On Parameter Differentiation for Integral Representations of Associated Legendre Functions

Howard S. Cohl

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

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-half-integer degrees, for general complex-orders, and derivatives with respect to the order are evaluated at integer-orders, for general complex-degrees. We also discuss the properties of the complex function $f:\mathbb C\setminus\{-1,1\}\to\mathbb C$ given by $f(z)=z/(\sqrt{z+1}\sqrt{z-1})$.

Détail

Keywords: Legendre functions; modified Bessel functions; derivatives.

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