Formula:5568

Digital Library of Mathematical Functions ID 14.15.E24

${\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{1}{\left(\nu+\frac{1}{2}\right)^{1/4}2^{(\nu+\mu)/2}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)}\left(\frac{\zeta^{2}-\alpha^{2}}{x^{2}-a^{2}}\right)^{1/4}\*\left(U\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)+O\left(\nu^{-2/3}\right)\mathrm{env}\mskip-1.0mu U\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)\right),}}$

Symbols List

• ${\displaystyle{\displaystyle O\left(\NVar{x}\right)}}$: order not exceeding
• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)}}$: Ferrers function of the first kind
• ${\displaystyle{\displaystyle\mathrm{env}\mskip-1.0mu U\left(\NVar{c},\NVar{x}\right)}}$:
• ${\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}$:
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle\mu}}$: general order
• ${\displaystyle{\displaystyle\nu}}$: general degree
• ${\displaystyle{\displaystyle a}}$: a
• ${\displaystyle{\displaystyle\zeta}}$: ζ
• ${\displaystyle{\displaystyle\alpha}}$: α
 

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