14 Legendre and Related FunctionsReal Arguments14.4 Graphics14.6 Integer Order

§14.5 Special Values

ⓘ

Keywords:

Ferrers functions, special values

Permalink:

http://dlmf.nist.gov/14.5

See also:

Annotations for Ch.14

Contents

1. §14.5(i) $x=0$
2. §14.5(ii) $\mu =0$, $\nu =0,1$
3. §14.5(iii) $\mu =\pm \frac{1}{2}$
4. §14.5(iv) $\mu =-\nu$
5. §14.5(v) $\mu =0$, $\nu =\pm \frac{1}{2}$
6. §14.5(vi) Addendum to §14.5(ii): $\mu =0$, $\nu =2$

§14.5(i) $x=0$

ⓘ

Notes:

These results may be derived from (14.3.11)–(14.3.14) and also (14.10.4), with $𝖯$ replaced by $𝖰$.

Permalink:

http://dlmf.nist.gov/14.5.i

See also:

Annotations for §14.5 and Ch.14

14.5.1		$${𝖯}_{\nu }^{\mu }\left(0\right)=\frac{2^{\mu }{\pi }^{1/2}}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +1\right)\mathrm{\Gamma }\left(\frac{1}{2}-\frac{1}{2}\nu -\frac{1}{2}\mu \right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mu$: general order and $\nu$: general degree A&S Ref: 8.6.1 (modified) Referenced by: §10.59, §14.30(ii) Permalink: http://dlmf.nist.gov/14.5.E1 Encodings: TeX, pMML, png See also: Annotations for §14.5(i), §14.5 and Ch.14

14.5.2		$${\frac{d{𝖯}_{\nu }^{\mu }\left(x\right)}{dx}|}_{x=0}=-\frac{2^{\mu +1}{\pi }^{1/2}}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(-\frac{1}{2}\nu -\frac{1}{2}\mu \right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.6.3 (modified) Permalink: http://dlmf.nist.gov/14.5.E2 Encodings: TeX, pMML, png See also: Annotations for §14.5(i), §14.5 and Ch.14

14.5.3		$${𝖰}_{\nu }^{\mu }\left(0\right)=-\frac{2^{\mu -1}{\pi }^{1/2}\sin \left(\frac{1}{2}(\nu +\mu )\pi \right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +1\right)},$$
$\nu +\mu \ne -1,-2,-3,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $\mu$: general order and $\nu$: general degree A&S Ref: 8.6.2 Referenced by: Erratum (V1.1.3) for Equations (14.5.3), (14.5.4), Erratum (V1.1.3) for Equations (14.5.3), (14.5.4) Permalink: http://dlmf.nist.gov/14.5.E3 Encodings: TeX, pMML, png Correction (effective with 1.1.3): The constraint has been corrected to exclude all negative integers. See also: Annotations for §14.5(i), §14.5 and Ch.14

14.5.4		$${\frac{d{𝖰}_{\nu }^{\mu }\left(x\right)}{dx}|}_{x=0}=\frac{2^{\mu }{\pi }^{1/2}\cos \left(\frac{1}{2}(\nu +\mu )\pi \right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)},$$
$\nu +\mu \ne -1,-2,-3,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.6.4 Referenced by: Erratum (V1.1.3) for Equations (14.5.3), (14.5.4), Erratum (V1.1.3) for Equations (14.5.3), (14.5.4) Permalink: http://dlmf.nist.gov/14.5.E4 Encodings: TeX, pMML, png Correction (effective with 1.1.3): The constraint has been corrected to exclude all negative integers. See also: Annotations for §14.5(i), §14.5 and Ch.14

§14.5(ii) $\mu =0$, $\nu =0,1$

ⓘ

Notes:

Use §14.7(i).

Referenced by:

§14.5(vi), Erratum (V1.0.17) for Subsections 14.5(ii), 14.5(vi), Erratum (V1.0.17) for Subsections 14.5(ii), 14.5(vi), Erratum (V1.0.7) for Subsection 14.5(vi)

Permalink:

http://dlmf.nist.gov/14.5.ii

Clarification (effective with 1.0.17):

The title of this subsection has been changed back to its original name.

Addition (effective with 1.0.7):

The title of this subsection has been revised and the sentence at its end has been added.

See also:

Annotations for §14.5 and Ch.14

14.5.5		$${𝖯}_0\left(x\right)=P_0\left(x\right)=1,$$
ⓘ Symbols: ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $x$: real variable A&S Ref: 8.4.1 Permalink: http://dlmf.nist.gov/14.5.E5 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14

14.5.6		$${𝖯}_1\left(x\right)=P_1\left(x\right)=x.$$
ⓘ Symbols: ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $x$: real variable A&S Ref: 8.4.3 Permalink: http://dlmf.nist.gov/14.5.E6 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14

14.5.7		${𝖰}_0\left(x\right)$	$={\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{1+x}{1-x}}\right),$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and $x$: real variable A&S Ref: 8.4.2 Permalink: http://dlmf.nist.gov/14.5.E7 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14
14.5.8		${𝖰}_1\left(x\right)$	$={\displaystyle \frac{x}{2}}\ln \left({\displaystyle \frac{1+x}{1-x}}\right)-1.$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and $x$: real variable A&S Ref: 8.4.4 Permalink: http://dlmf.nist.gov/14.5.E8 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14

14.5.9		${𝑸}_0\left(x\right)$	$={\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{x+1}{x-1}}\right),$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable A&S Ref: 8.4.2 (modified) Permalink: http://dlmf.nist.gov/14.5.E9 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14
14.5.10		${𝑸}_1\left(x\right)$	$={\displaystyle \frac{x}{2}}\ln \left({\displaystyle \frac{x+1}{x-1}}\right)-1.$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable A&S Ref: 8.4.4 (modified) Permalink: http://dlmf.nist.gov/14.5.E10 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14

For the corresponding formulas when $\nu =2$ see §14.5(vi).

§14.5(iii) $\mu =\pm \frac{1}{2}$

ⓘ

Keywords:

associated Legendre functions, special values

Notes:

See Erdélyi et al. (1953a, p. 150).

Permalink:

http://dlmf.nist.gov/14.5.iii

See also:

Annotations for §14.5 and Ch.14

In this subsection and the next two, and $\xi >0$.

14.5.11		${𝖯}_{\nu }^{1/2}\left(\cos \theta \right)$	$={\left({\displaystyle \frac{2}{\pi \sin \theta }}\right)}^{1/2}\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right),$
ⓘ Symbols: : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable A&S Ref: 8.6.12 Permalink: http://dlmf.nist.gov/14.5.E11 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14
14.5.12		${𝖯}_{\nu }^{-1/2}\left(\cos \theta \right)$	$={\left({\displaystyle \frac{2}{\pi \sin \theta }}\right)}^{1/2}{\displaystyle \frac{\sin \left(\left(\nu +\frac{1}{2}\right)\theta \right)}{\nu +\frac{1}{2}}},$
ⓘ Symbols: : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable A&S Ref: 8.6.14 Permalink: http://dlmf.nist.gov/14.5.E12 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14
14.5.13		${𝖰}_{\nu }^{1/2}\left(\cos \theta \right)$	$=-{\left({\displaystyle \frac{\pi }{2\sin \theta }}\right)}^{1/2}\sin \left(\left(\nu +\frac{1}{2}\right)\theta \right),$
ⓘ Symbols: : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable A&S Ref: 8.6.13 Permalink: http://dlmf.nist.gov/14.5.E13 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14

14.5.14		$${𝖰}_{\nu }^{-1/2}\left(\cos \theta \right)={\left(\frac{\pi }{2\sin \theta }\right)}^{1/2}\frac{\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right)}{\nu +\frac{1}{2}}.$$
ⓘ Symbols: : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable A&S Ref: 8.6.15 (corrected) Referenced by: Erratum (V1.0.16) for Equation (14.5.14) Permalink: http://dlmf.nist.gov/14.5.E14 Encodings: TeX, pMML, png Errata (effective with 1.0.16): Originally this equation was incorrect because of a minus sign in front of the right-hand side. Reported 2017-04-10 by André Greiner-Petter See also: Annotations for §14.5(iii), §14.5 and Ch.14

14.5.15		$P_{\nu }^{1/2}\left(\cosh \xi \right)$	$={\left({\displaystyle \frac{2}{\pi \sinh \xi }}\right)}^{1/2}\cosh \left(\left(\nu +\frac{1}{2}\right)\xi \right),$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable A&S Ref: 8.6.8 (modified) Permalink: http://dlmf.nist.gov/14.5.E15 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14
14.5.16		$P_{\nu }^{-1/2}\left(\cosh \xi \right)$	$={\left({\displaystyle \frac{2}{\pi \sinh \xi }}\right)}^{1/2}{\displaystyle \frac{\sinh \left(\left(\nu +\frac{1}{2}\right)\xi \right)}{\nu +\frac{1}{2}}},$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable A&S Ref: 8.6.9 (modified) Permalink: http://dlmf.nist.gov/14.5.E16 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14
14.5.17		${𝑸}_{\nu }^{\pm 1/2}\left(\cosh \xi \right)$	$={\left({\displaystyle \frac{\pi }{2\sinh \xi }}\right)}^{1/2}{\displaystyle \frac{\exp \left(-\left(\nu +\frac{1}{2}\right)\xi \right)}{\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}}.$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable A&S Ref: 8.6.10 (corrected) 8.6.11 (corrected) Permalink: http://dlmf.nist.gov/14.5.E17 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14

§14.5(iv) $\mu =-\nu$

ⓘ

Notes:

See Erdélyi et al. (1953a, p. 150).

Permalink:

http://dlmf.nist.gov/14.5.iv

See also:

Annotations for §14.5 and Ch.14

14.5.18		${𝖯}_{\nu }^{-\nu }\left(\cos \theta \right)$	$={\displaystyle \frac{{(\sin \theta )}^{\nu }}{2^{\nu }\mathrm{\Gamma }\left(\nu +1\right)}},$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable A&S Ref: 8.6.17 Permalink: http://dlmf.nist.gov/14.5.E18 Encodings: TeX, pMML, png See also: Annotations for §14.5(iv), §14.5 and Ch.14
14.5.19		$P_{\nu }^{-\nu }\left(\cosh \xi \right)$	$={\displaystyle \frac{{(\sinh \xi )}^{\nu }}{2^{\nu }\mathrm{\Gamma }\left(\nu +1\right)}}.$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable A&S Ref: 8.6.16 (modified) Permalink: http://dlmf.nist.gov/14.5.E19 Encodings: TeX, pMML, png See also: Annotations for §14.5(iv), §14.5 and Ch.14

§14.5(v) $\mu =0$, $\nu =\pm \frac{1}{2}$

ⓘ

Keywords:

Ferrers functions, associated Legendre functions, elliptic integrals, relations to other functions, special values

Notes:

See Magnus et al. (1966, p. 173). These results may be verified by comparing the hypergeometric representations of the Legendre functions and elliptic integrals (§§14.3 and 19.5).

Referenced by:

§14.19(i)

Permalink:

http://dlmf.nist.gov/14.5.v

See also:

Annotations for §14.5 and Ch.14

In this subsection $K\left(k\right)$ and $E\left(k\right)$ denote the complete elliptic integrals of the first and second kinds; see §19.2(ii).

14.5.20		$${𝖯}_{\frac{1}{2}}\left(\cos \theta \right)=\frac{2}{\pi }\left(2E\left(\sin \left(\frac{1}{2}\theta \right)\right)-K\left(\sin \left(\frac{1}{2}\theta \right)\right)\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function and : variable A&S Ref: 8.13.11 (in slightly different form) Permalink: http://dlmf.nist.gov/14.5.E20 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14

14.5.21		${𝖯}_{-\frac{1}{2}}\left(\cos \theta \right)$	$={\displaystyle \frac{2}{\pi }}K\left(\sin \left(\frac{1}{2}\theta \right)\right),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function and : variable A&S Ref: 8.13.9 Permalink: http://dlmf.nist.gov/14.5.E21 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14
14.5.22		${𝖰}_{\frac{1}{2}}\left(\cos \theta \right)$	$=K\left(\cos \left(\frac{1}{2}\theta \right)\right)-2E\left(\cos \left(\frac{1}{2}\theta \right)\right),$
ⓘ Symbols: $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and : variable A&S Ref: 8.13.12 (in slightly different form) Permalink: http://dlmf.nist.gov/14.5.E22 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14
14.5.23		${𝖰}_{-\frac{1}{2}}\left(\cos \theta \right)$	$=K\left(\cos \left(\frac{1}{2}\theta \right)\right).$
ⓘ Symbols: $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and : variable A&S Ref: 8.13.10 (in slightly different form) Permalink: http://dlmf.nist.gov/14.5.E23 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14

14.5.24		$P_{\frac{1}{2}}\left(\cosh \xi \right)$	$={\displaystyle \frac{2}{\pi }}{\mathrm{e}}^{\xi /2}E\left({\left(1-{\mathrm{e}}^{-2\xi }\right)}^{1/2}\right),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $\xi >0$: variable A&S Ref: 8.13.6 Permalink: http://dlmf.nist.gov/14.5.E24 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14
14.5.25		$P_{-\frac{1}{2}}\left(\cosh \xi \right)$	$={\displaystyle \frac{2}{\pi \cosh \left(\frac{1}{2}\xi \right)}}K\left(\tanh \left(\frac{1}{2}\xi \right)\right),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $\xi >0$: variable A&S Ref: 8.13.2 Permalink: http://dlmf.nist.gov/14.5.E25 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14

14.5.26		$${𝑸}_{\frac{1}{2}}\left(\cosh \xi \right)=\begin{array}{l}2{\pi }^{-1/2}\cosh \xi \mathrm{sech}\left(\frac{1}{2}\xi \right)K\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right)\\ -4{\pi }^{-1/2}\cosh \left(\frac{1}{2}\xi \right)E\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right),\end{array}$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable A&S Ref: 8.13.7 (in slightly different form) Permalink: http://dlmf.nist.gov/14.5.E26 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14

14.5.27		$${𝑸}_{-\frac{1}{2}}\left(\cosh \xi \right)=2{\pi }^{-1/2}{\mathrm{e}}^{-\xi /2}K\left({\mathrm{e}}^{-\xi }\right).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable A&S Ref: 8.13.4 (modified) Permalink: http://dlmf.nist.gov/14.5.E27 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14

§14.5(vi) Addendum to §14.5(ii): $\mu =0$, $\nu =2$

ⓘ

Referenced by:

§14.5(ii), Erratum (V1.0.17) for Subsections 14.5(ii), 14.5(vi), Erratum (V1.0.7) for Subsection 14.5(vi)

Permalink:

http://dlmf.nist.gov/14.5.vi

Clarification (effective with 1.0.17):

The title of this subsection has been restricted to be more descriptive.

Addition (effective with 1.0.7):

This subsection has been added.

See also:

Annotations for §14.5 and Ch.14

14.5.28		${𝖯}_2\left(x\right)$	$=P_2\left(x\right)={\displaystyle \frac{3x^2-1}{2}},$
ⓘ Symbols: ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $x$: real variable Referenced by: (14.5.28) Permalink: http://dlmf.nist.gov/14.5.E28 Encodings: TeX, pMML, png Addition (effective with 1.0.7): (14.5.28) has been added to this section. See also: Annotations for §14.5(vi), §14.5 and Ch.14
14.5.29		${𝖰}_2\left(x\right)$	$={\displaystyle \frac{3x^2-1}{4}}\ln \left({\displaystyle \frac{1+x}{1-x}}\right)-{\displaystyle \frac{3}{2}}x,$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and $x$: real variable Referenced by: (14.5.29) Permalink: http://dlmf.nist.gov/14.5.E29 Encodings: TeX, pMML, png Addition (effective with 1.0.7): (14.5.29) has been added to this section. See also: Annotations for §14.5(vi), §14.5 and Ch.14
14.5.30		${𝑸}_2\left(x\right)$	$={\displaystyle \frac{3x^2-1}{8}}\ln \left({\displaystyle \frac{x+1}{x-1}}\right)-{\displaystyle \frac{3}{4}}x.$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable Referenced by: (14.5.30) Permalink: http://dlmf.nist.gov/14.5.E30 Encodings: TeX, pMML, png Addition (effective with 1.0.7): (14.5.30) has been added to this section. See also: Annotations for §14.5(vi), §14.5 and Ch.14

14.4 Graphics14.6 Integer Order

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