Airy Functions and Derivatives

The Airy functions \(\operatorname{Ai}(x)\) and \(\operatorname{Bi}(x)\) are defined by the integral representations,

\[\begin{split}\operatorname{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos(\frac{1}{3} t^3 + xt) dt \\ \operatorname{Bi}(x) = \frac{1}{\pi} \int_0^\infty (e^{-\frac{1}{3} t^3 + xt} + \sin(\frac{1}{3} t^3 + xt)) dt\end{split}\]

For further information see Abramowitz & Stegun, Section 10.4.

Airy Functions

gsl_sf_airy_Ai(x)

This routine computes the Airy function \(\operatorname{Ai}(x)\).

gsl_sf_airy_Bi(x)

This routine computes the Airy function \(\operatorname{Bi}(x)\).

gsl_sf_airy_Ai_scaled(x)

This routine computes a scaled version of the Airy function \(\operatorname{S_A}(x) \operatorname{Ai}(x)\). For \(x > 0\) the scaling factor \(\operatorname{S_A}(x)\) is \(\exp(+(2/3) x^{3/2})\), and is \(1\) for \(x < 0\).

gsl_sf_airy_Bi_scaled(x)

This routine computes a scaled version of the Airy function \(\operatorname{S_B}(x) \operatorname{Bi}(x)\). For \(x > 0\) the scaling factor \(\operatorname{S_B}(x)\) is \(\exp(-(2/3) x^{3/2})\), and is \(1\) for \(x < 0\).

Zeros of Airy Functions

gsl_sf_airy_zero_Ai(s)

This routine computes the location of the \(s\)-th zero of the Airy function \(\operatorname{Ai}(x)\).

gsl_sf_airy_zero_Bi(s)

This routine computes the location of the \(s\)-th zero of the Airy function \(\operatorname{Bi}(x)\).

Zeros of Derivatives of Airy Functions

gsl_sf_airy_zero_Ai_deriv(s)

This routine computes the location of the \(s\)-th zero of the Airy function derivative \(\operatorname{Ai}'(x)\).

gsl_sf_airy_zero_Bi_deriv(s)

This routine computes the location of the \(s\)-th zero of the Airy function derivative \(\operatorname{Bi}'(x)\).

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