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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974 MR 0435697
  • [2] M. Pourahmadi, Taylor expansion of $ \exp \left( {\sum\nolimits_{k = 0}^\infty {{a_k}{z^k}} } \right)$ and some applications, Amer. Math. Monthly 91, 303-308 (1984) MR 740245
  • [3] G. Szegö, Über einige asymptotische Entwicklungen der Legendreschen Funktionen, Proc. Lond. Math. Soc., (2), 36, 427-450 (1932) MR 1575968
  • [4] F. Ursell, Integrals with a large parameter: Legendre functions of large degree and fixed order, Math. Proc. Camb. Phil. Soc. 95, 367-380 (1984) MR 735379
  • [5] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944 MR 0010746

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 33A45, 41A60

Retrieve articles in all journals with MSC: 33A45, 41A60


Additional Information

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

American Mathematical Society