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<title>GNU Scientific Library &ndash; Reference Manual: Lambert W Functions</title>

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<a name="Lambert-W-Functions"></a>
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<p>
Next: <a href="Legendre-Functions-and-Spherical-Harmonics.html#Legendre-Functions-and-Spherical-Harmonics" accesskey="n" rel="next">Legendre Functions and Spherical Harmonics</a>, Previous: <a href="Laguerre-Functions.html#Laguerre-Functions" accesskey="p" rel="previous">Laguerre Functions</a>, Up: <a href="Special-Functions.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Lambert-W-Functions-1"></a>
<h3 class="section">7.23 Lambert W Functions</h3>
<a name="index-W-function"></a>
<a name="index-Lambert-function"></a>

<p>Lambert&rsquo;s W functions, <em>W(x)</em>, are defined to be solutions
of the equation <em>W(x) \exp(W(x)) = x</em>. This function has
multiple branches for <em>x &lt; 0</em>; however, it has only
two real-valued branches. We define <em>W_0(x)</em> to be the
principal branch, where <em>W &gt; -1</em> for <em>x &lt; 0</em>, and 
<em>W_{-1}(x)</em> to be the other real branch, where
<em>W &lt; -1</em> for <em>x &lt; 0</em>.  The Lambert functions are
declared in the header file <samp>gsl_sf_lambert.h</samp>.
</p>

<dl>
<dt><a name="index-gsl_005fsf_005flambert_005fW0"></a>Function: <em>double</em> <strong>gsl_sf_lambert_W0</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flambert_005fW0_005fe"></a>Function: <em>int</em> <strong>gsl_sf_lambert_W0_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These compute the principal branch of the Lambert W function, <em>W_0(x)</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005flambert_005fWm1"></a>Function: <em>double</em> <strong>gsl_sf_lambert_Wm1</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flambert_005fWm1_005fe"></a>Function: <em>int</em> <strong>gsl_sf_lambert_Wm1_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These compute the secondary real-valued branch of the Lambert W function, 
<em>W_{-1}(x)</em>.
</p></dd></dl>





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