Editing Airy function (section)

==Properties==
The values of {{math|Ai(''x'')}} and {{math|Bi(''x'')}} and their derivatives at {{math|1=''x'' = 0}} are given by
<math display="block">\begin{align}
 \operatorname{Ai}(0) &{}= \frac{1}{3^{2/3} \, \Gamma\!\left(\frac{2}{3}\right)}, & \quad \operatorname{Ai}'(0) &{}= -\frac{1}{3^{1/3} \, \Gamma\!\left(\frac{1}{3}\right)}, \\
 \operatorname{Bi}(0) &{}= \frac{1}{3^{1/6} \, \Gamma\!\left(\frac{2}{3}\right)}, & \quad \operatorname{Bi}'(0) &{}= \frac{3^{1/6}}{\Gamma\!\left(\frac{1}{3}\right)}.
\end{align}</math>
Here, {{math|Γ}} denotes the [[Gamma function]]. It follows that the [[Wronskian]] of {{math|Ai(''x'')}} and {{math|Bi(''x'')}} is {{math|1/''π''}}.

When {{mvar|x}} is positive, {{math|Ai(''x'')}} is positive, [[convex function|convex]], and decreasing exponentially to zero, while {{math|Bi(''x'')}} is positive, convex, and increasing exponentially. When {{mvar|x}} is negative, {{math|Ai(''x'')}} and {{math|Bi(''x'')}} oscillate around zero with ever-increasing frequency and ever-decreasing amplitude. This is supported by the asymptotic formulae below for the Airy functions.

The Airy functions are orthogonal<ref>{{cite journal | last=Aspnes | first=David E. | title=Electric-Field Effects on Optical Absorption near Thresholds in Solids | journal=Physical Review | volume=147 | issue=2 | date=1966 | issn=0031-899X | doi=10.1103/PhysRev.147.554 | pages=554–566}}</ref> in the sense that
<math display="block"> \int_{-\infty}^\infty \operatorname{Ai}(t+x) \operatorname{Ai}(t+y) dt = \delta(x-y)</math>
again using an improper Riemann integral.

;Real zeros of {{math|Ai(''x'')}} and its derivative {{math|Ai'(''x'')}}
Neither {{math|Ai(''x'')}} nor its [[derivative]] {{math|Ai'(''x'')}} have positive real zeros. The "first" real zeros (i.e. nearest to x=0) are:<ref>{{cite web |url=https://dlmf.nist.gov/9.9 |title=Airy and Related Function |website=dlmf.nist.gov |access-date=9 October 2022}}</ref>
* "first" zeros of {{math|Ai(''x'')}} are at {{math|''x'' ≈ −2.33811, −4.08795, −5.52056, −6.78671, ...}}
* "first" zeros of its derivative {{math|Ai'(''x'')}} are at {{math|''x'' ≈ −1.01879, −3.24820, −4.82010, −6.16331, ...}}