# §9.5 Integral Representations

## §9.5(i) Real Variable

 9.5.1 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\cos\/}\nolimits\!\left(\tfrac{1}{3}t^{3}+xt\right)\mathrm{d}t.$
 9.5.2 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(-x\right)=\frac{x^{\ifrac{1}{2}}}{\pi}% \int_{-1}^{\infty}\mathop{\cos\/}\nolimits\!\left(x^{\ifrac{3}{2}}(\tfrac{1}{3% }t^{3}+t^{2}-\tfrac{2}{3})\right)\mathrm{d}t,$ $x>0$.
 9.5.3 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\exp\/}\nolimits\!\left(-{\tfrac{1}{3}}t^{3}+xt\right)\mathrm{d}t+% \frac{1}{\pi}\int_{0}^{\infty}\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{3}t^{3% }+xt\right)\mathrm{d}t.$

See also (9.10.19), (9.11.3), (36.9.2), and Vallée and Soares (2010, §2.1.3).

## §9.5(ii) Complex Variable

 9.5.4 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi i}\int_{\infty e% ^{-\pi i/3}}^{\infty e^{\pi i/3}}\mathop{\exp\/}\nolimits\!\left(\tfrac{1}{3}t% ^{3}-zt\right)\mathrm{d}t,$
 9.5.5 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi}\int_{-\infty}^{% \infty e^{\pi i/3}}\mathop{\exp\/}\nolimits\!\left(\tfrac{1}{3}t^{3}-zt\right)% \mathrm{d}t+\dfrac{1}{2\pi}\int_{-\infty}^{\infty e^{-\pi i/3}}\mathop{\exp\/}% \nolimits\!\left(\tfrac{1}{3}t^{3}-zt\right)\mathrm{d}t.$
 9.5.6 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{\sqrt{3}}{2\pi}\int_{0}^% {\infty}\mathop{\exp\/}\nolimits\!\left(-\frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}% \right)\mathrm{d}t.$
 9.5.7 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{e^{-\zeta}}{\pi}\int_{0}% ^{\infty}\mathop{\exp\/}\nolimits\!\left(-z^{\ifrac{1}{2}}t^{2}\right)\mathop{% \cos\/}\nolimits\!\left(\tfrac{1}{3}t^{3}\right)\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$.
 9.5.8 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{e^{-\zeta}\zeta^{\ifrac{% -1}{6}}}{\sqrt{\pi}(48)^{\ifrac{1}{6}}\mathop{\Gamma\/}\nolimits\!\left(\frac{% 5}{6}\right)}\int_{0}^{\infty}e^{-t}t^{-\ifrac{1}{6}}\left(2+\frac{t}{\zeta}% \right)^{-\ifrac{1}{6}}\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{2}{3}\pi$.

In (9.5.7) and (9.5.8) $\zeta=\frac{2}{3}z^{\ifrac{3}{2}}$.