# 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 $$\operatorname{Bi}'(x)$$.