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)
Argumentsz
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:

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
History6.1.0airy() introduced.