naginterfaces.library.specfun.dawson_​vector

naginterfaces.library.specfun.dawson_vector(x)

dawson_vector returns an array of values for Dawson’s Integral, $F\left(x\right)$.

For full information please refer to the NAG Library document for s15at

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s15atf.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 $\left(n\right)$

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

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Notes

dawson_vector evaluates approximations for Dawson’s Integral

$$F\left(x\right)=e^{-x^2}{\int }_0^x{e}^{t^2}dt\text{,}$$

for an array of arguments $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

The function is based on two Chebyshev expansions:

For $0<\left|x\right|\le 4$,

$$F\left(x\right)=x\sum _{r=0}^{\prime }{a}_r{T}_r\left(t\right)\text{, where}t=2{\left(\frac{x}{4}\right)}^2-1\text{.}$$

For $\left|x\right|>4$,

$$F\left(x\right)=\frac{1}{x}\sum _{r=0}^{\prime }{b}_r{T}_r\left(t\right)\text{, where}t=2{\left(\frac{4}{x}\right)}^2-1\text{.}$$

For $\left|x\right|$ near zero, $F\left(x\right)\simeq x$, and for $\left|x\right|$ large, $F\left(x\right)\simeq \frac{1}{2x}$. These approximations are used for those values of $x$ for which the result is correct to machine precision.

References

NIST Digital Library of Mathematical Functions