naginterfaces.library.specfun.airy_​bi_​deriv

naginterfaces.library.specfun.airy_bi_deriv(x)

airy_bi_deriv returns a value for the derivative of the Airy function $Bi\left(x\right)$.

For full information please refer to the NAG Library document for s17ak

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s17akf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

bidfloat

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

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_bi_deriv calculates an approximate value for the derivative of the Airy function $Bi\left(x\right)$. It is based on a number of Chebyshev expansions.

For $x<-5$,

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

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 the variable $t=-2{\left(\frac{5}{x}\right)}^3-1$.

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

$${\text{Bi}}^{\prime }\left(x\right)=\sqrt{3}(x^{2}f\left(t\right)+g\left(t\right))\text{,}$$

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

For $0<x<4.5$,

$${\text{Bi}}^{\prime }\left(x\right)=e^{3x/2}y\left(t\right)\text{,}$$

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

For $4.5\le x<9$,

$${\text{Bi}}^{\prime }\left(x\right)=e^{21x/8}u\left(t\right)\text{,}$$

where $u\left(t\right)$ is an expansion in $t=4x/9-3$.

For $x\ge 9$,

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

where $z=\frac{2}{3}\sqrt{x^3}$ and $v\left(t\right)$ 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 ${\text{Bi}}^{\prime }\left(0\right)$. This saves time and avoids possible underflows in calculation.

For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy 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 ${\text{Bi}}^{\prime }$ grows in an essentially exponential manner, there is a danger of overflow so the function must fail.

References

NIST Digital Library of Mathematical Functions