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=pod
=begin html
<link rel="stylesheet" href="podstyle.css" type="text/css" />
=end html
=head1 NAME
w_of_z, im_w_of_x - Faddeeva's rescaled complex error function
=head1 SYNOPSIS
B<#include <cerf.h>>
B<double _Complex w_of_z ( double _Complex z );>
B<double im_w_of_x ( double x );>
=head1 DESCRIPTION
Faddeeva's rescaled complex error function w(z), also called the plasma dispersion function.
B<w_of_z> returns w(z) = exp(-z^2) * erfc(-i*z).
B<im_w_of_x> returns Im[w(x)].
=head1 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 B<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 Kleinknecht, "Code generation for piecewise Chebyshev approximation"
(submitted; preprint available from the authors).
=head1 SEE ALSO
This function is used to compute several other complex error functions:
B<dawson(3)>, B<voigt(3)>, B<cerf(3)>, B<erfcx(3)>, B<erfi(3)>.
Homepage: http://apps.jcns.fz-juelich.de/libcerf
=head1 AUTHORS
Steven G. Johnson, http://math.mit.edu/~stevenj,
Massachusetts Institute of Technology,
researched the numerics, and implemented the Faddeeva function.
Joachim Wuttke <j.wuttke@fz-juelich.de>, Forschungszentrum Juelich,
reorganized the code into a library, and wrote this man page.
Please report bugs to the authors.
=head1 COPYING
Copyright (c) 2012 Massachusetts Institute of Technology
Copyright (c) 2013 Forschungszentrum Juelich GmbH
Software: MIT License.
This documentation: Creative Commons Attribution Share Alike.
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