Formula:5738

Digital Library of Mathematical Functions ID 15.6.E3

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "TeX input is invalid."): {\displaystyle \hyperOlverF@{a}{b}{c}{z}=e^{-b\pi\iunit}\frac{\EulerGamma@{1-b}}{2\pi\iunit\EulerGamma@{c-b}}\int_{\infty}^{(0+)}\frac{t^{b-1}(t+1)^{a-c}}{(t-zt+1)^{a}}\diff{t},}

Constraint(s)

${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}$
${\displaystyle{\displaystyle\Re(c-b)>0}}$
${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}$
${\displaystyle{\displaystyle b\neq 1,2,3,\dots}}$
${\displaystyle{\displaystyle\Re(c-b)>0}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$: Olver’s hypergeometric function
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle a}}$: real or complex parameter
• ${\displaystyle{\displaystyle b}}$: real or complex parameter
• ${\displaystyle{\displaystyle c}}$: real or complex parameter
 

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