Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T08:59:02.280Z Has data issue: false hasContentIssue false
  • English
  • Français

The Stokes phenomenon and certain nth-order differential equations I. Preliminary investigation of the equations

Published online by Cambridge University Press:  24 October 2008

J. Heading
J. Heading
Affiliation:
West Ham College of TechnologyLondon, E. 15
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction. The object of this investigation is to obtain by means of contour integrals exact solutions of certain nth-order differential equations, together with their n independent power-series solutions, their asymptotic solutions and the relationship between these two types of solution. The Stokes phenomenon associated with the changing constants in these asymptotic solutions will then be investigated by various methods. Paper I is concerned with the properties of the various contour-integral solutions of the equations under consideration. The manner in which the arbitrary constants in the general asymptotic solution must change as the argument of the independent variable z varies is dealt with in paper II. This phenomenon is considered in a way that has proved profitable for the approximate solution of more general linear differential equations of the nth order that are approximately identical with those considered here hi certain regions of the complex z–plane. In later publications, these approximations will be described together with their application to explicit physical problems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 2 , April 1957 , pp. 399 - 418
Copyright
Copyright © Cambridge Philosophical Society 1957

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)Barnes, E. W.Proc. Lond. Math. Soc. (2) 5 (1907), 59.CrossRefGoogle Scholar
(2)Bremmer, H.Phillips Res. Rep. 4 (1949), 189.Google Scholar
(3)Bremmer, H.Terrestrial radio waves (Elsevier, 1949).Google Scholar
(4)Försterling, K.Ann. Phys., Lpz., (6) 8 (1950), 129.CrossRefGoogle Scholar
(5)Gans, R.Ann. Phys., Lpz., (4) 47 (1915), 709.CrossRefGoogle Scholar
(6)Tables of the modified Hankel functions of order one-third (Harvard, 1945).Google Scholar
(7)Heading, J. and Whipple, R. T. P.Phil. Trans. A, 244 (1952), 469.Google Scholar
(8)Jeffreys, H.Proc. Lond. Math. Soc. (2) 23 (1923), 428.Google Scholar
(9)Jeffreys, H. and Jeffreys, B.Methods of mathematical physics, 3rd ed. (Cambridge, 1956).Google Scholar
(10)Miller, J. C. P.British Association mathematical tables, Part-volume B. The Airy integral (London, 1946).Google Scholar
(11)Molins, H.Mém. Acad. Sci. Toulouse, (7) 8 (1876), 167.Google Scholar
(12)Stokes, G. G.Proc. Camb. Phil. Soc. 6 (1889), 362.Google Scholar
(13)Watson, G. N.A treatise on the theory of Bessel functions, 2nd ed. (Cambridge, 1944).Google Scholar
(14)Whittaker, E. T. and Watson, G. N.A course of modern analysis, 4th ed. (Cambridge, 1927).Google Scholar
(15)Woodward, P. M. and Woodward, A. M.Phil. Mag. (7), 37 (1946), 236.CrossRefGoogle Scholar