Formula:5409

Digital Library of Mathematical Functions ID 14.3.E23

${\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(1-\mu\right)}\left(\frac{x+1}{x-1}\right)^{\mu/2}\phi^{(-\mu,\mu)}_{-\mathrm{i}(2\nu+1)}\left(\operatorname{arcsinh}\left((\tfrac{1}{2}x-\tfrac{1}{2})^{\ifrac{1}{2}}\right)\right).}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\phi^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{\lambda}}\left(\NVar{t}\right)}}$: Jacobi function
• ${\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}$: associated Legendre function of the first kind
• ${\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}$:
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle\mu}}$: general order
• ${\displaystyle{\displaystyle\nu}}$: general degree
 

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