Airy Function - Relation To Other Special Functions

Relation To Other Special Functions

For positive arguments, the Airy functions are related to the modified Bessel functions:

\begin{align} \mathrm{Ai}(x) &{}= \frac1\pi \sqrt{\frac13 x} \, K_{1/3}\left(\tfrac23 x^{3/2}\right), \\ \mathrm{Bi}(x) &{}= \sqrt{\frac13 x} \left(I_{1/3}\left(\tfrac23 x^{3/2}\right) + I_{-1/3}\left(\tfrac23 x^{3/2}\right)\right). \end{align}

Here, I±1/3 and K1/3 are solutions of

The first derivative of Airy function is

\mathrm{Ai'}(x) = - \frac{x} {\pi \sqrt{3}} \, K_{2/3}\left(\tfrac23 x^{3/2}\right) .

Functions and can be represented in terms of rapidly converged integrals (see also modified Bessel functions )

For negative arguments, the Airy function are related to the Bessel functions:

\begin{align} \mathrm{Ai}(-x) &{}= \frac13 \sqrt{x} \left(J_{1/3}\left(\tfrac23 x^{3/2}\right) + J_{-1/3}\left(\tfrac23 x^{3/2}\right)\right), \\ \mathrm{Bi}(-x) &{}= \sqrt{\frac13 x} \left(J_{-1/3}\left(\tfrac23 x^{3/2}\right) - J_{1/3}\left(\tfrac23 x^{3/2}\right)\right). \end{align}

Here, J±1/3 are solutions of .

The Scorer's functions solve the equation . They can also be expressed in terms of the Airy functions:

\begin{align} \mathrm{Gi}(x) &{}= \mathrm{Bi}(x) \int_x^\infty \mathrm{Ai}(t) \, dt + \mathrm{Ai}(x) \int_0^x \mathrm{Bi}(t) \, dt, \\ \mathrm{Hi}(x) &{}= \mathrm{Bi}(x) \int_{-\infty}^x \mathrm{Ai}(t) \, dt - \mathrm{Ai}(x) \int_{-\infty}^x \mathrm{Bi}(t) \, dt. \end{align}

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