§14.8 Behavior at Singularities

Keywords:: Ferrers functions
Permalink:: http://dlmf.nist.gov/14.8

§14.8(i) $x\to 1-$ or $x\to-1+$
§14.8(ii) $x\to 1+$
§14.8(iii) $x\to\infty$

§14.8(i) $x\to 1-$ or $x\to-1+$

Notes:: For (14.8.1), (14.8.4), and (14.8.6) see Olver (1997b, p. 186). For (14.8.2) and (14.8.3) see Erdélyi et al. (1953a, p. 163). (14.8.5) may be derived from (14.8.1) and (14.9.2).
Keywords:: associated Legendre functions
Referenced by:: §14.2(iii), §14.20(iii), §14.20(iii)
Permalink:: http://dlmf.nist.gov/14.8.i

As $x\to 1-$ ,

14.8.1 $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1-\mu\right)}\left(\frac{2}{1-x}\right)^{{\mu/2}},$ $\mu\neq 1,2,3,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\sim$ : asymptotic equality, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.30(ii), §14.8(i)
Permalink:: http://dlmf.nist.gov/14.8.E1
Encodings:: TeX, pMML, png

14.8.2 $\mathop{\mathsf{P}^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim(-1)^{m}\frac{\left(\nu-m+1\right)_{{2m}}}{m!}\left(\frac{1-x}{2}\right)^{{m/2}},$ $m=1,2,3,\dots$ , $\nu\neq m-1,m-2,\dots,-m$ ,

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree
Referenced by:: §14.30(ii), §14.8(i)
Permalink:: http://dlmf.nist.gov/14.8.E2
Encodings:: TeX, pMML, png

14.8.3 $\mathop{\mathsf{Q}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{2}{1-x}\right)-\EulerConstant-\mathop{\psi\/}\nolimits\!\left(\nu+1\right)+\mathop{O\/}\nolimits\!\left(1-x\right),$ $\nu\neq-1,-2,-3,\dots$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\EulerConstant$ : Euler’s constant, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable and $\nu$ : general degree
Referenced by:: §14.8(i)
Permalink:: http://dlmf.nist.gov/14.8.E3
Encodings:: TeX, pMML, png

where $\EulerConstant$ is Euler’s constant (§5.2(ii)). In the next three relations $\realpart{\mu}>0$ .

14.8.4 $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim\frac{1}{2}\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)\mathop{\Gamma\/}\nolimits\!\left(\mu\right)\left(\frac{2}{1-x}\right)^{{\mu/2}},$ $\mu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$ ,

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, $\sim$ : asymptotic equality, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.8(i)
Permalink:: http://dlmf.nist.gov/14.8.E4
Encodings:: TeX, pMML, png

14.8.5 $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim(-1)^{{\mu+(1/2)}}\frac{\pi\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{2\mathop{\Gamma\/}\nolimits\!\left(\mu+1\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}\left(\frac{1-x}{2}\right)^{{\mu/2}},$ $\mu=\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$ , $\nu\pm\mu\neq-1,-2,-3,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\sim$ : asymptotic equality, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.8(i)
Permalink:: http://dlmf.nist.gov/14.8.E5
Encodings:: TeX, pMML, png

14.8.6 $\mathop{\mathsf{Q}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}{2\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}\left(\frac{2}{1-x}\right)^{{\mu/2}},$ $\nu\pm\mu\neq-1,-2,-3,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\sim$ : asymptotic equality, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.8(i)
Permalink:: http://dlmf.nist.gov/14.8.E6
Encodings:: TeX, pMML, png

The behavior of $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ as $x\to-1+$ follows from the above results and the connection formulas (14.9.8) and (14.9.10).

§14.8(ii) $x\to 1+$

Notes:: See Erdélyi et al. (1953a, p. 163). (14.8.10) may be derived from (14.8.7) and (14.9.12).
Permalink:: http://dlmf.nist.gov/14.8.ii

14.8.7 $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1-\mu\right)}\left(\frac{2}{x-1}\right)^{{\mu/2}},$ $\mu\neq 1,2,3,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\sim$ : asymptotic equality, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.21(iii), §14.24, §14.8(ii)
Permalink:: http://dlmf.nist.gov/14.8.E7
Encodings:: TeX, pMML, png

14.8.8 $\mathop{P^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+m+1\right)}{m!\mathop{\Gamma\/}\nolimits\!\left(\nu-m+1\right)}\left(\frac{x-1}{2}\right)^{{m/2}},$ $m=1,2,3,\dots$ , $\nu\pm m\neq-1,-2,-3,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.8.E8
Encodings:: TeX, pMML, png

14.8.9 $\mathop{\boldsymbol{Q}_{{\nu}}\/}\nolimits\!\left(x\right)=-\frac{\mathop{\ln\/}\nolimits\!\left(x-1\right)}{2\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}+\frac{\frac{1}{2}\mathop{\ln\/}\nolimits 2-\EulerConstant-\mathop{\psi\/}\nolimits\!\left(\nu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}+\mathop{O\/}\nolimits\!\left(x-1\right),$ $\nu\neq-1,-2,-3,\dots$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\EulerConstant$ : Euler’s constant, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.8.E9
Encodings:: TeX, pMML, png

14.8.10 $\mathop{\boldsymbol{Q}_{{-n}}\/}\nolimits\!\left(x\right)\to(-1)^{{n+1}}(n-1)!,$ $n=1,2,3,\dots$ ,

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $!$ : $n!$ : factorial, $x$ : real variable and $n$ : nonnegative integer
Referenced by:: §14.8(ii)
Permalink:: http://dlmf.nist.gov/14.8.E10
Encodings:: TeX, pMML, png

14.8.11 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}{2\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}\left(\frac{2}{x-1}\right)^{{\mu/2}},$ $\realpart{\mu}>0$ , $\nu+\mu\neq-1,-2,-3,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\realpart{}$ : real part, $\sim$ : asymptotic equality, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.24
Permalink:: http://dlmf.nist.gov/14.8.E11
Encodings:: TeX, pMML, png

§14.8(iii) $x\to\infty$

Notes:: See Olver (1997b, pp. 171 and 173). (14.8.16) may be derived from (14.8.15) and (14.9.12).
Keywords:: Ferrers functions
Permalink:: http://dlmf.nist.gov/14.8.iii

14.8.12 $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)}{\pi^{{1/2}}\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}(2x)^{{\nu}},$ $\realpart{\nu}>-\tfrac{1}{2}$ , $\mu-\nu\neq 1,2,3,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\realpart{}$ : real part, $\sim$ : asymptotic equality, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.8.E12
Encodings:: TeX, pMML, png

14.8.13 $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim\frac{\mathop{\Gamma\/}\nolimits\!\left(-\nu-\frac{1}{2}\right)}{\pi^{{1/2}}\mathop{\Gamma\/}\nolimits\!\left(-\mu-\nu\right)(2x)^{{\nu+1}}},$ $\realpart{\nu}<-\tfrac{1}{2}$ , $\nu+\mu\neq 0,1,2,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\realpart{}$ : real part, $\sim$ : asymptotic equality, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.8.E13
Encodings:: TeX, pMML, png

14.8.14 $\mathop{P^{{\mu}}_{{-1/2}}\/}\nolimits\!\left(x\right)\sim\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu\right)}\left(\frac{2}{\pi x}\right)^{{1/2}}\mathop{\ln\/}\nolimits x,$ $\mu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality, $x$ : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.8.E14
Encodings:: TeX, pMML, png

14.8.15 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\sim\frac{\pi^{{1/2}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{3}{2}\right)(2x)^{{\nu+1}}},$ $\nu\neq-\tfrac{3}{2},-\tfrac{5}{2},-\tfrac{7}{2},\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\sim$ : asymptotic equality, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.8(iii)
Permalink:: http://dlmf.nist.gov/14.8.E15
Encodings:: TeX, pMML, png

14.8.16 ${\mathop{\boldsymbol{Q}^{{\mu}}_{{-n-(1/2)}}\/}\nolimits\!\left(x\right)\sim\frac{\pi^{{1/2}}\mathop{\Gamma\/}\nolimits\!\left(\mu+n+\frac{1}{2}\right)}{n!\mathop{\Gamma\/}\nolimits\!\left(\mu-n+\frac{1}{2}\right)(2x)^{{n+(1/2)}}}},$ $n=1,2,3,\dots$ , $\mu-n+\frac{1}{2}\neq 0,-1,-2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality, $x$ : real variable, $n$ : nonnegative integer and $\mu$ : general order
Referenced by:: §14.21(iii), §14.8(iii)
Permalink:: http://dlmf.nist.gov/14.8.E16
Encodings:: TeX, pMML, png

§14.8 Behavior at Singularities

Contents

§14.8(i) or

§14.8(ii)

§14.8(iii)

§14.8(i) $x\to 1-$ or $x\to-1+$

§14.8(ii) $x\to 1+$

§14.8(iii) $x\to\infty$