AiryAi, AiryBi - Maple Help

AiryAi, AiryBi

The Airy Ai and Bi wave functions

 Calling Sequence AiryAi(x) AiryBi(x) AiryAi(n, x) AiryBi(n, x)

Parameters

 n - algebraic expression (the order or index) x - algebraic expression (the argument)

Description

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

$\mathrm{AiryAi}\left(z\right)={c}_{1}{}_{0}F_{1}\left(;\frac{2}{3};\frac{{z}^{3}}{9}\right)-{c}_{2}z{}_{0}F_{1}\left(;\frac{4}{3};\frac{{z}^{3}}{9}\right)$

$\mathrm{AiryBi}\left(z\right)={3}^{\frac{1}{2}}\left[{c}_{1}{}_{0}F_{1}\left(;\frac{2}{3};\frac{{z}^{3}}{9}\right)+{c}_{2}z{}_{0}F_{1}\left(;\frac{4}{3};\frac{{z}^{3}}{9}\right)\right]$

 where ${}_{0}F_{1}$ 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 $\frac{1}{3}n$ for $n=-2,-1,1,2$ (see the examples below).

Examples

 > $\mathrm{AiryAi}\left(0\right)$
 $\frac{{1}}{{3}}{}\frac{{{3}}^{{1}{/}{3}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}$ (1)
 > $\mathrm{AiryBi}\left(0\right)$
 $\frac{{1}}{{3}}{}\frac{{{3}}^{{5}{/}{6}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}$ (2)
 > $\mathrm{AiryAi}\left(1.23\right)$
 ${0.1021992656}$ (3)
 > $\mathrm{AiryBi}\left(-3.45+2.75I\right)$
 ${-}{16.85910551}{-}{32.61659997}{}{I}$ (4)
 > $\mathrm{AiryAi}\left(1,x\right)$
 ${\mathrm{AiryAi}}{}\left({1}{,}{x}\right)$ (5)
 > $\mathrm{AiryBi}\left(2,x\right)$
 ${\mathrm{AiryBi}}{}\left({2}{,}{x}\right)$ (6)
 > $\mathrm{convert}\left(\mathrm{AiryAi}\left(x\right),\mathrm{Bessel}\right)$
 ${-}\frac{{1}}{{3}}{}\frac{{x}{}{\mathrm{BesselI}}{}\left(\frac{{1}}{{3}}{,}\frac{{2}}{{3}}{}\sqrt{{{x}}^{{3}}}\right)}{{\left({{x}}^{{3}}\right)}^{{1}{/}{6}}}{+}\frac{{1}}{{3}}{}{\left({{x}}^{{3}}\right)}^{{1}{/}{6}}{}{\mathrm{BesselI}}{}\left({-}\frac{{1}}{{3}}{,}\frac{{2}}{{3}}{}\sqrt{{{x}}^{{3}}}\right)$ (7)
 > $\mathrm{convert}\left(\mathrm{AiryBi}\left(1,x\right),\mathrm{Bessel}\right)$
 $\frac{{1}}{{3}}{}\frac{\sqrt{{3}}{}\left({{x}}^{{2}}{}{\mathrm{BesselI}}{}\left(\frac{{2}}{{3}}{,}\frac{{2}}{{3}}{}\sqrt{{{x}}^{{3}}}\right){+}{\left({{x}}^{{3}}\right)}^{{2}{/}{3}}{}{\mathrm{BesselI}}{}\left({-}\frac{{2}}{{3}}{,}\frac{{2}}{{3}}{}\sqrt{{{x}}^{{3}}}\right)\right)}{{\left({{x}}^{{3}}\right)}^{{1}{/}{3}}}$ (8)
 > $\frac{ⅆ}{ⅆx}\mathrm{AiryAi}\left(\mathrm{sin}\left(x\right)\right)$
 ${\mathrm{cos}}{}\left({x}\right){}{\mathrm{AiryAi}}{}\left({1}{,}{\mathrm{sin}}{}\left({x}\right)\right)$ (9)
 > $\frac{\partial }{\partial x}\mathrm{AiryBi}\left(n,x\right)$
 ${\mathrm{AiryBi}}{}\left({n}{+}{1}{,}{x}\right)$ (10)
 > ${\mathrm{D}}^{\left(5\right)}\left(\mathrm{AiryBi}\right)$
 ${z}{→}{4}{}{z}{}{\mathrm{AiryBi}}{}\left({z}\right){+}{{z}}^{{2}}{}{\mathrm{AiryBi}}{}\left({1}{,}{z}\right)$ (11)
 > $\mathrm{series}\left(\mathrm{AiryAi}\left(x\right),x,4\right)$
 $\frac{{1}}{{3}}{}\frac{{{3}}^{{1}{/}{3}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{-}\frac{{1}}{{2}}{}\frac{{{3}}^{{1}{/}{6}}{}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{{\mathrm{π}}}{}{x}{+}\frac{{1}}{{18}}{}\frac{{{3}}^{{1}{/}{3}}}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{}{{x}}^{{3}}{+}{\mathrm{O}}\left({{x}}^{{4}}\right)$ (12)