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
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gsl_sf_airy_Ai(x)
This routine computes the Airy function \(\operatorname{Ai}(x)\).
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gsl_sf_airy_Bi(x)
This routine computes the Airy function \(\operatorname{Bi}(x)\).
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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 \(x < 0\).
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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
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gsl_sf_airy_zero_Ai(s)
This routine computes the location of the \(s\)-th zero of the Airy
function \(\operatorname{Ai}(x)\).
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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
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gsl_sf_airy_zero_Ai_deriv(s)
This routine computes the location of the \(s\)-th zero of the Airy
function derivative \(\operatorname{Ai}'(x)\).
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gsl_sf_airy_zero_Bi_deriv(s)
This routine computes the location of the \(s\)-th zero of the Airy
function derivative \(\operatorname{Bi}'(x)\).