naginterfaces.library.specfun.kelvin_​bei

naginterfaces.library.specfun.kelvin_bei(x)

kelvin_bei returns a value for the Kelvin function $bei\left(x\right)$.

For full information please refer to the NAG Library document for s19ab

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s19abf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

beifloat

The value of the function at $x$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $\left|x\right|\le ⟨value⟩$.

$\left|x\right|$ is too large for an accurate result to be returned and the function returns zero.

Notes

kelvin_bei evaluates an approximation to the Kelvin function $bei\left(x\right)$.

Note: $bei(-x)=bei\left(x\right)$, so the approximation need only consider $x\ge 0.0$.

The function is based on several Chebyshev expansions:

For $0\le x\le 5$,

$$bei\left(x\right)=\frac{x^2}{4}\sum _{r=0}^{\prime }{a}_r{T}_r\left(t\right)\text{, with}t=2{\left(\frac{x}{5}\right)}^4-1\text{;}$$

For $x>5$,

$$bei\left(x\right)=\frac{e^{x/\sqrt{2}}}{\sqrt{2\pi x}}[(1+\frac{1}{x}a\left(t\right))\sin \left(\alpha \right)-\frac{1}{x}b\left(t\right)\cos \left(\alpha \right)]$$

$$+\frac{e^{x/\sqrt{2}}}{\sqrt{2\pi x}}[(1+\frac{1}{x}c\left(t\right))\cos \left(\beta \right)-\frac{1}{x}d\left(t\right)\sin \left(\beta \right)]$$

where $\alpha =\frac{x}{\sqrt{2}}-\frac{\pi }{8}$, $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$,

and $a\left(t\right)$, $b\left(t\right)$, $c\left(t\right)$, and $d\left(t\right)$ are expansions in the variable $t=\frac{10}{x}-1$.

When $x$ is sufficiently close to zero, the result is computed as $bei\left(x\right)=\frac{x^2}{4}$. If this result would underflow, the result returned is $bei\left(x\right)=0.0$.

For large $x$, there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner;, therefore, the function must fail.

References

NIST Digital Library of Mathematical Functions