×

Integrals with a large parameter: Legendre functions of large degree and fixed order. (English) Zbl 0539.33005

The Legendre functions \(P_ n^{-m}(\cosh z), Q_ n^{-m}(\cosh z)\) are considered for large values of n, m fixed, and z in a domain containing \(z=0\). It is known that the asymptotic expansion has modified Bessel functions \(I_ m(uz), K_ m(uz)\) as approximants \((u=n+1/2).\) The author gives a new approach for deriving the expansion (with as starting point integral representation). Moreover he gives a new method (based on the maximum-modulus theorem for analytic functions) for obtaining information on the remainders in the asymptotic expansions.
Reviewer: N.M.Temme

MSC:

33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
30E15 Asymptotic representations in the complex plane
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Watson, Bessel Functions (1944)
[2] Ursell, Proc. Cambridge Philos. Soc. 72 pp 49.65– (1972)
[3] Ursell, Proc. Cambridge Philos. Soc. 61 pp 113– (1965)
[4] A., Higher Transcendental Functions: Bateman Manuscript Project (1953) · Zbl 0052.29502
[5] DOI: 10.1112/plms/s2-36.1.427 · Zbl 0008.16301 · doi:10.1112/plms/s2-36.1.427
[6] Olver, Asymptotics and Special Functions (1974)
[7] Martin, Proc. Cambridge Philos. Soc. 76 pp 211– (1974)
[8] Titchmarsh, Theory of Functions (1939)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.