
SIGMA 7 (2011), 050, 16 pages arXiv:1101.3756
http://dx.doi.org/10.3842/SIGMA.2011.050
Contribution to the Special Issue “Symmetry, Separation, Superintegrability and Special Functions (S^{4})”
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
oddhalfinteger degrees, for general complexorders, and derivatives
with respect to the order are evaluated at integerorders,
for general complexdegrees. 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)
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