On Parameter Differentiation for Integral Representations of Associated Legendre Functions

Details On Parameter Differentiation for Integral Representations of Associated Legendre Functions On Parameter Differentiation for Integral Representations of Associated Legendre Functions

2011-01-19

arxiv.org/abs/1101.3756v2

SIGMA 7 (2011), 050, 16 pages

Mathematics

Classical Analysis and ODEs

Scientific paper

10.3842/SIGMA.2011.050

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 $f(z)=z/(\sqrt{z+1}\sqrt{z-1})$.

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