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<title>GNU Scientific Library – Reference Manual: Complete Fermi-Dirac Integrals</title>
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<a name="Complete-Fermi_002dDirac-Integrals"></a>
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<p>
Next: <a href="Incomplete-Fermi_002dDirac-Integrals.html#Incomplete-Fermi_002dDirac-Integrals" accesskey="n" rel="next">Incomplete Fermi-Dirac Integrals</a>, Up: <a href="Fermi_002dDirac-Function.html#Fermi_002dDirac-Function" accesskey="u" rel="up">Fermi-Dirac Function</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Complete-Fermi_002dDirac-Integrals-1"></a>
<h4 class="subsection">7.18.1 Complete Fermi-Dirac Integrals</h4>
<a name="index-complete-Fermi_002dDirac-integrals"></a>
<a name="index-Fj_0028x_0029_002c-Fermi_002dDirac-integral"></a>
<p>The complete Fermi-Dirac integral <em>F_j(x)</em> is given by,
</p>
<div class="example">
<pre class="example">F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
</pre></div>
<p>Note that the Fermi-Dirac integral is sometimes defined without the
normalisation factor in other texts.
</p>
<dl>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005fm1"></a>Function: <em>double</em> <strong>gsl_sf_fermi_dirac_m1</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005fm1_005fe"></a>Function: <em>int</em> <strong>gsl_sf_fermi_dirac_m1_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the complete Fermi-Dirac integral with an index of <em>-1</em>.
This integral is given by
<em>F_{-1}(x) = e^x / (1 + e^x)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005f0"></a>Function: <em>double</em> <strong>gsl_sf_fermi_dirac_0</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005f0_005fe"></a>Function: <em>int</em> <strong>gsl_sf_fermi_dirac_0_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the complete Fermi-Dirac integral with an index of <em>0</em>.
This integral is given by <em>F_0(x) = \ln(1 + e^x)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005f1"></a>Function: <em>double</em> <strong>gsl_sf_fermi_dirac_1</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005f1_005fe"></a>Function: <em>int</em> <strong>gsl_sf_fermi_dirac_1_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the complete Fermi-Dirac integral with an index of <em>1</em>,
<em>F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1))</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005f2"></a>Function: <em>double</em> <strong>gsl_sf_fermi_dirac_2</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005f2_005fe"></a>Function: <em>int</em> <strong>gsl_sf_fermi_dirac_2_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the complete Fermi-Dirac integral with an index
of <em>2</em>,
<em>F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1))</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005fint"></a>Function: <em>double</em> <strong>gsl_sf_fermi_dirac_int</strong> <em>(int <var>j</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005fint_005fe"></a>Function: <em>int</em> <strong>gsl_sf_fermi_dirac_int_e</strong> <em>(int <var>j</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the complete Fermi-Dirac integral with an integer
index of <em>j</em>,
<em>F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1))</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005fmhalf"></a>Function: <em>double</em> <strong>gsl_sf_fermi_dirac_mhalf</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005fmhalf_005fe"></a>Function: <em>int</em> <strong>gsl_sf_fermi_dirac_mhalf_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the complete Fermi-Dirac integral
<em>F_{-1/2}(x)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005fhalf"></a>Function: <em>double</em> <strong>gsl_sf_fermi_dirac_half</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005fhalf_005fe"></a>Function: <em>int</em> <strong>gsl_sf_fermi_dirac_half_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the complete Fermi-Dirac integral
<em>F_{1/2}(x)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005f3half"></a>Function: <em>double</em> <strong>gsl_sf_fermi_dirac_3half</strong> <em>(double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ffermi_005fdirac_005f3half_005fe"></a>Function: <em>int</em> <strong>gsl_sf_fermi_dirac_3half_e</strong> <em>(double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the complete Fermi-Dirac integral
<em>F_{3/2}(x)</em>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Next: <a href="Incomplete-Fermi_002dDirac-Integrals.html#Incomplete-Fermi_002dDirac-Integrals" accesskey="n" rel="next">Incomplete Fermi-Dirac Integrals</a>, Up: <a href="Fermi_002dDirac-Function.html#Fermi_002dDirac-Function" accesskey="u" rel="up">Fermi-Dirac Function</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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