build::libcerf::src::libcerf::man::w_of_z(3)

NAME
w_of_z, im_w_of_x - Faddeeva's rescaled complex error function

SYNOPSIS
#include <cerf.h>

double _Complex w_of_z ( double _Complex z );

double im_w_of_x ( double x );

DESCRIPTION
Faddeeva's rescaled complex error function w(z), also called the plasma dispersion function.

w_of_z returns w(z) = exp(-z^2) * erfc(-i*z).

im_w_of_x returns Im[w(x)].

REFERENCES
To compute w(z), a combination of two algorithms is used:

For sufficiently large |z|, a continued-fraction expansion similar to those described by Gautschi (1970) and Poppe & Wijers (1990).

Otherwise, Algorithm 916 by Zaghloul & Ali (2011), which is generally competitive at small |z|, and more accurate than the Poppe & Wijers expansion in some regions, e.g. in the vicinity of z=1+i.

To compute Im[w(x)], Chebyshev polynomials and continous fractions are used.

Milton Abramowitz and Irene M. Stegun, "Handbook of Mathematical Functions", National Bureau of Standards (1964): Formula (7.1.3) introduces the nameless function w(z).

Walter Gautschi, "Efficient computation of the complex error function," SIAM J. Numer. Anal. 7, 187 (1970).

G. P. M. Poppe and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Trans. Math. Soft. 16, 38 (1990).

Mofreh R. Zaghloul and Ahmed N. Ali, "Algorithm 916: Computing the Faddeyeva and Voigt Functions," ACM Trans. Math. Soft. 38, 15 (2011).

Steven G. Johnson, http://ab-initio.mit.edu/Faddeeva

The implementation of im_w_of_x is self-contained. It has been entirely rewritten for libcerf-2.5, and now guarantees a relative accuracy of 2.37e-16 for |x| < 12.

The piecewise polynomial approximation for intermediate values of |x| is explained in Wuttke and Kleinsorge, "Code generation for piecewise Chebyshev approximation", ACM Trans. Math. Softw. 52, 13 (2026), https://doi.org/10.1145/3805698.

SEE ALSO
This function is used to compute several other complex error functions: dawson(3), voigt(3), cerf(3), erfcx(3), erfi(3).

Homepage: https://jugit.fz-juelich.de/mlz/lib/cerf

AUTHORS
Steven G. Johnson, Massachusetts Institute of Technology,
  researched the numerics, and implemented the Faddeeva function.

Joachim Wuttke, Forschungszentrum Juelich,
  reorganized the code into a library, and wrote this man page.

CONTACT
Please report bugs to the maintainer:

Joachim Wuttke <j.wuttke@fz-juelich.de>

COPYING
Copyright (c) 2012 Massachusetts Institute of Technology

Copyright (c) 2013-25 Forschungszentrum Juelich GmbH

Software: MIT License.

This documentation: Creative Commons Attribution Share Alike.