Airy function

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In mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function named after the British astronomer George Biddell Airy. The function Ai ⁡ ( x ) {\displaystyle \operatorname {Ai} (x)} and the related function B i ( x ) {\displaystyle \mathbf {Bi({\boldsymbol {x}})} } are linearly independent solutions to the differential equation d 2 y d x 2 − x y = 0 , {\displaystyle {\frac {d^{2}y}{dx^{2}}}-xy=0,} known as the Airy equation or the Stokes equation. Because the solution of the linear differential equation d 2 y d x 2 − k y = 0 {\displaystyle {\frac {d^{2}y}{dx^{2}}}-ky=0} is oscillatory for k < 0 {\displaystyle k<0} and exponential for k > 0 {\displaystyle k>0} , the Airy functions are oscillatory for x < 0 {\displaystyle x<0} and exponential for x > 0 {\displaystyle 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).