Formula:5782

Digital Library of Mathematical Functions ID 15.8.E28

${\displaystyle{\displaystyle\frac{2\sqrt{z}\Gamma\left(-\tfrac{1}{2}\right)\Gamma\left(a+b-\tfrac{1}{2}\right)}{\Gamma\left(a-\tfrac{1}{2}\right)\Gamma\left(b-\tfrac{1}{2}\right)}F\left(a,b;\tfrac{3}{2};z\right)=F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}-\tfrac{1}{2}\sqrt{z}\right)-F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}+\tfrac{1}{2}\sqrt{z}\right),}}$

Constraint(s)

${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}$
${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle a}}$: real or complex parameter
• ${\displaystyle{\displaystyle b}}$: real or complex parameter
 

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