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<title>Complete Fermi-Dirac Integrals - GNU Scientific Library -- Reference Manual</title>
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Next: <a rel="next" accesskey="n" href="Incomplete-Fermi_002dDirac-Integrals.html">Incomplete Fermi-Dirac Integrals</a>,
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<h4 class="subsection">7.18.1 Complete Fermi-Dirac Integrals</h4>
<p><a name="index-complete-Fermi_002dDirac-integrals-557"></a><a name="index-Fj_0028x_0029_002c-Fermi_002dDirac-integral-558"></a>The complete Fermi-Dirac integral F_j(x) is given by,
<pre class="example"> F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
</pre>
<p class="noindent">Note that the Fermi-Dirac integral is sometimes defined without the
normalisation factor in other texts.
<div class="defun">
— Function: double <b>gsl_sf_fermi_dirac_m1</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fm1-559"></a></var><br>
— Function: int <b>gsl_sf_fermi_dirac_m1_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fm1_005fe-560"></a></var><br>
<blockquote><p>These routines compute the complete Fermi-Dirac integral with an index of -1.
This integral is given by
<!-- {$F_{-1}(x) = e^x / (1 + e^x)$} -->
F_{-1}(x) = e^x / (1 + e^x).
<!-- Exceptional Return Values: GSL_EUNDRFLW -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_fermi_dirac_0</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f0-561"></a></var><br>
— Function: int <b>gsl_sf_fermi_dirac_0_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f0_005fe-562"></a></var><br>
<blockquote><p>These routines compute the complete Fermi-Dirac integral with an index of 0.
This integral is given by F_0(x) = \ln(1 + e^x).
<!-- Exceptional Return Values: GSL_EUNDRFLW -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_fermi_dirac_1</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f1-563"></a></var><br>
— Function: int <b>gsl_sf_fermi_dirac_1_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f1_005fe-564"></a></var><br>
<blockquote><p>These routines compute the complete Fermi-Dirac integral with an index of 1,
F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)).
<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_fermi_dirac_2</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f2-565"></a></var><br>
— Function: int <b>gsl_sf_fermi_dirac_2_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f2_005fe-566"></a></var><br>
<blockquote><p>These routines compute the complete Fermi-Dirac integral with an index
of 2,
F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)).
<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_fermi_dirac_int</b> (<var>int j, double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fint-567"></a></var><br>
— Function: int <b>gsl_sf_fermi_dirac_int_e</b> (<var>int j, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fint_005fe-568"></a></var><br>
<blockquote><p>These routines compute the complete Fermi-Dirac integral with an integer
index of j,
F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1)).
<!-- Complete integral F_j(x) for integer j -->
<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_fermi_dirac_mhalf</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fmhalf-569"></a></var><br>
— Function: int <b>gsl_sf_fermi_dirac_mhalf_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fmhalf_005fe-570"></a></var><br>
<blockquote><p>These routines compute the complete Fermi-Dirac integral
<!-- {$F_{-1/2}(x)$} -->
F_{-1/2}(x).
<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_fermi_dirac_half</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fhalf-571"></a></var><br>
— Function: int <b>gsl_sf_fermi_dirac_half_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fhalf_005fe-572"></a></var><br>
<blockquote><p>These routines compute the complete Fermi-Dirac integral
<!-- {$F_{1/2}(x)$} -->
F_{1/2}(x).
<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_fermi_dirac_3half</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f3half-573"></a></var><br>
— Function: int <b>gsl_sf_fermi_dirac_3half_e</b> (<var>double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f3half_005fe-574"></a></var><br>
<blockquote><p>These routines compute the complete Fermi-Dirac integral
<!-- {$F_{3/2}(x)$} -->
F_{3/2}(x).
<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
</p></blockquote></div>
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