14 Legendre and Related Functions Real Arguments 14.10 Recurrence Relations and Derivatives 14.12 Integral Representations

§14.11 Derivatives with Respect to Degree or Order

ⓘ

Keywords:: Ferrers functions, associated Legendre functions, derivatives, with respect to degree or order, with respect to degree or order
Notes:: (14.11.1) may be derived from (14.3.1). (14.11.2) may be derived from (14.9.10) and (14.11.1). (14.11.4) may be derived from (14.3.1) and the hypergeometric expansion for $\mathsf{Q}_{\nu}\left(x\right)$ (Hobson (1931, §132)). (14.11.5) may be derived from (14.9.8), (14.9.10), and (14.11.4).
Referenced by:: Erratum (V1.0.5) for References, Erratum (V1.0.7) for References
Permalink:: http://dlmf.nist.gov/14.11
Addition (effective with 1.0.7):: The references to Cohl (2010, 2011) have been added at the end of this section.
Addition (effective with 1.0.5):: The references to Szmytkowski (2006, 2009, 2011, 2012) have been added at the end of this section.
See also:: Annotations for Ch.14

14.11.1		$\frac{\partial}{\partial\nu}\mathsf{P}^{\mu}_{\nu}\left(x\right)=\pi\cot\left(% \nu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x\right)-\frac{1}{\pi}\mathsf{A}_{\nu% }^{\mu}(x),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $\mathsf{A}_{\nu}^{\mu}(x)$ Referenced by: §14.11, §14.11 Permalink: http://dlmf.nist.gov/14.11.E1 Encodings: TeX, pMML, png See also: Annotations for §14.11 and Ch.14

14.11.2		$\frac{\partial}{\partial\nu}\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\tfrac{1}{2}% \pi^{2}\mathsf{P}^{\mu}_{\nu}\left(x\right)+\frac{\pi\sin\left(\mu\pi\right)}{% \sin\left(\nu\pi\right)\sin\left((\nu+\mu)\pi\right)}\mathsf{Q}^{\mu}_{\nu}% \left(x\right)-\tfrac{1}{2}\cot\left((\nu+\mu)\pi\right)\mathsf{A}_{\nu}^{\mu}% (x)+\tfrac{1}{2}\csc\left((\nu+\mu)\pi\right)\mathsf{A}_{\nu}^{\mu}(-x),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cot\NVar{z}$ : cotangent function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $\mathsf{A}_{\nu}^{\mu}(x)$ Referenced by: §14.11 Permalink: http://dlmf.nist.gov/14.11.E2 Encodings: TeX, pMML, png See also: Annotations for §14.11 and Ch.14

where

14.11.3		$\mathsf{A}_{\nu}^{\mu}(x)=\sin\left(\nu\pi\right)\left(\frac{1+x}{1-x}\right)^% {\mu/2}\\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}-\frac{1}{2}x\right)^{k}% \Gamma\left(k-\nu\right)\Gamma\left(k+\nu+1\right)}{k!\Gamma\left(k-\mu+1% \right)}\\left(\psi\left(k+\nu+1\right)-\psi\left(k-\nu\right)\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $\mathsf{A}_{\nu}^{\mu}(x)$ Referenced by: §14.11 Permalink: http://dlmf.nist.gov/14.11.E3 Encodings: TeX, pMML, png See also: Annotations for §14.11 and Ch.14

14.11.4	$\displaystyle\left.\frac{\partial}{\partial\mu}\mathsf{P}^{\mu}_{\nu}\left(x% \right)\right\|_{\mu=0}$	$\displaystyle=\left(\psi\left(-\nu\right)-\pi\cot\left(\nu\pi\right)\right)% \mathsf{P}_{\nu}\left(x\right)+\mathsf{Q}_{\nu}\left(x\right),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.11, §14.11 Permalink: http://dlmf.nist.gov/14.11.E4 Encodings: TeX, pMML, png See also: Annotations for §14.11 and Ch.14
14.11.5	$\displaystyle\left.\frac{\partial}{\partial\mu}\mathsf{Q}^{\mu}_{\nu}\left(x% \right)\right\|_{\mu=0}$	$\displaystyle=-\tfrac{1}{4}\pi^{2}\mathsf{P}_{\nu}\left(x\right)+\left(\psi% \left(-\nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{Q}_{\nu}\left(x% \right).$
ⓘ Symbols: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.11 Permalink: http://dlmf.nist.gov/14.11.E5 Encodings: TeX, pMML, png See also: Annotations for §14.11 and Ch.14

(14.11.1) holds if $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ is replaced by $P^{\mu}_{\nu}\left(x\right)$ , provided that the factor $(\ifrac{(1+x)}{(1-x)})^{\mu/2}$ in (14.11.3) is replaced by $(\ifrac{(x+1)}{(x-1)})^{\mu/2}$ . (14.11.4) holds if $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ , $\mathsf{P}_{\nu}\left(x\right)$ , and $\mathsf{Q}_{\nu}\left(x\right)$ are replaced by $P^{\mu}_{\nu}\left(x\right)$ , $P_{\nu}\left(x\right)$ , and $Q_{\nu}\left(x\right)$ , respectively.

See also Szmytkowski (2006, 2009, 2011, 2012), Cohl (2010, 2011) and Magnus et al. (1966, pp. 177–178).

14.10 Recurrence Relations and Derivatives 14.12 Integral Representations

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§14.11 Derivatives with Respect to Degree or Order

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