# §14.5 Special Values

## §14.5(i) $x=0$

 14.5.1 $\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(0\right)=\frac{2^{\mu}\pi^{1% /2}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\frac{1}{2}\nu-\frac{1}{2}\mu% \right)},$
 14.5.2 $\left.\frac{\mathrm{d}\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x% \right)}{\mathrm{d}x}\right|_{x=0}=-\frac{2^{\mu+1}\pi^{1/2}}{\mathop{\Gamma\/% }\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)\mathop{% \Gamma\/}\nolimits\!\left(-\frac{1}{2}\nu-\frac{1}{2}\mu\right)},$
 14.5.3 $\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(0\right)=-\frac{2^{\mu-1}\pi% ^{1/2}\mathop{\sin\/}\nolimits\!\left(\frac{1}{2}(\nu+\mu)\pi\right)\mathop{% \Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)},$ $\nu+\mu\neq-1,-3,-5,\dots$,
 14.5.4 $\left.\frac{\mathrm{d}\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x% \right)}{\mathrm{d}x}\right|_{x=0}=\frac{2^{\mu}\pi^{1/2}\mathop{\cos\/}% \nolimits\!\left(\frac{1}{2}(\nu+\mu)\pi\right)\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)},$ $\nu+\mu\neq-2,-4,-6,\dots$.

## §14.5(ii) $\mu=0$, $\nu=0,1$, and 2

 14.5.5 $\mathop{\mathsf{P}_{0}\/}\nolimits\!\left(x\right)=\mathop{P_{0}\/}\nolimits\!% \left(x\right)=1,$
 14.5.6 $\mathop{\mathsf{P}_{1}\/}\nolimits\!\left(x\right)=\mathop{P_{1}\/}\nolimits\!% \left(x\right)=x.$
 14.5.7 $\displaystyle\mathop{\mathsf{Q}_{0}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{1+x}{1-x}\right),$ 14.5.8 $\displaystyle\mathop{\mathsf{Q}_{1}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{x}{2}\mathop{\ln\/}\nolimits\!\left(\frac{1+x}{1-x}\right)% -1.$
 14.5.9 $\displaystyle\mathop{\boldsymbol{Q}_{0}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{x+1}{x-1}\right),$ 14.5.10 $\displaystyle\mathop{\boldsymbol{Q}_{1}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{x}{2}\mathop{\ln\/}\nolimits\!\left(\frac{x+1}{x-1}\right)% -1.$

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 $\displaystyle\mathop{\mathsf{P}^{1/2}_{\nu}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)$ $\displaystyle=\left(\frac{2}{\pi\mathop{\sin\/}\nolimits\theta}\right)^{1/2}% \mathop{\cos\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)\theta\right),$ 14.5.12 $\displaystyle\mathop{\mathsf{P}^{-1/2}_{\nu}\/}\nolimits\!\left(\mathop{\cos\/% }\nolimits\theta\right)$ $\displaystyle=\left(\frac{2}{\pi\mathop{\sin\/}\nolimits\theta}\right)^{1/2}% \frac{\mathop{\sin\/}\nolimits\!\left(\left(\nu+\frac{1}{2}\right)\theta\right% )}{\nu+\frac{1}{2}},$ 14.5.13 $\displaystyle\mathop{\mathsf{Q}^{1/2}_{\nu}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)$ $\displaystyle=-\left(\frac{\pi}{2\mathop{\sin\/}\nolimits\theta}\right)^{1/2}% \mathop{\sin\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)\theta\right),$
 14.5.14 $\mathop{\mathsf{Q}^{-1/2}_{\nu}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)=-\left(\frac{\pi}{2\mathop{\sin\/}\nolimits\theta}\right)^{1/2}% \frac{\mathop{\cos\/}\nolimits\!\left(\left(\nu+\frac{1}{2}\right)\theta\right% )}{\nu+\frac{1}{2}}.$
 14.5.15 $\displaystyle\mathop{P^{1/2}_{\nu}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)$ $\displaystyle=\left(\frac{2}{\pi\mathop{\sinh\/}\nolimits\xi}\right)^{1/2}% \mathop{\cosh\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)\xi\right),$ 14.5.16 $\displaystyle\mathop{P^{-1/2}_{\nu}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)$ $\displaystyle=\left(\frac{2}{\pi\mathop{\sinh\/}\nolimits\xi}\right)^{1/2}% \frac{\mathop{\sinh\/}\nolimits\!\left(\left(\nu+\frac{1}{2}\right)\xi\right)}% {\nu+\frac{1}{2}},$ 14.5.17 $\displaystyle\mathop{\boldsymbol{Q}^{\pm 1/2}_{\nu}\/}\nolimits\!\left(\mathop% {\cosh\/}\nolimits\xi\right)$ $\displaystyle=\left(\frac{\pi}{2\mathop{\sinh\/}\nolimits\xi}\right)^{1/2}% \frac{\mathop{\exp\/}\nolimits\!\left(-\left(\nu+\frac{1}{2}\right)\xi\right)}% {\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{3}{2}\right)}.$

## §14.5(iv) $\mu=-\nu$

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

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

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

 14.5.20 $\mathop{\mathsf{P}_{\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)=\frac{2}{\pi}\left(2\!\mathop{E\/}\nolimits\!\left(\mathop{\sin% \/}\nolimits\!\left(\tfrac{1}{2}\theta\right)\right)-\mathop{K\/}\nolimits\!% \left(\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\theta\right)\right)\right),$
 14.5.21 $\displaystyle\mathop{\mathsf{P}_{-\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cos% \/}\nolimits\theta\right)$ $\displaystyle=\frac{2}{\pi}\mathop{K\/}\nolimits\!\left(\mathop{\sin\/}% \nolimits\!\left(\tfrac{1}{2}\theta\right)\right),$ 14.5.22 $\displaystyle\mathop{\mathsf{Q}_{\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cos% \/}\nolimits\theta\right)$ $\displaystyle=\mathop{K\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\!\left(% \tfrac{1}{2}\theta\right)\right)-2\!\mathop{E\/}\nolimits\!\left(\mathop{\cos% \/}\nolimits\!\left(\tfrac{1}{2}\theta\right)\right),$ 14.5.23 $\displaystyle\mathop{\mathsf{Q}_{-\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cos% \/}\nolimits\theta\right)$ $\displaystyle=\mathop{K\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\!\left(% \tfrac{1}{2}\theta\right)\right).$
 14.5.24 $\displaystyle\mathop{P_{\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)$ $\displaystyle=\frac{2}{\pi}e^{\xi/2}\mathop{E\/}\nolimits\!\left(\left(1-e^{-2% \xi}\right)^{1/2}\right),$ 14.5.25 $\displaystyle\mathop{P_{-\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)$ $\displaystyle=\frac{2}{\pi\mathop{\cosh\/}\nolimits\!\left(\frac{1}{2}\xi% \right)}\mathop{K\/}\nolimits\!\left(\mathop{\tanh\/}\nolimits\!\left(\tfrac{1% }{2}\xi\right)\right),$
 14.5.26 $\mathop{\boldsymbol{Q}_{\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)=2\pi^{-1/2}\mathop{\cosh\/}\nolimits\xi\mathop{\mathrm{% sech}\/}\nolimits\!\left(\tfrac{1}{2}\xi\right)\mathop{K\/}\nolimits\!\left(% \mathop{\mathrm{sech}\/}\nolimits\!\left(\tfrac{1}{2}\xi\right)\right)-4\pi^{-% 1/2}\mathop{\cosh\/}\nolimits\!\left(\tfrac{1}{2}\xi\right)\mathop{E\/}% \nolimits\!\left(\mathop{\mathrm{sech}\/}\nolimits\!\left(\tfrac{1}{2}\xi% \right)\right),$
 14.5.27 $\mathop{\boldsymbol{Q}_{-\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)=2\pi^{-1/2}e^{-\xi/2}\mathop{K\/}\nolimits\!\left(e^{-\xi}% \right).$

## §14.5(vi) Addendum to §14.5(ii)$\mu=0,\nu=0,1$, and 2

 14.5.28 $\displaystyle\mathop{\mathsf{P}_{2}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{P_{2}\/}\nolimits\!\left(x\right)=\frac{3x^{2}-1}{2},$ Symbols: $\mathop{\mathsf{P}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)=\mathop{% \mathsf{P}^{0}_{\nu}\/}\nolimits\!\left(x\right)$: Ferrers function of the first kind, $\mathop{P_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)=\mathop{P^{0}_{\nu}% \/}\nolimits\!\left(z\right)$: 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. Reported 2012-07-30 See also: Annotations for 14.5(vi) 14.5.29 $\displaystyle\mathop{\mathsf{Q}_{2}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{3x^{2}-1}{4}\mathop{\ln\/}\nolimits\!\left(\frac{1+x}{1-x}% \right)-\frac{3}{2}x,$ Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\mathop{\mathsf{Q}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)=\mathop{% \mathsf{Q}^{0}_{\nu}\/}\nolimits\!\left(x\right)$: 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. Reported 2012-07-30 See also: Annotations for 14.5(vi) 14.5.30 $\displaystyle\mathop{\boldsymbol{Q}_{2}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{3x^{2}-1}{8}\mathop{\ln\/}\nolimits\!\left(\frac{x+1}{x-1}% \right)-\frac{3}{4}x.$ Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\mathop{\boldsymbol{Q}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)=\mathop% {\boldsymbol{Q}^{0}_{\nu}\/}\nolimits\!\left(z\right)$: 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. Reported 2012-07-30 See also: Annotations for 14.5(vi)