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SIGMA 7 (2011), 050, 16 pages arXiv:1101.3756
https://doi.org/10.3842/SIGMA.2011.050
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”
On Parameter Differentiation for Integral Representations of Associated Legendre Functions
Howard S. Cohl a, b
a) Applied and Computational Mathematics Division,
Information Technology Laboratory, National Institute of Standards and Technology,
Gaithersburg, Maryland, USA
b) Department of Mathematics, University of Auckland, 38 Princes Str., Auckland, New Zealand
Received January 19, 2011, in final form May 04, 2011; Published online May 24, 2011
Abstract
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: C\{−1,1}→C given by
f(z)=z/((z+1)1/2(z−1)1/2).
Key words:
Legendre functions; modified Bessel functions; derivatives.
pdf (496 kb)
tex (134 kb)
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