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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

##### 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
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