Airy integral with complex coefficients

Question

In the book by Olivier Vallee and Manuel Soares about Airy function following integral has been given: $$ \int_{-\infty}^{\infty}\exp\left({\rm i} \left({{t^{3} \over 3} + at^{2} + bt}\right)\right) {\rm d}t = 2\pi\exp\left({\rm i}a\left({2a^{2} \over 3} - b\right)\right)\operatorname{Ai}\left(b - a^{2}\right) $$ I wonder if this is valid or not when $a$ is complex. Anyone knows $?$.