Olver associated Legendre function

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1: 14.21 Definitions and Basic Properties
14.21.1 $\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{% \mathrm{d}w}{\mathrm{d}z}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w=0.$
2: 14.24 Analytic Continuation
14.24.1 $P^{-\mu}_{\nu}\left(ze^{s\pi i}\right)=e^{s\nu\pi i}P^{-\mu}_{\nu}\left(z% \right)+\frac{2i\sin\left(\left(\nu+\frac{1}{2}\right)s\pi\right)e^{-s\pi i/2}% }{\cos\left(\nu\pi\right)\Gamma\left(\mu-\nu\right)}\boldsymbol{Q}^{\mu}_{\nu}% \left(z\right),$
14.24.2 $\boldsymbol{Q}^{\mu}_{\nu}\left(ze^{s\pi i}\right)=(-1)^{s}e^{-s\nu\pi i}% \boldsymbol{Q}^{\mu}_{\nu}\left(z\right),$
14.24.4 $\boldsymbol{Q}^{\mu}_{\nu,s}\left(z\right)=e^{-s\mu\pi i}\boldsymbol{Q}^{\mu}_% {\nu}\left(z\right)-\frac{\pi i\sin\left(s\mu\pi\right)}{\sin\left(\mu\pi% \right)\Gamma\left(\nu-\mu+1\right)}P^{-\mu}_{\nu}\left(z\right),$
3: 14.12 Integral Representations
14.12.9 $\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{n!}\int_{0}^{u}\left(x-\left(x^{% 2}-1\right)^{1/2}\cosh t\right)^{n}\cosh\left(mt\right)\,\mathrm{d}t,$
14.12.11 $\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{\left(x^{2}-1\right)^{m/2}}{2^{n+1}% n!}\int_{-1}^{1}\frac{\left(1-t^{2}\right)^{n}}{(x-t)^{n+m+1}}\,\mathrm{d}t,$
14.12.12 $\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{(n-m)!}P^{m}_{n}\left(x\right)% \int_{x}^{\infty}\frac{\,\mathrm{d}t}{\left(t^{2}-1\right)\left(\displaystyle P% ^{m}_{n}\left(t\right)\right)^{2}},$ $n\geq m$.
14.12.13 $\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{2(n!)}\int_{-1}^{1}\frac{P_{n}\left(% t\right)}{x-t}\,\mathrm{d}t.$
14.12.14 $\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{n!}\int_{0}^{\infty}\frac{\,\mathrm{% d}t}{\left(x+(x^{2}-1)^{1/2}\cosh t\right)^{n+1}}.$
5: 14.19 Toroidal (or Ring) Functions
14.19.3 $\boldsymbol{Q}^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\pi^{1/2}% \left(1-e^{-2\xi}\right)^{\mu}}{e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\mu+% \tfrac{1}{2},\nu+\mu+\tfrac{1}{2};\nu+1;e^{-2\xi}\right).$
14.19.5 $\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+% \frac{1}{2}\right)}{\Gamma\left(n+m+\tfrac{1}{2}\right)\Gamma\left(n-m+\frac{1% }{2}\right)}\*\int_{0}^{\infty}\frac{\cosh\left(mt\right)}{(\cosh\xi+\cosh t% \sinh\xi)^{n+(1/2)}}\,\mathrm{d}t,$ $m.
14.19.6 $\boldsymbol{Q}^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right)+2\sum_{n=1}^{\infty}% \frac{\Gamma\left(\mu+n+\tfrac{1}{2}\right)}{\Gamma\left(\mu+\tfrac{1}{2}% \right)}\boldsymbol{Q}^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right)\cos\left(n% \phi\right)=\dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu% }}{\left(\cosh\xi-\cos\phi\right)^{\mu+(1/2)}},$ $\Re\mu>-\tfrac{1}{2}$.
14.19.7 $P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+m+\tfrac{1}{2}% \right)}{\Gamma\left(n-m+\tfrac{1}{2}\right)}\*\left(\frac{2}{\pi\sinh\xi}% \right)^{1/2}\boldsymbol{Q}^{n}_{m-\frac{1}{2}}\left(\coth\xi\right),$
14.19.8 $\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(m-n+% \tfrac{1}{2}\right)}{\Gamma\left(m+n+\tfrac{1}{2}\right)}\*\left(\frac{\pi}{2% \sinh\xi}\right)^{1/2}P^{n}_{m-\frac{1}{2}}\left(\coth\xi\right).$
6: 14.9 Connection Formulas
14.9.12 $\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right)=-\frac{\boldsymbol{Q}^{\mu% }_{\nu}\left(x\right)}{\Gamma\left(\mu-\nu\right)}+\frac{\boldsymbol{Q}^{\mu}_% {-\nu-1}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}.$
14.9.15 $\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=% \frac{P^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}-\frac{P^{-\mu% }_{\nu}\left(x\right)}{\Gamma\left(\nu-\mu+1\right)}.$
14.9.16 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\left(\tfrac{1}{2}\pi\right)^{1/2}% \left(x^{2}-1\right)^{-1/4}\*P^{-\nu-(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}-1% \right)^{-1/2}\right).$
14.9.17 $P^{\mu}_{\nu}\left(x\right)=(2/\pi)^{1/2}\left(x^{2}-1\right)^{-1/4}\*% \boldsymbol{Q}^{\nu+(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}-1\right)^{-1/2}% \right).$
7: 14.25 Integral Representations
14.25.2 $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)=\frac{\pi^{1/2}\left(z^{2}-1\right)^{% \mu/2}}{2^{\mu}\Gamma\left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}% \*\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{\left(z+(z^{2}-1)^{1/2}\cosh t% \right)^{\nu+\mu+1}}\,\mathrm{d}t,$ $\Re\left(\nu+1\right)>\Re\mu>-\tfrac{1}{2}$,
8: 14.3 Definitions and Hypergeometric Representations
14.3.6 $P^{\mu}_{\nu}\left(x\right)=\left(\frac{x+1}{x-1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).$
14.3.8 $P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(% \nu-m+1\right)}\left(x^{2}-1\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;% \tfrac{1}{2}-\tfrac{1}{2}x\right).$
14.3.10 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i}\frac{Q^{\mu}_{\nu}\left% (x\right)}{\Gamma\left(\nu+\mu+1\right)}.$
14.3.19 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{2^{\nu}\Gamma\left(\nu+1\right)% (x+1)^{\mu/2}}{(x-1)^{(\mu/2)+\nu+1}}\mathbf{F}\left(\nu+1,\nu+\mu+1;2\nu+2;% \frac{2}{1-x}\right),$
14.3.20 $\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=% \frac{(x+1)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)(x-1)^{\mu/2}}\mathbf{F}\left% (\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{(x-1)^{\mu/2}}{% \Gamma\left(\nu-\mu+1\right)(x+1)^{\mu/2}}\mathbf{F}\left(\nu+1,-\nu;\mu+1;% \tfrac{1}{2}-\tfrac{1}{2}x\right).$
9: 14.23 Values on the Cut
14.23.2 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)=\frac{e^{\pm\mu\pi i/2}}{\Gamma% \left(\nu+\mu+1\right)}\left(\mathsf{Q}^{\mu}_{\nu}\left(x\right)\mp\tfrac{1}{% 2}\pi i\mathsf{P}^{\mu}_{\nu}\left(x\right)\right).$
14.23.3 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)=\frac{e^{\mp\nu\pi i/2}\pi^{3/2% }\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\frac{x\mathbf{F}\left(\frac{1}% {2}\mu-\frac{1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};% x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)% \Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}\mp i\frac{% \mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+% \frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}% \mu+1\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}\right).$
14.23.5 $\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\tfrac{1}{2}\Gamma\left(\nu+\mu+1\right)% \left(e^{-\mu\pi i/2}\boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)+e^{\mu\pi i/2% }\boldsymbol{Q}^{\mu}_{\nu}\left(x-i0\right)\right),$
14.23.6 $\mathsf{Q}^{\mu}_{\nu}\left(x\right)=e^{\mp\mu\pi i/2}\Gamma\left(\nu+\mu+1% \right)\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)\pm\tfrac{1}{2}\pi ie^{% \pm\mu\pi i/2}P^{\mu}_{\nu}\left(x\pm i0\right).$
10: 14.8 Behavior at Singularities
14.8.9 $\boldsymbol{Q}_{\nu}\left(x\right)=-\frac{\ln\left(x-1\right)}{2\Gamma\left(% \nu+1\right)}+\frac{\frac{1}{2}\ln 2-\gamma-\psi\left(\nu+1\right)}{\Gamma% \left(\nu+1\right)}+O\left(\left(x-1\right)\ln\left(x-1\right)\right),$ $\nu\neq-1,-2,-3,\dots$,
14.8.11 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)\sim\frac{\Gamma\left(\mu\right)}{2% \Gamma\left(\nu+\mu+1\right)}\left(\frac{2}{x-1}\right)^{\mu/2},$ $\Re\mu>0$, $\nu+\mu\neq-1,-2,-3,\dots$.
14.8.15 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)\sim\frac{\pi^{1/2}}{\Gamma\left(\nu+% \frac{3}{2}\right)(2x)^{\nu+1}},$ $\nu\neq-\tfrac{3}{2},-\tfrac{5}{2},-\tfrac{7}{2},\dots$,
14.8.16 ${\boldsymbol{Q}^{\mu}_{-n-(1/2)}\left(x\right)\sim\frac{\pi^{1/2}\Gamma\left(% \mu+n+\frac{1}{2}\right)}{n!\Gamma\left(\mu-n+\frac{1}{2}\right)(2x)^{n+(1/2)}% }},$ $n=1,2,3,\dots$, $\mu-n+\frac{1}{2}\neq 0,-1,-2,\dots$.