Editing Airy function (section)

==Asymptotic formulae==
[[File:Mplwp airyai asymptotic.svg|thumb|320px|Ai(blue) and sinusoidal/exponential asymptotic form of Ai(magenta)]]
[[File:Mplwp airybi asymptotic.svg|thumb|320px|Bi(blue) and sinusoidal/exponential asymptotic form of Bi(magenta)]]
As explained below, the Airy functions can be extended to the complex plane, giving [[entire function]]s. The asymptotic behaviour of the Airy functions as {{mvar|{{abs|z}}}} goes to infinity at a constant value of {{math|[[arg (mathematics)|arg]](''z'')}} depends on {{math|arg(''z'')}}: this is called the [[Stokes phenomenon]]. For {{math|{{abs|arg(''z'')}} < ''π''}} we have the following [[asymptotic formula]] for {{math|Ai(''z'')}}:<ref name=":0">{{harvtxt|Abramowitz|Stegun|1983|p=448|ignore-err=yes}}, Eqns 10.4.59, 10.4.61</ref>

<math display="block">
\operatorname{Ai}(z)\sim 
\dfrac{1}{2\sqrt\pi\,z^{1/4}} 
\exp\left(-\frac{2}{3}z^{3/2}\right) 
\left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(n+\frac{5}{6}\right) \, \Gamma\!\left(n+\frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].</math>
or<math display="block">
\operatorname{Ai}(z)\sim 
\dfrac{e^{-\zeta}}{4\pi^{3/2}\,z^{1/4}} 
\left[ \sum_{n=0}^{\infty} \dfrac{\Gamma\!\left(n+\frac{5}{6}\right) \, \Gamma\!\left(n+\frac{1}{6}\right)}{n! (-2\zeta)^n} \right].</math>
where <math>\zeta = \tfrac 23 z^{3/2}.</math> In particular, the first few terms are<ref>{{Cite web |title=DLMF: §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions |url=https://dlmf.nist.gov/9.7 |access-date=2023-05-11 |website=dlmf.nist.gov}}</ref><math display="block">\operatorname{Ai}(z) = \frac{e^{-\zeta}}{2\pi^{1/2}z^{1/4}}\left(1 - \frac{5}{72 \zeta} + \frac{385}{10368 \zeta^2} + O(\zeta^{-3})\right)
</math>
There is a similar one for {{math|Bi(''z'')}}, but only applicable when {{math|{{abs|arg(''z'')}} < ''π''/3}}:

<math display="block">
\operatorname{Bi}(z)\sim 
\frac{1}{\sqrt\pi\,z^{1/4}}
\exp\left(\frac{2}{3}z^{3/2}\right)
\left[ \sum_{n=0}^{\infty} \dfrac{ \Gamma\!\left(n+\frac{5}{6}\right) \, \Gamma\!\left(n+\frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].</math>
A more accurate formula for {{math|Ai(''z'')}} and a formula for {{math|Bi(''z'')}} when {{math|''π''/3 < {{abs|arg(''z'')}} < ''π''}} or, equivalently, for {{math|Ai(−''z'')}} and {{math|Bi(−''z'')}} when {{math|{{abs|arg(''z'')}} < 2''π''/3}} but not zero, are:<ref name=":0" /><ref name=":1">{{harvtxt|Abramowitz|Stegun|1983|p=448|ignore-err=yes}}, Eqns 10.4.60 and 10.4.64</ref><math display="block">\begin{align}
 \operatorname{Ai}(-z) \sim&{} \ 
 \frac{1}{\sqrt\pi\,z^{1/4}} 
 \sin\left( \frac{2}{3}z^{3/2} + \frac{\pi}{4} \right) 
 \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6} \right) \, \Gamma\!\left(2n+\frac{1}{6}\right) \left(\frac{3}{4} \right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt]

 &{}-\frac{1}{\sqrt\pi \, z^{1/4}} 
\cos\left(\frac{2}{3}z^{3/2}+\frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right] \\[6pt]
 
 \operatorname{Bi}(-z) \sim&{} 
 \frac{1}{\sqrt\pi \, z^{1/4}}
 \cos \left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6}\right) \, \Gamma\!\left(2n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt]
 
 &{}+ \frac{1}{\sqrt\pi\,z^{\frac{1}{4}}} 
 \sin\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right].
\end{align}</math>

When {{math|1={{abs|arg(''z'')}} = 0}} these are good approximations but are not asymptotic because the ratio between {{math|Ai(−''z'')}} or {{math|Bi(−''z'')}} and the above approximation goes to infinity whenever the sine or cosine goes to zero.
[[Asymptotic analysis|Asymptotic expansions]] for these limits are also available. These are listed in (Abramowitz and Stegun, 1983) and (Olver, 1974).

One is also able to obtain asymptotic expressions for the derivatives {{math|Ai'(z)}} and {{math|Bi'(z)}}. Similarly to before, when {{math|{{abs|arg(''z'')}} < ''π''}}:<ref name=":1" />

<math display="block">
\operatorname{Ai}'(z)\sim 
-\dfrac{z^{1/4}}{2\sqrt\pi\,} 
\exp\left(-\frac{2}{3}z^{3/2}\right)
\left[ \sum_{n=0}^{\infty} \frac{1+6n}{1-6n} \dfrac{(-1)^n \, \Gamma\!\left(n + \frac{5}{6}\right) \, \Gamma\!\left(n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].</math>

When {{math|{{abs|arg(''z'')}} < ''π''/3}} we have:<ref name=":1" />

<math display="block">
\operatorname{Bi}'(z)\sim 
\frac{z^{1/4}}{\sqrt\pi\,}
\exp\left(\frac{2}{3}z^{3/2}\right)
\left[ \sum_{n=0}^{\infty} \frac{1+6n}{1-6n} \dfrac{ \Gamma\!\left(n + \frac{5}{6}\right) \, \Gamma\!\left(n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^n}{2\pi \, n! \, z^{3n/2}} \right].</math>

Similarly, an expression for {{math|Ai'(−''z'')}} and {{math|Bi'(−''z'')}} when {{math|{{abs|arg(''z'')}} < 2''π''/3}} but not zero, are<ref name=":1" />

<math display="block">\begin{align}
 \operatorname{Ai}'(-z) \sim&{} 
 -\frac{z^{1/4}}{\sqrt\pi\,} 
 \cos\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \frac{1+12n}{1-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6}\right) \, \Gamma\!\left(2n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt]

 &{}-\frac{z^{1/4}}{\sqrt\pi\,} 
 \sin\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \frac{7+12n}{-5-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right] \\[6pt]
 
 \operatorname{Bi}'(-z) \sim&{} \ 
 \frac{z^{1/4}}{\sqrt\pi\,} 
 \sin\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \frac{1+12n}{1-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{5}{6}\right) \, \Gamma\!\left(2n + \frac{1}{6}\right) \left(\frac{3}{4}\right)^{2n}}{2\pi \, (2n)! \, z^{3n}} \right] \\[6pt]
 
 &{}-\frac{z^{1/4}}{\sqrt\pi\,} 
 \cos\left(\frac{2}{3}z^{3/2} + \frac{\pi}{4} \right)
 \left[ \sum_{n=0}^{\infty} \frac{7+12n}{-5-12n} \dfrac{(-1)^n \, \Gamma\!\left(2n + \frac{11}{6}\right) \, \Gamma\!\left(2n + \frac{7}{6}\right) \left(\frac{3}{4}\right)^{2n+1}}{2\pi \, (2n+1)! \, z^{3n\,+\,3/2}} \right] \\
\end{align}</math>