Editing Airy function (section)

{{short description|Special function in the physical sciences}}
{{About|the Airy special function|the Airy stress function employed in solid mechanics|Stress functions|the Airy disk function that describes the optics diffraction pattern through a circular aperture|Airy disk|generic Airy distribution arising from optical resonance between two mirrors|Fabry–Pérot interferometer|the Airy equation as an example of a linear dispersive partial differential equation|Dispersive partial differential equation}}

In the physical sciences, the '''Airy function''' (or '''Airy function of the first kind''') {{math|'''Ai(''x'')'''}} is a [[special function]] named after the British astronomer [[George Biddell Airy]] (1801–1892). The function Ai(''x'') and the related function '''Bi(''x'')''', are [[Linear independence|linearly independent]] solutions to the [[differential equation]]
<math display="block">\frac{d^2y}{dx^2} - xy = 0 , </math>
known as the '''Airy equation''' or the '''Stokes equation'''.

Because the solution of the linear differential equation
<math display="block">\frac{d^2y}{dx^2} - ky = 0</math>
is oscillatory for {{math|''k''<0}} and exponential for {{math|''k''>0}}, the Airy functions are oscillatory for {{math|''x''<0}} and exponential for {{math|''x''>0}}.  In fact, the Airy equation is the simplest second-order [[linear differential equation]] with a turning point (a point where the character of the solutions changes from oscillatory to exponential).

[[File:Plot of the Airy function Ai(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 {{math|Ai(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the Airy function {{math|Ai(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]

[[File:Plot of the derivative of the Airy function Ai'(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 {{math|Ai'(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the derivative of the Airy function {{math|Ai'(''z'')}} in the complex plane from {{math|-2 - 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]