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.IX Title "build::libcerf::src::libcerf::man::w_of_z 3"
.TH build::libcerf::src::libcerf::man::w_of_z 3 2025-10-25 "perl v5.42.0" "libcerf manual"
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.SH NAME
w_of_z, im_w_of_x \- Faddeeva\*(Aqs rescaled complex error function
.SH SYNOPSIS
.IX Header "SYNOPSIS"
\&\fB#include <cerf.h\fR>
.PP
\&\fBdouble _Complex w_of_z ( double _Complex z );\fR
.PP
\&\fBdouble im_w_of_x ( double x );\fR
.SH DESCRIPTION
.IX Header "DESCRIPTION"
Faddeeva\*(Aqs rescaled complex error function w(z), also called the plasma dispersion function.
.PP
\&\fBw_of_z\fR returns w(z) = exp(\-z^2) * erfc(\-i*z).
.PP
\&\fBim_w_of_x\fR returns Im[w(x)].
.SH REFERENCES
.IX Header "REFERENCES"
To compute w(z), a combination of two algorithms is used:
.PP
For sufficiently large |z|, a continued\-fraction expansion similar
to those described by Gautschi (1970) and Poppe & Wijers (1990).
.PP
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.
.PP
To compute Im[w(x)], Chebyshev polynomials and continous fractions are used.
.PP
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).
.PP
Walter Gautschi, "Efficient computation of the complex error function," SIAM J. Numer. Anal. 7, 187 (1970).
.PP
G. P. M. Poppe and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Trans. Math. Soft. 16, 38 (1990).
.PP
Mofreh R. Zaghloul and Ahmed N. Ali, "Algorithm 916: Computing the Faddeyeva and Voigt Functions," ACM Trans. Math. Soft. 38, 15 (2011).
.PP
Steven G. Johnson, http://ab\-initio.mit.edu/Faddeeva
.PP
The implementation of \fBim_w_of_x\fR 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.
.PP
The piecewise polynomial approximation for intermediate values of |x| is explained
in Wuttke and Kleinknecht, "Code generation for piecewise Chebyshev approximation"
(submitted; preprint available from the authors).
.SH "SEE ALSO"
.IX Header "SEE ALSO"
This function is used to compute several other complex error functions:
\&\fBdawson\|(3)\fR, \fBvoigt\|(3)\fR, \fBcerf\|(3)\fR, \fBerfcx\|(3)\fR, \fBerfi\|(3)\fR.
.PP
Homepage: http://apps.jcns.fz\-juelich.de/libcerf
.SH AUTHORS
.IX Header "AUTHORS"
Steven G. Johnson, http://math.mit.edu/~stevenj,
  Massachusetts Institute of Technology,
  researched the numerics, and implemented the Faddeeva function.
.PP
Joachim Wuttke <j.wuttke@fz\-juelich.de>, Forschungszentrum Juelich,
  reorganized the code into a library, and wrote this man page.
.PP
Please report bugs to the authors.
.SH COPYING
.IX Header "COPYING"
Copyright (c) 2012 Massachusetts Institute of Technology
.PP
Copyright (c) 2013\-25 Forschungszentrum Juelich GmbH
.PP
Software: MIT License.
.PP
This documentation: Creative Commons Attribution Share Alike.