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Transformations between second order linear differential equations and a reinterpretation of recurrence relations between Bessel functions with algebraic coefficients

Published online by Cambridge University Press:  14 February 2012

John Heading
John Heading
Affiliation:
Department of Applied Mathematics, University College of Wales, Aberystwyth
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Synopsis

A scheme devised by Chandrasekhar for investigating the transformations between various differential equations of the second order governing perturbations of the Schwarzschild black hole demands further investigation. The transformation between two differential equations in normal form is considered, and a wide survey of the properties of the transformation is given. It is shown how Chandrasekhar's equations fit into the scheme, after which some examples with particular properties are considered. A detailed investigation of Bessel's equation is undertaken using various devices, in particular by employing asymptotic methods for products of Bessel functions, and employing matrix methods for dealing with large numbers of matrix equations which necessitates an interesting method of solution, the results being reinterpretations of the standard recurrence relations for Bessel functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 79 , Issue 1-2 , 1977 , pp. 87 - 105
Copyright
Copyright © Royal Society of Edinburgh 1977

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References

References:

1Borůvka, O.. Linear differential transformations of the second order (London: English Univ. Press, 1971).Google Scholar
2Chandrasekhar, S.. On the equations governing the perturbations of the Schwarzchild black hole. Proc. Roy. Soc. London Ser. A 343 (1975), 289–298.Google Scholar
3Heading, J.. Refractive index profiles based on the hypergeometric equation and the confluent hypergeometric equation. Proc. Cambridge Philos. Soc. 61 (1965), 897–913.CrossRefGoogle Scholar
4Heading, J.. Resolution of the mystery behind Chandrasekhar's black hole transformations. J. Phys., A10, (1977), 885–897.Google Scholar
5Watson, N.. A treatise on the theory of Bessel functions (Cambridge: Univ. Press, 1944).Google Scholar