Error bounds for a uniform asymptotic expansion of the Legendre function $P_n^{-m}(\textrm {cosh}\ z)$
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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Article copyright:
© Copyright 1988
American Mathematical Society