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Conical functions with one or both parameters large

Published online by Cambridge University Press:  14 November 2011

T. M. Dunster
T. M. Dunster
Affiliation:
Department of Mathematics, San Diego State University, San Diego, CA 92182-0314, U.S.A.
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Synopsis

Uniform asymptotic expansions are derived for conical functions, Legendre functions of order µ and degree −½ + iτ, where µ and τ are non-negative real parameters. As τ → ∞, expansions are furnished for the conical functions which involve Bessel functions of order µ. These expansions are uniformly valid for 0 ≦ µ ≦ Aτ (A an arbitrary positive constant), and are also uniformly valid for Re (z) ≧ 0 in the complex argument case, and 0 ≦ z < ∞ in the real argument case. The case µ → ∞ is also considered, and expansions are furnished which are uniformly valid in the same z regions for 0 ≦ τ ≧ Bµ (B an arbitrary positive constant); in the cases where Re(z) ≧ 0 and 1 ≦ z < ∞, the expansions involve Bessel functions of purely imaginary order iτ, and in the case where 0 ≦ z < 1 the expansions involve elementary functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 3-4 , 1991 , pp. 311 - 327
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

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