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<title>libcerf - Complex error, Faddeeva, Dawson, and Voigt functions</title>
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<li><a href="#name">NAME</a></li>
<li><a href="#synopsis">SYNOPSIS</a></li>
<li><a href="#description">DESCRIPTION</a></li>
<li><a href="#resources">RESOURCES</a></li>
<li><a href="#references">REFERENCES</a></li>
<li><a href="#bug_reports">BUG REPORTS</a></li>
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<h1><a name="name">NAME</a></h1>
<p>voigt - Voigt's function, convolution of Gaussian and Lorentzian</p>
<p>
</p>
<hr />
<h1><a name="synopsis">SYNOPSIS</a></h1>
<p><strong>#include <cerf.h</strong>></p>
<p><strong>double voigt ( double x, double sigma, double gamma );</strong></p>
<p>
</p>
<hr />
<h1><a name="description">DESCRIPTION</a></h1>
<p>The function <strong>voigt</strong> returns Voigt's convolution</p>
<pre>
voigt(x,sigma,gamma) = integral G(t,sigma) L(x-t,gamma) dt</pre>
<p>of a Gaussian</p>
<pre>
G(x,sigma) = 1/sqrt(2*pi)/|sigma| * exp(-x^2/2/sigma^2)</pre>
<p>and a Lorentzian</p>
<pre>
L(x,gamma) = |gamma| / pi / ( x^2 + gamma^2 ),</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>
<p>
</p>
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<h1><a name="resources">RESOURCES</a></h1>
<p>Project web site: <a href="http://apps.jcns.fz-juelich.de/libcerf">http://apps.jcns.fz-juelich.de/libcerf</a></p>
<p>
</p>
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<h1><a name="references">REFERENCES</a></h1>
<p>Formula (7.4.13) in Abramowitz & Stegun (1964) relates Voigt's convolution integral to Faddeeva's function <strong>w_of_z</strong>, upon which this implementation is based:</p>
<pre>
voigt(x,sigma,gamma) = Re[w(z)] / sqrt(2*pi) / |sigma|</pre>
<p>with</p>
<pre>
z = (x+i*|gamma|) / sqrt(2) / |sigma|.</pre>
<p>
</p>
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<h1><a name="bug_reports">BUG REPORTS</a></h1>
<p>Please report bugs to the authors.</p>
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</p>
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<h1><a name="authors">AUTHORS</a></h1>
<p>Joachim Wuttke <<a href="mailto:j.wuttke@fz-juelich.de">j.wuttke@fz-juelich.de</a>>, Forschungszentrum Juelich,
based on the w_of_z implementation by Steven G. Johnson [http://math.mit.edu/~stevenj], Massachusetts Institute of Technology.</p>
<p>
</p>
<hr />
<h1><a name="see_also">SEE ALSO</a></h1>
<p>Related complex error functions in liberfc:</p>
<p><strong>w_of_z(3)</strong>, <strong>dawson(3)</strong>, <strong>cerf(3)</strong>, <strong>erfcx(3)</strong>, <strong>erfi(3)</strong>.</p>
<p>
</p>
<hr />
<h1><a name="copying">COPYING</a></h1>
<p>Copyright (c) 2013 Forschungszentrum Juelich GmbH</p>
<p>Software: MIT License.</p>
<p>This documentation: Creative Commons Attribution Share Alike.</p>
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