Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Error bounds for a uniform asymptotic expansion of the Legendre function $ P_n^{-m}({\rm cosh}\ z)$

Authors: P. N. Shivakumar and R. Wong
Journal: Quart. Appl. Math. 46 (1988), 473-488
MSC: Primary 33A45; Secondary 41A60
DOI: https://doi.org/10.1090/qam/963583
MathSciNet review: 963583
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Abstract: For fixed $ m$ with $ m + \frac{1}{2} > 0$, an asymptotic expansion for large $ n$ is derived for the Legendre function $ P_n^{ - m}\left( {\cosh z} \right)$,which is uniformly valid for $ z$ in the unbounded interval $ \left[ {0, \infty } \right)$. Our method is based on an integral representation of this function. The coefficients in the expansion satisfy a recurrence relation. Simple computable bounds are also constructed for the error terms associated with the expansion.

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DOI: https://doi.org/10.1090/qam/963583
Article copyright: © Copyright 1988 American Mathematical Society

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