<?xml version="1.0" ?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>libcerf\ manual</title> <meta http-equiv="content-type" content="text/html; charset=utf-8" /> <link rev="made" href="mailto:root@localhost" /> </head> <body> <link rel="stylesheet" href="podstyle.css" type="text/css" /> <h1 id="NAME">NAME</h1> <p>voigt - Voigt's function, convolution of Gaussian and Lorentzian</p> <h1 id="SYNOPSIS">SYNOPSIS</h1> <p><b>#include <cerf.h</b>></p> <p><b>double voigt ( double x, double sigma, double gamma );</b></p> <h1 id="DESCRIPTION">DESCRIPTION</h1> <p>The function <b>voigt</b> returns Voigt's convolution</p> <pre><code> voigt(x,sigma,gamma) = integral G(t,sigma) L(x-t,gamma) dt</code></pre> <p>of a Gaussian</p> <pre><code> G(x,sigma) = 1/sqrt(2*pi)/|sigma| * exp(-x^2/2/sigma^2)</code></pre> <p>and a Lorentzian</p> <pre><code> L(x,gamma) = |gamma| / pi / ( x^2 + gamma^2 ),</code></pre> <p>with the integral extending from -infinity to +infinity.</p> <p>If sigma=0, L(x,gamma) is returned. Conversely, if gamma=0, G(x,sigma) is returned.</p> <p>If sigma=gamma=0, the return value is Inf for x=0, and 0 for all other x. It is advisable to test input arguments to exclude this irregular case.</p> <h1 id="REFERENCES">REFERENCES</h1> <p>Formula (7.4.13) in Abramowitz & Stegun (1964) relates Voigt's convolution integral to Faddeeva's function <b>w_of_z</b>, upon which this implementation is based:</p> <pre><code> voigt(x,sigma,gamma) = Re[w(z)] / sqrt(2*pi) / |sigma|</code></pre> <p>with</p> <pre><code> z = (x+i*|gamma|) / sqrt(2) / |sigma|.</code></pre> <h1 id="SEE-ALSO">SEE ALSO</h1> <p><b>voigt_hwhm(3)</b></p> <p>Related complex error functions: <b>w_of_z(3)</b>, <b>dawson(3)</b>, <b>cerf(3)</b>, <b>erfcx(3)</b>, <b>erfi(3)</b>.</p> <p>Homepage: http://apps.jcns.fz-juelich.de/libcerf</p> <h1 id="AUTHORS">AUTHORS</h1> <p>Joachim Wuttke <j.wuttke@fz-juelich.de>, Forschungszentrum Juelich, based on the w_of_z implementation by Steven G. Johnson, http://math.mit.edu/~stevenj, Massachusetts Institute of Technology.</p> <p>Please report bugs to the authors.</p> <h1 id="COPYING">COPYING</h1> <p>Copyright (c) 2013 Forschungszentrum Juelich GmbH</p> <p>Software: MIT License.</p> <p>This documentation: Creative Commons Attribution Share Alike.</p> </body> </html>
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