airy Airy functions of the first and second kind, and their derivatives Syntax a = airy(z) a = airy(fun, z) a = airy(fun, z, scaled) Arguments z array of decimal or complex numbers of any size, from scalr to hypermatrix. fun Selected Airy function to evaluate. It can be either a string among "Ai" "dAi" "Bi" "dBi", or an equivalent integer in [0, 3] (for compatibility with Octave and Julia)
 fun Description 0 "Ai" Airy function of the first kind (default) 1 "dAi" Derivative Ai' of Ai 2 "Bi" Airy function of the second kind 3 "dBi" Derivative Bi' of Bi
scaled Single boolean or integer 0|1. Default %F. When scaled is %T or set to 1, the raw result is scaled by the following factors before being returned:
 Ai, dAi : exp\left(2\,z^{3/2}/ 3\right) Bi, dBi : exp\left(-2\left|{\mathrm Re}\!\left(z^{3/2}\right)\right|/ 3\right)
Description For a real x variable, the Airy functions of the first and second kind -- respectively Ai(x) and Bi(x) -- are independent real solutions y(x) of the Airy differential equation y'' = x.y. They are defined as the convergent integrals Ai(x) = {1 \over \pi} \int_0^\infty cos\left({t^3 \over 3} + xt\right)dt and Bi(x) = {1 \over \pi} \int_0^\infty \left[ sin\left({t^3 \over 3} + xt\right) + exp\left(- {t^3 \over 3} + xt\right)\right] dt These definitions can be extended to the complex plane, for any z complex variable, as Ai(z) = {1 \over 2\pi} \int_{-\infty}^\infty e^{i\left({t^3 \over 3} + zt\right)}dt Let us note the properties Ai(\overline z) = \overline{Ai(z)} \quad Bi(\overline z) = \overline{Bi(z)} \quad Ai'(\overline z) = \overline{Ai'(z)} \quad Bi'(\overline z) = \overline{Bi'(z)} In Scilab, Ai, Bi, and their first derivative are computed through Bessel and gamma functions. Examples With real numbers With scaling With complex numbers See also besseli besselj gamma History 6.1.0 airy() introduced.