Formula:3595

Digital Library of Mathematical Functions ID 9.10.E14

${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-pt}\mathrm{Ai}\left(t\right)\mathrm{d}t=e^{-p^{3}/3}\left(\frac{1}{3}-\frac{p{{}_{1}F_{1}}\left(\tfrac{1}{3};\tfrac{4}{3};\tfrac{1}{3}p^{3}\right)}{3^{4/3}\Gamma\left(\tfrac{4}{3}\right)}+\frac{p^{2}{{}_{1}F_{1}}\left(\tfrac{2}{3};\tfrac{5}{3};\tfrac{1}{3}p^{3}\right)}{3^{5/3}\Gamma\left(\tfrac{5}{3}\right)}\right),}}$

Constraint(s)

${\displaystyle{\displaystyle p\in\mathbb{C}}}$

Symbols List

• ${\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}$:
• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle{{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)}}$: = M ( a , b , z ) notation for the Kummer confluent hypergeometric function
• ${\displaystyle{\displaystyle\mathbb{C}}}$:
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\in}}$:
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle p}}$: parameter
 

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