naginterfaces.library.specfun.airy_​bi_​real

naginterfaces.library.specfun.airy_bi_real(x)

airy_bi_real returns a value of the Airy function, $Bi\left(x\right)$.

For full information please refer to the NAG Library document for s17ah

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s17ahf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

bifloat

The value 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_real evaluates an approximation to the Airy function $Bi\left(x\right)$. It is based on a number of Chebyshev expansions.

For $x<-5$,

$$Bi\left(x\right)=\frac{a\left(t\right)\cos \left(z\right)+b\left(t\right)\sin \left(z\right)}{{(-x)}^{1/4}}\text{,}$$

where $z=\frac{\pi }{4}+\frac{2}{3}\sqrt{-x^3}$ 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 $-5\le x\le 0$,

$$Bi\left(x\right)=\sqrt{3}(f\left(t\right)+xg\left(t\right))\text{,}$$

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

For $0<x<4.5$,

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

where $y$ is an expansion in $t=4x/9-1$.

For $4.5\le x<9$,

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

where $v$ is an expansion in $t=4x/9-3$.

For $x\ge 9$,

$$Bi\left(x\right)=\frac{e^{z}u\left(t\right)}{x^{1/4}}\text{,}$$

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

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

For large negative arguments, it becomes impossible to calculate the phase of the oscillating function with any accuracy so the function must fail. This occurs if $x<-{\left(\frac{3}{2ϵ}\right)}^{2/3}$, where $ϵ$ is the machine precision.

For large positive arguments, there is a danger of causing overflow since Bi grows in an essentially exponential manner, so the function must fail.

References

NIST Digital Library of Mathematical Functions