15.3 Airy Functions

The Airy functions \({\rm Ai}(x)\) and \({\rm Bi}(x)\) are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Section 10.4 and DLMF 9.

The Airy differential equation is:

The numerically satisfactory pair of solutions (DLMF 9.2#T1) on the real line are \(y = {\rm Ai}(x)\) and \(y = {\rm Bi}(x).\)

These two solutions are oscillatory for \(x < 0\). \({\rm Ai}(x)\) is the solution subject to the condition that \(y\rightarrow 0\) as \(x\rightarrow +\infty,\) and \({\rm Bi}(x)\) is the second solution with the same amplitude as \({\rm Ai}(x)\) as \(x\rightarrow-\infty\) which differs in phase by \(\pi/2.\) Also, \({\rm Bi}(x)\) is unbounded as \(x\rightarrow +\infty.\)

If the argument \(x\) is a real or complex floating point number, the numerical value of the function is returned.

Function: airy_ai (x) ¶

The Airy function \({\rm Ai}(x).\) See A&S eqn 10.4.2 and DLMF 9.

See also airy_bi, airy_dai, and airy_dbi.

Categories: Airy functions · Special functions ·

Function: airy_dai (x) ¶

The derivative of the Airy function \({\rm Ai}(x)\) :

\[{\rm airy\_dai}(x) = {d\over dx}{\rm Ai}(x) \]

See airy_ai.

Categories: Airy functions · Special functions ·

Function: airy_bi (x) ¶

The Airy function \({\rm Bi}(x)\) . See A&S eqn 10.4.3 and DLMF 9.

See airy_ai, and airy_dbi.

Categories: Airy functions · Special functions ·

Function: airy_dbi (x) ¶

The derivative of the Airy function \({\rm Bi}(x)\) :

\[{\rm airy\_dbi}(x) = {d\over dx}{\rm Bi}(x) \]

See airy_ai, and airy_bi.

Categories: Airy functions · Special functions ·