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The asymptotic solution of differential equations with a turning point and singularities†

Published online by Cambridge University Press:  24 October 2008

R. C. Thorne
R. C. Thorne
Affiliation:
California Institute of TechnologyPasadena 4, California
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Abstract

Asymptotic solutions of the differential equation

for large positive values of u, are examined; z is a complex variable in a domain Dz in which P1(z) and z2q(z) are regular and p1(z) does not vanish. In this paper it is shown that there exist Airy-type expansions of the solutions of this equation which are valid uniformly with respect to z in a domain in which z = 0 and z = z0 are interior points. If Dz is unbounded and the equation has a regular singularity at infinity, Airy-type expansions exist which are valid at z = 0, z = z0 and z = δ. If p(z) = constant + O (│z│-1) as │ z │ → ∞ in Dz, similar expansions also exist. The results given here are new.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 2 , April 1957 , pp. 382 - 398
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

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