Formula:5809

Digital Library of Mathematical Functions ID 15.9.E22

${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}};z\right)=\frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left(b+\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{\pm\pi\mathrm{i}(a+b-(\ifrac{1}{2}))}P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(-\sqrt{z}\right)+P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{z}\right)\right),}}$

Constraint(s)

${\displaystyle{\displaystyle 0<|\operatorname{ph}z|<\pi}}$
${\displaystyle{\displaystyle a,b\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\ldots}}$
${\displaystyle{\displaystyle 0<|\operatorname{ph}z|<\pi}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}$: associated Legendre function of the first kind
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$: Olver’s hypergeometric function
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle a}}$: real or complex parameter
• ${\displaystyle{\displaystyle b}}$: real or complex parameter
 

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