naginterfaces.library.specfun.kelvin_​ker

naginterfaces.library.specfun.kelvin_ker(x)

kelvin_ker returns a value for the Kelvin function $ker\left(x\right)$.

For full information please refer to the NAG Library document for s19ac

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s19acf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

kerfloat

The value of the function at $x$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$. The function returns zero.

Constraint: $x\le ⟨value⟩$.

$x$ is too large, the result underflows and the function returns zero.

(errno $2$)

On entry, $x=⟨value⟩$.

Constraint: $x>0.0$.

The function is undefined and returns zero.

Notes

kelvin_ker evaluates an approximation to the Kelvin function $ker\left(x\right)$.

Note: for $x<0$ the function is undefined and at $x=0$ it is infinite so we need only consider $x>0$.

The function is based on several Chebyshev expansions:

For $0<x\le 1$,

$$ker\left(x\right)=-f\left(t\right)\log \left(x\right)+\frac{\pi }{16}{x}^{2}g\left(t\right)+y\left(t\right)$$

where $f\left(t\right)$, $g\left(t\right)$ and $y\left(t\right)$ are expansions in the variable $t=2x^4-1$.

For $1<x\le 3$,

$$ker\left(x\right)=exp(-\frac{11}{16}x)q\left(t\right)$$

where $q\left(t\right)$ is an expansion in the variable $t=x-2$.

For $x>3$,

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

where $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$, and $c\left(t\right)$ and $d\left(t\right)$ are expansions in the variable $t=\frac{6}{x}-1$.

When $x$ is sufficiently close to zero, the result is computed as

$$ker\left(x\right)=-\gamma -\log \left(\frac{x}{2}\right)+(\pi -\frac{3}{8}{x}^2)\frac{x^2}{16}$$

and when $x$ is even closer to zero, simply as $ker\left(x\right)=-\gamma -\log \left(\frac{x}{2}\right)$.

For large $x$, $ker\left(x\right)$ is asymptotically given by $\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}$ and this becomes so small that it cannot be computed without underflow and the function fails.

References

NIST Digital Library of Mathematical Functions