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 known as the Airy equation or the Stokes equation. This 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). For real values of x, the Airy function of the first kind can be defined by the improper Riemann integral: which converges by Dirichlet's test. For any real number x there is a positive real number M such that function is increasing, unbounded and convex with continuous and unbounded derivative on interval The convergence of the integral on this interval can be proven by Dirichlet's test after substitution y = Ai(x) satisfies the Airy equation This equation has two linearly independent solutions. Up to scalar multiplication, Ai(x) is the solution subject to the condition y → 0 as 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 Ai(x) as x → −∞ which differs in phase by π/2: The values of Ai(x) and Bi(x) and their derivatives at x = 0 are given by Here, Γ denotes the Gamma function. It follows that the Wronskian of Ai(x) and Bi(x) is 1/π. When x is positive, Ai(x) is positive, convex, and decreasing exponentially to zero, while Bi(x) is positive, convex, and increasing exponentially. When x is negative, Ai(x) and Bi(x) oscillate around zero with ever-increasing frequency and ever-decreasing amplitude. This is supported by the asymptotic formulae below for the Airy functions. The Airy functions are orthogonal in the sense that again using an improper Riemann integral. Real zeros of Ai(x) and its derivative Ai'(x) Neither Ai(x) nor its derivative Ai(x) have positive real zeros.