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 $${\displaystyle {\frac {d^{2}y}{dx^{2}}}-xy=0,}$$ known as the Airy equation or the Stokes equation.

Because the solution of the linear differential equation $${\displaystyle {\frac {d^{2}y}{dx^{2}}}-ky=0}$$ 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).