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Asymptotic formulae for solutions of linear second-order differential equations with a large parameter

Published online by Cambridge University Press:  09 April 2009

R. C. Thorne
R. C. Thorne
Affiliation:
The University of Sydney
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Abstract

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Uniform asymptotic formulae are obtained for solutions of the differential equation for large positive values of the parameter u. Here p is a positive integer, θ an arbitrary parameter and z a complex variable whose domain of variation may be unbounded. The function ƒ (u, θ, z) is a regular function of ζ having an asymptotic expansion of the form for large u.

The results obtained include and extend those of earlier writers which are applicable to this equation.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 4 , December 1960 , pp. 439 - 464
Copyright
Copyright © Australian Mathematical Society 1960

References

[1] Cherry, T. M., Trans. Amer. Math. Soc. 68 (1950) 224–257.CrossRefGoogle Scholar
[2] Erdélyi, A., Asymptotic expansions. New York: Dover (1956).Google Scholar
[3] Langer, R. E., Trans. Amer. Math. Soc. 33 (1931) 23–64.CrossRefGoogle Scholar
[4] Langer, R. E., Trans. Amer. Math. Soc. 37 (1935) 397–416.Google Scholar
[5] Langer, R. E., Trans. Amer. Math. Soc. 67 (1949) 461–490.CrossRefGoogle Scholar
[6] Liouville, J., J. Math. Pures appl. 2 (1837) 16–35.Google Scholar
[7] Olver, F. W. J., Phil. Trans. A, 247 (1954) 307–327.Google Scholar
[8] Olver, F. W. J., Phil. Trans. A, 247 (1954) 328–368.Google Scholar
[9] Olver, F. W. J., Phil. Trans. A, 249 (1956) 65–97.Google Scholar
[10] Olver, F. W. J., Phil. Trans. A, 250 (1958) 479–517.Google Scholar
[11] Thorne, R. C., Proc. Camb. Phil. Soc. 53 (1957) 382–398.CrossRefGoogle Scholar
[12] Thorne, R. C., Phil. Trans. A, 249 (1957) 585–596.Google Scholar