Editing Airy function (section)

==Definitions==
[[File:Airy Functions.svg|right|thumb|400px|Plot of {{math|Ai(''x'')}} in red and {{math|Bi(''x'')}} in blue]]
For real values of {{mvar|x}}, the Airy function of the first kind can be defined by the [[improper integral|improper]] [[Riemann integral]]:
<math display="block">\operatorname{Ai}(x) = \dfrac{1}{\pi}\int_0^\infty\cos\left(\dfrac{t^3}{3} + xt\right)\, dt\equiv \dfrac{1}{\pi} \lim_{b\to\infty} \int_0^b \cos\left(\dfrac{t^3}{3} + xt\right)\, dt,</math>
which converges by [[Dirichlet's test]]. For any [[real number]] {{mvar|x}} there is a positive real number {{mvar|M}} such that function <math display="inline">\tfrac{t^3}3 + xt</math> is increasing, unbounded and convex with continuous and unbounded derivative on interval <math>[M,\infty).</math> The convergence of the integral on this interval can be proven by Dirichlet's test after substitution <math display="inline">u=\tfrac{t^3}3 + xt.</math>

{{math|1=''y'' = Ai(''x'')}} satisfies the Airy equation
<math display="block">y'' - xy = 0.</math>
This equation has two [[linear independence|linearly independent]] solutions.
Up to scalar multiplication, {{math|Ai(''x'')}} is the solution subject to the condition {{math|''y'' → 0}} as {{math|''x'' → ∞}}.
The standard choice for the other solution is the Airy function of the second kind, denoted Bi(''x''). It is defined as the solution with the same amplitude of oscillation as {{math|Ai(''x'')}} as {{math|''x'' → −∞}} which differs in phase by {{math|''π''/2}}:
[[File:Plot of the Airy function Bi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the Airy function Bi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the Airy function {{math|Bi(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]
<math display="block">\operatorname{Bi}(x) = \frac{1}{\pi} \int_0^\infty \left[\exp\left(-\tfrac{t^3}{3} + xt\right) + \sin\left(\tfrac{t^3}{3} + xt\right)\,\right]dt.</math>
[[File:Plot of the derivative of the Airy function Bi'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the derivative of the Airy function Bi'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the derivative of the Airy function {{math|Bi'(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]