§14.5 Special Values

Keywords:: Ferrers functions
Permalink:: http://dlmf.nist.gov/14.5

§14.5(i) $x=0$
§14.5(ii) $\mu=0$ , $\nu=0,1$
§14.5(iii) $\mu=\pm\frac{1}{2}$
§14.5(iv) $\mu=-\nu$
§14.5(v) $\mu=0,\nu=\pm\frac{1}{2}$

§14.5(i) $x=0$

Notes:: These results may be derived from (14.3.11)–(14.3.14) and also (14.10.4), with $\mathop{\mathsf{P}\/}\nolimits$ replaced by $\mathop{\mathsf{Q}\/}\nolimits$ .
Permalink:: http://dlmf.nist.gov/14.5.i

14.5.1 $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(0\right)=\frac{2^{{\mu}}\pi^{{1/2}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\frac{1}{2}\nu-\frac{1}{2}\mu\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mu$ : 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

14.5.2 $\left.\frac{d\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)}{dx}\right|_{{x=0}}=-\frac{2^{{\mu+1}}\pi^{{1/2}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(-\frac{1}{2}\nu-\frac{1}{2}\mu\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\frac{df}{dx}$ : 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

14.5.3 $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(0\right)=-\frac{2^{{\mu-1}}\pi^{{1/2}}\mathop{\sin\/}\nolimits\!\left(\frac{1}{2}(\nu+\mu)\pi\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)},$ $\nu+\mu\neq-1,-3,-5,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\sin\/}\nolimits z$ : sine function, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.6.2
Permalink:: http://dlmf.nist.gov/14.5.E3
Encodings:: TeX, pMML, png

14.5.4 $\left.\frac{d\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)}{dx}\right|_{{x=0}}=\frac{2^{{\mu}}\pi^{{1/2}}\mathop{\cos\/}\nolimits\!\left(\frac{1}{2}(\nu+\mu)\pi\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)},$ $\nu+\mu\neq-2,-4,-6,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : 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
Permalink:: http://dlmf.nist.gov/14.5.E4
Encodings:: TeX, pMML, png

§14.5(ii) $\mu=0$ , $\nu=0,1$

Notes:: Use §14.7(i).
Permalink:: http://dlmf.nist.gov/14.5.ii

14.5.5 $\mathop{\mathsf{P}_{{0}}\/}\nolimits\!\left(x\right)=\mathop{P_{{0}}\/}\nolimits\!\left(x\right)=1,$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated 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

14.5.6 $\mathop{\mathsf{P}_{{1}}\/}\nolimits\!\left(x\right)=\mathop{P_{{1}}\/}\nolimits\!\left(x\right)=x.$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated 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

14.5.7 $\mathop{\mathsf{Q}_{{0}}\/}\nolimits\!\left(x\right)=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{1+x}{1-x}\right),$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 8.4.2
Permalink:: http://dlmf.nist.gov/14.5.E7
Encodings:: TeX, pMML, png

14.5.8 $\mathop{\mathsf{Q}_{{1}}\/}\nolimits\!\left(x\right)=\frac{x}{2}\mathop{\ln\/}\nolimits\!\left(\frac{1+x}{1-x}\right)-1.$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 8.4.4
Permalink:: http://dlmf.nist.gov/14.5.E8
Encodings:: TeX, pMML, png

14.5.9 $\mathop{\boldsymbol{Q}_{{0}}\/}\nolimits\!\left(x\right)=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{x+1}{x-1}\right),$

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 8.4.2 (modified)
Permalink:: http://dlmf.nist.gov/14.5.E9
Encodings:: TeX, pMML, png

14.5.10 $\mathop{\boldsymbol{Q}_{{1}}\/}\nolimits\!\left(x\right)=\frac{x}{2}\mathop{\ln\/}\nolimits\!\left(\frac{x+1}{x-1}\right)-1.$

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 8.4.4 (modified)
Permalink:: http://dlmf.nist.gov/14.5.E10
Encodings:: TeX, pMML, png

§14.5(iii) $\mu=\pm\frac{1}{2}$

Notes:: See Erdélyi et al. (1953a, p. 150).
Keywords:: associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.5.iii

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

14.5.11 $\mathop{\mathsf{P}^{{1/2}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\left(\frac{2}{\pi\mathop{\sin\/}\nolimits\theta}\right)^{{1/2}}\mathop{\cos\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)\theta\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\nu$ : general degree and $0<\theta<\pi$ : variable
A&S Ref:: 8.6.12
Permalink:: http://dlmf.nist.gov/14.5.E11
Encodings:: TeX, pMML, png

14.5.12 $\mathop{\mathsf{P}^{{-1/2}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\left(\frac{2}{\pi\mathop{\sin\/}\nolimits\theta}\right)^{{1/2}}\frac{\mathop{\sin\/}\nolimits\!\left(\left(\nu+\frac{1}{2}\right)\theta\right)}{\nu+\frac{1}{2}},$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\nu$ : general degree and $0<\theta<\pi$ : variable
A&S Ref:: 8.6.14
Permalink:: http://dlmf.nist.gov/14.5.E12
Encodings:: TeX, pMML, png

14.5.13 $\mathop{\mathsf{Q}^{{1/2}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=-\left(\frac{\pi}{2\mathop{\sin\/}\nolimits\theta}\right)^{{1/2}}\mathop{\sin\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)\theta\right),$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\nu$ : general degree and $0<\theta<\pi$ : variable
A&S Ref:: 8.6.13
Permalink:: http://dlmf.nist.gov/14.5.E13
Encodings:: TeX, pMML, png

14.5.14 $\mathop{\mathsf{Q}^{{-1/2}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=-\left(\frac{\pi}{2\mathop{\sin\/}\nolimits\theta}\right)^{{1/2}}\frac{\mathop{\cos\/}\nolimits\!\left(\left(\nu+\frac{1}{2}\right)\theta\right)}{\nu+\frac{1}{2}}.$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\nu$ : general degree and $0<\theta<\pi$ : variable
A&S Ref:: 8.6.15 (corrected)
Permalink:: http://dlmf.nist.gov/14.5.E14
Encodings:: TeX, pMML, png

14.5.15 $\mathop{P^{{1/2}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)=\left(\frac{2}{\pi\mathop{\sinh\/}\nolimits\xi}\right)^{{1/2}}\mathop{\cosh\/}\nolimits\!\left(\left(\nu+\tfrac{1}{2}\right)\xi\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : 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

14.5.16 $\mathop{P^{{-1/2}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)=\left(\frac{2}{\pi\mathop{\sinh\/}\nolimits\xi}\right)^{{1/2}}\frac{\mathop{\sinh\/}\nolimits\!\left(\left(\nu+\frac{1}{2}\right)\xi\right)}{\nu+\frac{1}{2}},$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : 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

14.5.17 $\mathop{\boldsymbol{Q}^{{\pm 1/2}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)=\left(\frac{\pi}{2\mathop{\sinh\/}\nolimits\xi}\right)^{{1/2}}\frac{\mathop{\exp\/}\nolimits\!\left(-\left(\nu+\frac{1}{2}\right)\xi\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{3}{2}\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : 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

§14.5(iv) $\mu=-\nu$

Notes:: See Erdélyi et al. (1953a, p. 150).
Permalink:: http://dlmf.nist.gov/14.5.iv

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\nu$ : general degree and $0<\theta<\pi$ : variable
A&S Ref:: 8.6.17
Permalink:: http://dlmf.nist.gov/14.5.E18
Encodings:: TeX, pMML, png

14.5.19 $\mathop{P^{{-\nu}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)=\frac{(\mathop{\sinh\/}\nolimits\xi)^{{\nu}}}{2^{{\nu}}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : 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

§14.5(v) $\mu=0,\nu=\pm\frac{1}{2}$

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).
Keywords:: Ferrers functions, associated Legendre functions, elliptic integrals
Referenced by:: §14.19(i)
Permalink:: http://dlmf.nist.gov/14.5.v

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 $\mathop{\mathsf{P}_{{\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\frac{2}{\pi}\left(2\!\mathop{E\/}\nolimits\!\left(\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\theta\right)\right)-\mathop{K\/}\nolimits\!\left(\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\theta\right)\right)\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $0<\theta<\pi$ : variable
A&S Ref:: 8.13.11 (in slightly different form)
Permalink:: http://dlmf.nist.gov/14.5.E20
Encodings:: TeX, pMML, png

14.5.21 $\mathop{\mathsf{P}_{{-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\frac{2}{\pi}\mathop{K\/}\nolimits\!\left(\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\theta\right)\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $0<\theta<\pi$ : variable
A&S Ref:: 8.13.9
Permalink:: http://dlmf.nist.gov/14.5.E21
Encodings:: TeX, pMML, png

14.5.22 $\mathop{\mathsf{Q}_{{\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\mathop{K\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\theta\right)\right)-2\!\mathop{E\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\theta\right)\right),$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function and $0<\theta<\pi$ : variable
A&S Ref:: 8.13.12 (in slightly different form)
Permalink:: http://dlmf.nist.gov/14.5.E22
Encodings:: TeX, pMML, png

14.5.23 $\mathop{\mathsf{Q}_{{-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\mathop{K\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\theta\right)\right).$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function 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

14.5.24 $\mathop{P_{{\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)=\frac{2}{\pi}e^{{\xi/2}}\mathop{E\/}\nolimits\!\left(\left(1-e^{{-2\xi}}\right)^{{1/2}}\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\xi>0$ : variable
A&S Ref:: 8.13.6
Permalink:: http://dlmf.nist.gov/14.5.E24
Encodings:: TeX, pMML, png

14.5.25 $\mathop{P_{{-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)=\frac{2}{\pi\mathop{\cosh\/}\nolimits\!\left(\frac{1}{2}\xi\right)}\mathop{K\/}\nolimits\!\left(\mathop{\tanh\/}\nolimits\!\left(\tfrac{1}{2}\xi\right)\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and $\xi>0$ : variable
A&S Ref:: 8.13.2
Permalink:: http://dlmf.nist.gov/14.5.E25
Encodings:: TeX, pMML, png

14.5.26 $\mathop{\boldsymbol{Q}_{{\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)=2\pi^{{-1/2}}\mathop{\cosh\/}\nolimits\xi\mathop{\mathrm{sech}\/}\nolimits\!\left(\tfrac{1}{2}\xi\right)\mathop{K\/}\nolimits\!\left(\mathop{\mathrm{sech}\/}\nolimits\!\left(\tfrac{1}{2}\xi\right)\right)-4\pi^{{-1/2}}\mathop{\cosh\/}\nolimits\!\left(\tfrac{1}{2}\xi\right)\mathop{E\/}\nolimits\!\left(\mathop{\mathrm{sech}\/}\nolimits\!\left(\tfrac{1}{2}\xi\right)\right),$

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant 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

14.5.27 $\mathop{\boldsymbol{Q}_{{-\frac{1}{2}}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)=2\pi^{{-1/2}}e^{{-\xi/2}}\mathop{K\/}\nolimits\!\left(e^{{-\xi}}\right).$

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\xi>0$ : variable
A&S Ref:: 8.13.4 (modified)
Permalink:: http://dlmf.nist.gov/14.5.E27
Encodings:: TeX, pMML, png

§14.5 Special Values

Contents

§14.5(i)

§14.5(ii) ,

§14.5(iii)

§14.5(iv)

§14.5(v)

§14.5(i) $x=0$

§14.5(ii) $\mu=0$ , $\nu=0,1$

§14.5(iii) $\mu=\pm\frac{1}{2}$

§14.5(iv) $\mu=-\nu$

§14.5(v) $\mu=0,\nu=\pm\frac{1}{2}$