# §14.5 Special Values

## §14.5(i) $x=0$

 14.5.1 $\mathsf{P}^{\mu}_{\nu}\left(0\right)=\frac{2^{\mu}\pi^{1/2}}{\Gamma\left(\frac% {1}{2}\nu-\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}-\frac{1}{2}\nu-\frac{% 1}{2}\mu\right)},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mu$: 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 14
 14.5.2 $\left.\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{\mathrm{d}x}\right% |_{x=0}=-\frac{2^{\mu+1}\pi^{1/2}}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+% \frac{1}{2}\right)\Gamma\left(-\frac{1}{2}\nu-\frac{1}{2}\mu\right)},$
 14.5.3 $\mathsf{Q}^{\mu}_{\nu}\left(0\right)=-\frac{2^{\mu-1}\pi^{1/2}\sin\left(\frac{% 1}{2}(\nu+\mu)\pi\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}% \right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)},$ $\nu+\mu\neq-1,-3,-5,\dots$,
 14.5.4 $\left.\frac{\mathrm{d}\mathsf{Q}^{\mu}_{\nu}\left(x\right)}{\mathrm{d}x}\right% |_{x=0}=\frac{2^{\mu}\pi^{1/2}\cos\left(\frac{1}{2}(\nu+\mu)\pi\right)\Gamma% \left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}{\Gamma\left(\frac{1}{2}\nu-\frac% {1}{2}\mu+\frac{1}{2}\right)},$ $\nu+\mu\neq-2,-4,-6,\dots$.

## §14.5(ii) $\mu=0$, $\nu=0,1$

 14.5.5 $\mathsf{P}_{0}\left(x\right)=P_{0}\left(x\right)=1,$ ⓘ Symbols: $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$: Ferrers function of the first kind, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$: 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 14
 14.5.6 $\mathsf{P}_{1}\left(x\right)=P_{1}\left(x\right)=x.$ ⓘ Symbols: $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$: Ferrers function of the first kind, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$: 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 14
 14.5.7 $\displaystyle\mathsf{Q}_{0}\left(x\right)$ $\displaystyle=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right),$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$: 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 14 14.5.8 $\displaystyle\mathsf{Q}_{1}\left(x\right)$ $\displaystyle=\frac{x}{2}\ln\left(\frac{1+x}{1-x}\right)-1.$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$: 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 14
 14.5.9 $\displaystyle\boldsymbol{Q}_{0}\left(x\right)$ $\displaystyle=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right),$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$: 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 14 14.5.10 $\displaystyle\boldsymbol{Q}_{1}\left(x\right)$ $\displaystyle=\frac{x}{2}\ln\left(\frac{x+1}{x-1}\right)-1.$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$: 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 14

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

## §14.5(iii) $\mu=\pm\frac{1}{2}$

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

 14.5.11 $\displaystyle\mathsf{P}^{1/2}_{\nu}\left(\cos\theta\right)$ $\displaystyle=\left(\frac{2}{\pi\sin\theta}\right)^{1/2}\cos\left(\left(\nu+% \tfrac{1}{2}\right)\theta\right),$ 14.5.12 $\displaystyle\mathsf{P}^{-1/2}_{\nu}\left(\cos\theta\right)$ $\displaystyle=\left(\frac{2}{\pi\sin\theta}\right)^{1/2}\frac{\sin\left(\left(% \nu+\frac{1}{2}\right)\theta\right)}{\nu+\frac{1}{2}},$ 14.5.13 $\displaystyle\mathsf{Q}^{1/2}_{\nu}\left(\cos\theta\right)$ $\displaystyle=-\left(\frac{\pi}{2\sin\theta}\right)^{1/2}\sin\left(\left(\nu+% \tfrac{1}{2}\right)\theta\right),$
 14.5.14 $\mathsf{Q}^{-1/2}_{\nu}\left(\cos\theta\right)=\left(\frac{\pi}{2\sin\theta}% \right)^{1/2}\frac{\cos\left(\left(\nu+\frac{1}{2}\right)\theta\right)}{\nu+% \frac{1}{2}}.$ ⓘ Symbols: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $\nu$: general degree and $0<\theta<\pi$: variable A&S Ref: 8.6.15 (corrected) Referenced by: 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 14
 14.5.15 $\displaystyle P^{1/2}_{\nu}\left(\cosh\xi\right)$ $\displaystyle=\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\cosh\left(\left(\nu+% \tfrac{1}{2}\right)\xi\right),$ 14.5.16 $\displaystyle P^{-1/2}_{\nu}\left(\cosh\xi\right)$ $\displaystyle=\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\frac{\sinh\left(\left(% \nu+\frac{1}{2}\right)\xi\right)}{\nu+\frac{1}{2}},$ 14.5.17 $\displaystyle\boldsymbol{Q}^{\pm 1/2}_{\nu}\left(\cosh\xi\right)$ $\displaystyle=\left(\frac{\pi}{2\sinh\xi}\right)^{1/2}\frac{\exp\left(-\left(% \nu+\frac{1}{2}\right)\xi\right)}{\Gamma\left(\nu+\frac{3}{2}\right)}.$

## §14.5(iv) $\mu=-\nu$

 14.5.18 $\displaystyle\mathsf{P}^{-\nu}_{\nu}\left(\cos\theta\right)$ $\displaystyle=\frac{(\sin\theta)^{\nu}}{2^{\nu}\Gamma\left(\nu+1\right)},$ 14.5.19 $\displaystyle P^{-\nu}_{\nu}\left(\cosh\xi\right)$ $\displaystyle=\frac{(\sinh\xi)^{\nu}}{2^{\nu}\Gamma\left(\nu+1\right)}.$

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

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 $\mathsf{P}_{\frac{1}{2}}\left(\cos\theta\right)=\frac{2}{\pi}\left(2\!E\left(% \sin\left(\tfrac{1}{2}\theta\right)\right)-K\left(\sin\left(\tfrac{1}{2}\theta% \right)\right)\right),$
 14.5.21 $\displaystyle\mathsf{P}_{-\frac{1}{2}}\left(\cos\theta\right)$ $\displaystyle=\frac{2}{\pi}K\left(\sin\left(\tfrac{1}{2}\theta\right)\right),$ 14.5.22 $\displaystyle\mathsf{Q}_{\frac{1}{2}}\left(\cos\theta\right)$ $\displaystyle=K\left(\cos\left(\tfrac{1}{2}\theta\right)\right)-2\!E\left(\cos% \left(\tfrac{1}{2}\theta\right)\right),$ 14.5.23 $\displaystyle\mathsf{Q}_{-\frac{1}{2}}\left(\cos\theta\right)$ $\displaystyle=K\left(\cos\left(\tfrac{1}{2}\theta\right)\right).$ ⓘ Symbols: $K\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$: cosine function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$: Ferrers function of the second kind and $0<\theta<\pi$: 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 14
 14.5.24 $\displaystyle P_{\frac{1}{2}}\left(\cosh\xi\right)$ $\displaystyle=\frac{2}{\pi}e^{\xi/2}E\left(\left(1-e^{-2\xi}\right)^{1/2}% \right),$ 14.5.25 $\displaystyle P_{-\frac{1}{2}}\left(\cosh\xi\right)$ $\displaystyle=\frac{2}{\pi\cosh\left(\frac{1}{2}\xi\right)}K\left(\tanh\left(% \tfrac{1}{2}\xi\right)\right),$
 14.5.26 $\boldsymbol{Q}_{\frac{1}{2}}\left(\cosh\xi\right)=2\pi^{-1/2}\cosh\xi% \operatorname{sech}\left(\tfrac{1}{2}\xi\right)K\left(\operatorname{sech}\left% (\tfrac{1}{2}\xi\right)\right)-4\pi^{-1/2}\cosh\left(\tfrac{1}{2}\xi\right)E% \left(\operatorname{sech}\left(\tfrac{1}{2}\xi\right)\right),$
 14.5.27 $\boldsymbol{Q}_{-\frac{1}{2}}\left(\cosh\xi\right)=2\pi^{-1/2}e^{-\xi/2}K\left% (e^{-\xi}\right).$

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

 14.5.28 $\displaystyle\mathsf{P}_{2}\left(x\right)$ $\displaystyle=P_{2}\left(x\right)=\frac{3x^{2}-1}{2},$ ⓘ Symbols: $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$: Ferrers function of the first kind, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$: 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 14 14.5.29 $\displaystyle\mathsf{Q}_{2}\left(x\right)$ $\displaystyle=\frac{3x^{2}-1}{4}\ln\left(\frac{1+x}{1-x}\right)-\frac{3}{2}x,$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$: 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 14 14.5.30 $\displaystyle\boldsymbol{Q}_{2}\left(x\right)$ $\displaystyle=\frac{3x^{2}-1}{8}\ln\left(\frac{x+1}{x-1}\right)-\frac{3}{4}x.$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$: 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 14