Airy Function - Properties

Properties

The values of Ai(x) and Bi(x) and their derivatives at x = 0 are given by

\begin{align} \mathrm{Ai}(0) &{}= \frac{1}{3^{2/3}\Gamma(\tfrac23)}, & \quad \mathrm{Ai}'(0) &{}= -\frac{1}{3^{1/3}\Gamma(\tfrac13)}, \\ \mathrm{Bi}(0) &{}= \frac{1}{3^{1/6}\Gamma(\tfrac23)}, & \quad \mathrm{Bi}'(0) &{}= \frac{3^{1/6}}{\Gamma(\tfrac13)}. \end{align}

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

Read more about this topic:  Airy Function

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