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The Airy wave functions AiryAi and AiryBi are linearly independent solutions for w in the equation $w\text{'}\text{'}-zw=0$. Specifically,

where $F_1^0$ is the generalized hypergeometric function, $c_1=\mathrm{AiryAi}\left(0\right)$ and $c_2=-\mathrm{AiryAi}\'\left(0\right)$.

The two argument forms are used to represent the derivatives, so AiryAi(1, x) = D(AiryAi)(x) and AiryBi(1, x) = D(AiryBi)(x). Note that all higher derivatives can be written in terms of the 0'th and 1st derivatives.

Note also that $\mathrm{AiryAi}\left(3\,x^2\right)$ is the 3rd derivative of $\mathrm{AiryAi}\left(x\right)$ evaluated at $x^2$, and not the 3rd derivative of $\mathrm{AiryAi}\left(x^2\right)$.

The Airy functions are related to Bessel functions of order $\frac{n}{3}$ for $n=\mathrm{-2},\mathrm{-1},1,2$ (see the examples below).

$\mathrm{AiryAi}\left(0\right)$

$\frac{3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3\mathrm{\Gamma }\left(\frac{2}{3}\right)}$

$\mathrm{AiryBi}\left(0\right)$

$\frac{3^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{3\mathrm{\Gamma }\left(\frac{2}{3}\right)}$

$\mathrm{AiryAi}\left(1.23\right)$

$0.1021992656$

$\mathrm{AiryBi}\left(-3.45+2.75\mathrm{I}\right)$

$\mathrm{-16.85910551}-32.61659997\mathrm{I}$

$\mathrm{AiryAi}\left(1\,x\right)$

$\mathrm{AiryAi}\left(1\,x\right)$

$\mathrm{AiryBi}\left(2\,x\right)$

$\mathrm{AiryBi}\left(2\,x\right)$

$\mathrm{convert}\left(\mathrm{AiryAi}\left(x\right)\,\mathrm{Bessel}\right)$

$-\frac{x\mathrm{BesselI}\left(\frac{1}{3}\,\frac{2\sqrt{x^3}}{3}\right)}{3{\left(x^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}+\frac{{\left(x^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\mathrm{BesselI}\left(-\frac{1}{3}\,\frac{2\sqrt{x^3}}{3}\right)}{3}$

$\mathrm{convert}\left(\mathrm{AiryBi}\left(1\,x\right)\,\mathrm{Bessel}\right)$

$\frac{\sqrt{3}\left(x^2\mathrm{BesselI}\left(\frac{2}{3}\,\frac{2\sqrt{x^3}}{3}\right)+{\left(x^3\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{BesselI}\left(-\frac{2}{3}\,\frac{2\sqrt{x^3}}{3}\right)\right)}{3{\left(x^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

$\mathrm{diff}\left(\mathrm{AiryAi}\left(\sin \left(x\right)\right)\,x\right)$

$\cos \left(x\right)\mathrm{AiryAi}\left(1\,\sin \left(x\right)\right)$

$\mathrm{diff}\left(\mathrm{AiryBi}\left(n\,x\right)\,x\right)$

$\mathrm{AiryBi}\left(n+1\,x\right)$

${\mathrm{D}}^{\left(5\right)}\left(\mathrm{AiryBi}\right)$

$z\mapsto 4\cdot z\cdot \mathrm{AiryBi}\left(z\right)+z^2\cdot \mathrm{AiryBi}\left(1\,z\right)$

$\mathrm{series}\left(\mathrm{AiryAi}\left(x\right)\,x\,4\right)$

$\frac{3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3\mathrm{\Gamma }\left(\frac{2}{3}\right)}-\frac{1}{2}\frac{3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\pi }}x+\frac{1}{18}\frac{3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{x}^3+O\left(x^4\right)$