naginterfaces.library.specfun.airy_​ai_​deriv

naginterfaces.library.specfun.airy_ai_deriv(x)

airy_ai_deriv returns a value of the derivative of the Airy function $Ai\left(x\right)$.

For full information please refer to the NAG Library document for s17aj

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s17ajf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

aidfloat

The value of the derivative of the Airy function $Ai\left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x\le ⟨value⟩$.

$x$ is too large and positive. The function returns zero.

(errno $2$)

On entry, $x=⟨value⟩$.

Constraint: $x\ge ⟨value⟩$.

$x$ is too large and negative. The function returns zero.

Notes

airy_ai_deriv evaluates an approximation to the derivative of the Airy function $Ai\left(x\right)$. It is based on a number of Chebyshev expansions.

For $x<-5$,

$${\text{Ai}}^{\prime }\left(x\right)=\sqrt[4]{4-x}[a\left(t\right)\cos \left(z\right)+\frac{b\left(t\right)}{\zeta }\sin \left(z\right)]\text{,}$$

where $z=\frac{\pi }{4}+\zeta$, $\zeta =\frac{2}{3}\sqrt{-x^3}$ and $a\left(t\right)$ and $b\left(t\right)$ are expansions in variable $t=-2{\left(\frac{5}{x}\right)}^3-1$.

For $-5\le x\le 0$,

$${\text{Ai}}^{\prime }\left(x\right)=x^{2}f\left(t\right)-g\left(t\right)\text{,}$$

where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^3-1$.

For $0<x<4.5$,

$${\text{Ai}}^{\prime }\left(x\right)=e^{-11x/8}y\left(t\right)\text{,}$$

where $y\left(t\right)$ is an expansion in $t=4\left(\frac{x}{9}\right)-1$.

For $4.5\le x<9$,

$${\text{Ai}}^{\prime }\left(x\right)=e^{-5x/2}v\left(t\right)\text{,}$$

where $v\left(t\right)$ is an expansion in $t=4\left(\frac{x}{9}\right)-3$.

For $x\ge 9$,

$${\text{Ai}}^{\prime }\left(x\right)=\sqrt[4]{4x}{e}^{-z}u\left(t\right)\text{,}$$

where $z=\frac{2}{3}\sqrt{x^3}$ and $u\left(t\right)$ is an expansion in $t=2\left(\frac{18}{z}\right)-1$.

For $\left|x\right|<$ the square of the machine precision, the result is set directly to ${\text{Ai}}^{\prime }\left(0\right)$. This both saves time and avoids possible intermediate underflows.

For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy and so the function must fail. This occurs for $x<-{\left(\frac{\sqrt{\pi }}{ϵ}\right)}^{4/7}$, where $ϵ$ is the machine precision.

For large positive arguments, where ${\text{Ai}}^{\prime }$ decays in an essentially exponential manner, there is a danger of underflow so the function must fail.

References

NIST Digital Library of Mathematical Functions