Formula:5072

Digital Library of Mathematical Functions ID 13.4.E12

${\displaystyle{\displaystyle{\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b\right)}{2\pi\mathrm{i}}z^{1-b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;\ifrac{1}{t}\right)\mathrm{d}t,}}$

Constraint(s)

${\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\frac{1}{2}\pi}}$
${\displaystyle{\displaystyle b\neq 0,-1,-2,\dots}}$
${\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\frac{1}{2}\pi}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)}}$: Olver’s confluent hypergeometric 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{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\right)\)\@add@PDF@RDFa@triples\end{document}}}$: scaled (or Olver’s) generalized hypergeometric function
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}$:
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle z}}$: complex variable
 

Resources that cite this formula

No records found.