.\" Automatically generated by Pod::Man 4.14 (Pod::Simple 3.43)
.\"
.\" Standard preamble:
.\" ========================================================================
.de Sp \" Vertical space (when we can't use .PP)
.if t .sp .5v
.if n .sp
..
.de Vb \" Begin verbatim text
.ft CW
.nf
.ne \\$1
..
.de Ve \" End verbatim text
.ft R
.fi
..
.\" Set up some character translations and predefined strings.  \*(-- will
.\" give an unbreakable dash, \*(PI will give pi, \*(L" will give a left
.\" double quote, and \*(R" will give a right double quote.  \*(C+ will
.\" give a nicer C++.  Capital omega is used to do unbreakable dashes and
.\" therefore won't be available.  \*(C` and \*(C' expand to `' in nroff,
.\" nothing in troff, for use with C<>.
.tr \(*W-
.ds C+ C\v'-.1v'\h'-1p'\s-2+\h'-1p'+\s0\v'.1v'\h'-1p'
.ie n \{\
.    ds -- \(*W-
.    ds PI pi
.    if (\n(.H=4u)&(1m=24u) .ds -- \(*W\h'-12u'\(*W\h'-12u'-\" diablo 10 pitch
.    if (\n(.H=4u)&(1m=20u) .ds -- \(*W\h'-12u'\(*W\h'-8u'-\"  diablo 12 pitch
.    ds L" ""
.    ds R" ""
.    ds C` ""
.    ds C' ""
'br\}
.el\{\
.    ds -- \|\(em\|
.    ds PI \(*p
.    ds L" ``
.    ds R" ''
.    ds C`
.    ds C'
'br\}
.\"
.\" Escape single quotes in literal strings from groff's Unicode transform.
.ie \n(.g .ds Aq \(aq
.el       .ds Aq '
.\"
.\" If the F register is >0, we'll generate index entries on stderr for
.\" titles (.TH), headers (.SH), subsections (.SS), items (.Ip), and index
.\" entries marked with X<> in POD.  Of course, you'll have to process the
.\" output yourself in some meaningful fashion.
.\"
.\" Avoid warning from groff about undefined register 'F'.
.de IX
..
.nr rF 0
.if \n(.g .if rF .nr rF 1
.if (\n(rF:(\n(.g==0)) \{\
.    if \nF \{\
.        de IX
.        tm Index:\\$1\t\\n%\t"\\$2"
..
.        if !\nF==2 \{\
.            nr % 0
.            nr F 2
.        \}
.    \}
.\}
.rr rF
.\"
.\" Accent mark definitions (@(#)ms.acc 1.5 88/02/08 SMI; from UCB 4.2).
.\" Fear.  Run.  Save yourself.  No user-serviceable parts.
.    \" fudge factors for nroff and troff
.if n \{\
.    ds #H 0
.    ds #V .8m
.    ds #F .3m
.    ds #[ \f1
.    ds #] \fP
.\}
.if t \{\
.    ds #H ((1u-(\\\\n(.fu%2u))*.13m)
.    ds #V .6m
.    ds #F 0
.    ds #[ \&
.    ds #] \&
.\}
.    \" simple accents for nroff and troff
.if n \{\
.    ds ' \&
.    ds ` \&
.    ds ^ \&
.    ds , \&
.    ds ~ ~
.    ds /
.\}
.if t \{\
.    ds ' \\k:\h'-(\\n(.wu*8/10-\*(#H)'\'\h"|\\n:u"
.    ds ` \\k:\h'-(\\n(.wu*8/10-\*(#H)'\`\h'|\\n:u'
.    ds ^ \\k:\h'-(\\n(.wu*10/11-\*(#H)'^\h'|\\n:u'
.    ds , \\k:\h'-(\\n(.wu*8/10)',\h'|\\n:u'
.    ds ~ \\k:\h'-(\\n(.wu-\*(#H-.1m)'~\h'|\\n:u'
.    ds / \\k:\h'-(\\n(.wu*8/10-\*(#H)'\z\(sl\h'|\\n:u'
.\}
.    \" troff and (daisy-wheel) nroff accents
.ds : \\k:\h'-(\\n(.wu*8/10-\*(#H+.1m+\*(#F)'\v'-\*(#V'\z.\h'.2m+\*(#F'.\h'|\\n:u'\v'\*(#V'
.ds 8 \h'\*(#H'\(*b\h'-\*(#H'
.ds o \\k:\h'-(\\n(.wu+\w'\(de'u-\*(#H)/2u'\v'-.3n'\*(#[\z\(de\v'.3n'\h'|\\n:u'\*(#]
.ds d- \h'\*(#H'\(pd\h'-\w'~'u'\v'-.25m'\f2\(hy\fP\v'.25m'\h'-\*(#H'
.ds D- D\\k:\h'-\w'D'u'\v'-.11m'\z\(hy\v'.11m'\h'|\\n:u'
.ds th \*(#[\v'.3m'\s+1I\s-1\v'-.3m'\h'-(\w'I'u*2/3)'\s-1o\s+1\*(#]
.ds Th \*(#[\s+2I\s-2\h'-\w'I'u*3/5'\v'-.3m'o\v'.3m'\*(#]
.ds ae a\h'-(\w'a'u*4/10)'e
.ds Ae A\h'-(\w'A'u*4/10)'E
.    \" corrections for vroff
.if v .ds ~ \\k:\h'-(\\n(.wu*9/10-\*(#H)'\s-2\u~\d\s+2\h'|\\n:u'
.if v .ds ^ \\k:\h'-(\\n(.wu*10/11-\*(#H)'\v'-.4m'^\v'.4m'\h'|\\n:u'
.    \" for low resolution devices (crt and lpr)
.if \n(.H>23 .if \n(.V>19 \
\{\
.    ds : e
.    ds 8 ss
.    ds o a
.    ds d- d\h'-1'\(ga
.    ds D- D\h'-1'\(hy
.    ds th \o'bp'
.    ds Th \o'LP'
.    ds ae ae
.    ds Ae AE
.\}
.rm #[ #] #H #V #F C
.\" ========================================================================
.\"
.IX Title "w_of_z 3"
.TH w_of_z 3 "2023-08-05" "perl v5.36.0" "libcerf manual"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
.\" way too many mistakes in technical documents.
.if n .ad l
.nh
.SH "NAME"
w_of_z, im_w_of_x \- Faddeeva's 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's 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, \*(L"Handbook of Mathematical Functions\*(R", National Bureau of Standards (1964): Formula (7.1.3) introduces the nameless function w(z).
.PP
Walter Gautschi, \*(L"Efficient computation of the complex error function,\*(R" \s-1SIAM J.\s0 Numer. Anal. 7, 187 (1970).
.PP
G. P. M. Poppe and C. M. J. Wijers, \*(L"More efficient computation of the complex error function,\*(R" \s-1ACM\s0 Trans. Math. Soft. 16, 38 (1990).
.PP
Mofreh R. Zaghloul and Ahmed N. Ali, \*(L"Algorithm 916: Computing the Faddeyeva and Voigt Functions,\*(R" \s-1ACM\s0 Trans. Math. Soft. 38, 15 (2011).
.PP
Steven G. Johnson, http://ab\-initio.mit.edu/Faddeeva
.SH "SEE ALSO"
.IX Header "SEE ALSO"
This function is used to compute several other complex error functions:
\&\fB\fBdawson\fB\|(3)\fR, \fB\fBvoigt\fB\|(3)\fR, \fB\fBcerf\fB\|(3)\fR, \fB\fBerfcx\fB\|(3)\fR, \fB\fBerfi\fB\|(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 Forschungszentrum Juelich GmbH
.PP
Software: \s-1MIT\s0 License.
.PP
This documentation: Creative Commons Attribution Share Alike.