Mathematics – Classical Analysis and ODEs
Scientific paper
2008-03-27
Mathematics
Classical Analysis and ODEs
LaTeX, 22 pages
Scientific paper
The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, $\partial P_{n}^{\mu}(z)/\partial\mu$, is studied. After deriving and investigating general formulas for $\mu$ arbitrary complex, a detailed discussion of $[\partial P_{n}^{\mu}(z)/\partial\mu]_{\mu=\pm m}$, where $m$ is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{\pm m}(z)$. In particular, we arrive at formulas which generalize to the case of $Q_{n}^{\pm m}(z)$ ($0\leqslant m\leqslant n$) the well-known Christoffel's representation of the Legendre function of the second kind, $Q_{n}(z)$. The derivatives $[\partial^{2} P_{n}^{\mu}(z)/\partial\mu^{2}]_{\mu=m}$, $[\partial Q_{n}^{\mu}(z)/\partial\mu]_{\mu=m}$ and $[\partial Q_{-n-1}^{\mu}(z)/\partial\mu]_{\mu=m}$, all with $m>n$, are also evaluated.
Szmytkowski Radoslaw
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