Formula:5879

Digital Library of Mathematical Functions ID 15.16.E5

${\displaystyle{\displaystyle F\left({\frac{1}{2}+\lambda,-\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)+F\left({\frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({-\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)-F\left({\frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)=\frac{\Gamma\left(1+\lambda+\mu\right)\Gamma\left(1+\nu+\mu\right)}{\Gamma\left(\lambda+\mu+\nu+\frac{3}{2}\right)\Gamma\left(\frac{1}{2}+\nu\right)},}}$

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
 

Resources that cite this formula

No records found.