olver, F. W. J., The asymptotic solution of linear differential equations of the second order for large values of a parameter. Phil. Trans. A, 247 (1954), 307–27.Google Scholar
(4)Olver, F. W. J., Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Stand. 63 B (1959), 131–69.CrossRefGoogle Scholar
(5)Watson, G. N., The harmonic functions associated with the parabolic cylinder. Proc. Lond. Math. Soc. 17 (1918), 116–48.Google Scholar
(6)Whittaker, E. T. and Watson, G. N., A course of modern analysis, 4th ed. (Cambridge, 1927).Google Scholar