naginterfaces.library.specfun.erfc_​complex

naginterfaces.library.specfun.erfc_complex(z)

erfc_complex computes values of the function $w\left(z\right)=e^{-z^2}erfc(-iz)$, for complex $z$.

For full information please refer to the NAG Library document for s15dd

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s15ddf.html

Parameters

zcomplex

The argument $z$ of the function.

Returns

erfczcomplex

The value of the function evaluated at $z$, $w\left(z\right)=e^{-z^2}erfc(-iz)$.

Raises

NagValueError

(errno $5$)

Result has no precision when entered with argument $z=(⟨value⟩,⟨value⟩)$.

Warns

NagAlgorithmicWarning

(errno $1$)

Real part of result overflows when entered with argument $z=(⟨value⟩,⟨value⟩)$.

(errno $2$)

Imaginary part of result overflows when entered with argument $z=(⟨value⟩,⟨value⟩)$.

(errno $3$)

Both real and imaginary parts of result overflow when entered with argument $z=(⟨value⟩,⟨value⟩)$.

(errno $4$)

Result has less than half precision when entered with argument $z=(⟨value⟩,⟨value⟩)$.

Notes

erfc_complex computes values of the function $w\left(z\right)=e^{-z^2}erfc(-iz)$, where $erfc\left(z\right)$ is the complementary error function

$$erfc\left(z\right)=\frac{2}{\sqrt{\pi }}{\int }_z^{\infty }{e}^{-t^2}dt\text{,}$$

for complex $z$. The method used is that in Gautschi (1970) for $z$ in the first quadrant of the complex plane, and is extended for $z$ in other quadrants via the relations $w(-z)=2e^{-z^2}-w\left(z\right)$ and $w\left(\overline{z}\right)=\overline{w(-z)}$. Following advice in Gautschi (1970) and van der Laan and Temme (1984), the code in Gautschi (1969) has been adapted to work in various precisions up to $18$ decimal places. The real part of $w\left(z\right)$ is sometimes known as the Voigt function.

References

Gautschi, W, 1969, Algorithm 363: Complex error function, Comm. ACM (12), 635

Gautschi, W, 1970, Efficient computation of the complex error function, SIAM J. Numer. Anal. (7), 187–198

van der Laan, C G and Temme, N M, 1984, Calculation of special functions: the gamma function, the exponential integrals and error-like functions, CWI Tract 10, Centre for Mathematics and Computer Science, Amsterdam