Book Review: Tables of associated Legendre functions

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On Parameter Differentiation for Integral Representations of Associated Legendre Functions

For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for associated Legendre functions of the first and second kind with respect to the degree are evaluated at odd-halfinteger degrees, for general complex-orders, ...

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Hilbert scales and Sobolev spaces defined by associated Legendre functions

In this paper we study the Hilbert scales defined by the associated Legendre functions for arbitrary integer values of the parameter. This problem is equivalent to study the left–definite spectral theory associated to the modified Legendre equation. We give several characterizations of the spaces as weighted Sobolev spaces and prove identities among the spaces corresponding to lower regularity ...

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Associated Legendre Functions and Dipole Transition Matrix Elements

The states also are labeled by some other principal quantum number n′ and n, that is needed to determine the radial wave functions Rnl(r). The volume element for these integrals is dr = rdrdΩ, where dΩ = −dφ d(cos θ) = dφ sin θ dθ, and the angular integrals are over the surface of a unit sphere. In these notes I am only interested in the angular integrals. The radial unit vector in Cartesian co...

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Book Review

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Evaluation of Associated Legendre Functions off the Cut and Parabolic Cylinder Functions∗

We review a set of algorithms to evaluate associated Legendre functions off the cut; in particular, we consider prolate spheroidal, oblate spheroidal and toroidal harmonics. A similar scheme can be applied to other families of special functions like Bessel and parabolic cylinder functions; we will describe the corresponding algorithm for the evaluation of parabolic cylinder functions.

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ژورنال

عنوان ژورنال: Bulletin of the American Mathematical Society

سال: 1946

ISSN: 0002-9904

DOI: 10.1090/s0002-9904-1946-08531-6