naginterfaces.library.specfun.kelvin_​kei

naginterfaces.library.specfun.kelvin_kei(x)

kelvin_kei returns a value for the Kelvin function $kei\left(x\right)$.

For full information please refer to the NAG Library document for s19ad

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s19adf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

keifloat

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: $x\ge 0.0$.

The function is undefined and returns zero.

Notes

kelvin_kei evaluates an approximation to the Kelvin function $kei\left(x\right)$.

Note: for $x<0$ the function is undefined, so we need only consider $x\ge 0$.

The function is based on several Chebyshev expansions:

For $0\le x\le 1$,

$$kei\left(x\right)=-\frac{\pi }{4}f\left(t\right)+\frac{x^2}{4}[-g\left(t\right)\log \left(x\right)+v\left(t\right)]$$

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

For $1<x\le 3$,

$$kei\left(x\right)=exp(-\frac{9}{8}x)u\left(t\right)$$

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

For $x>3$,

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

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

For $x<0$, the function is undefined, and hence the function fails and returns zero.

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

$$kei\left(x\right)=-\frac{\pi }{4}+(1-\gamma -\log \left(\frac{x}{2}\right))\frac{x^2}{4}$$

and when $x$ is even closer to zero simply as

$$kei\left(x\right)=-\frac{\pi }{4}\text{.}$$

For large $x$, $kei\left(x\right)$ 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