naginterfaces.library.specfun.kelvin_​kei_​vector

naginterfaces.library.specfun.kelvin_kei_vector(x)

kelvin_kei_vector returns an array of values for the Kelvin function $kei\left(x\right)$.

For full information please refer to the NAG Library document for s19ar

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s19arf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

The argument $x_{\text{i}}$ of the function, for $\text{i}=1,2,\dots ,n$.

Returns

ffloat, ndarray, shape $\left(n\right)$

$kei\left(x_i\right)$, the function values.

ivalidint, ndarray, shape $\left(n\right)$

$ivalid[\text{i}-1]$ contains the error code for $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

$x_i$ is too large, the result underflows. $f[\text{i}-1]$ contains zero. The threshold value is the same as for $errno$ = 1 in kelvin_kei().

$ivalid[i-1]=2$

$x_i<0.0$, the function is undefined. $f[\text{i}-1]$ contains $0.0$.

Raises

NagValueError

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $x$ was invalid.

Check $ivalid$ for more information.

Notes

kelvin_kei_vector evaluates an approximation to the Kelvin function $kei\left(x_i\right)$ for an array of arguments $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

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