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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

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