Airy function

In the physical sciences, the Airy function (or Airy function of the first kind) $Ai(x)$ 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 linearly independent solutions to the differential equation ${\frac {d^{2}y}{dx^{2}}}-xy=0,$ known as the Airy equation or the Stokes equation.

Because the solution of the linear differential equation ${\frac {d^{2}y}{dx^{2}}}-ky=0$ is oscillatory for $k <0$ and exponential for $k >0$ , the Airy functions are oscillatory for $x <0$ and exponential for $x >0$ . 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).