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<title>Complete Fermi-Dirac Integrals - GNU Scientific Library -- Reference Manual</title>
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<h4 class="subsection">7.18.1 Complete Fermi-Dirac Integrals</h4>
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<p><a name="index-complete-Fermi_002dDirac-integrals-528"></a><a name="index-Fj_0028x_0029_002c-Fermi_002dDirac-integral-529"></a>The complete Fermi-Dirac integral F_j(x) is given by,
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— Function: double <b>gsl_sf_fermi_dirac_m1</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fm1-530"></a></var><br>
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— 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-531"></a></var><br>
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<blockquote><p>These routines compute the complete Fermi-Dirac integral with an index of -1.
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This integral is given by
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<!-- {$F_{-1}(x) = e^x / (1 + e^x)$} -->
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F_{-1}(x) = e^x / (1 + e^x).
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<!-- Exceptional Return Values: GSL_EUNDRFLW -->
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</p></blockquote></div>
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— Function: double <b>gsl_sf_fermi_dirac_0</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f0-532"></a></var><br>
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— 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-533"></a></var><br>
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<blockquote><p>These routines compute the complete Fermi-Dirac integral with an index of 0.
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This integral is given by F_0(x) = \ln(1 + e^x).
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<!-- Exceptional Return Values: GSL_EUNDRFLW -->
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</p></blockquote></div>
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— Function: double <b>gsl_sf_fermi_dirac_1</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f1-534"></a></var><br>
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— 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-535"></a></var><br>
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<blockquote><p>These routines compute the complete Fermi-Dirac integral with an index of 1,
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F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)).
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<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
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</p></blockquote></div>
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— Function: double <b>gsl_sf_fermi_dirac_2</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f2-536"></a></var><br>
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— 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-537"></a></var><br>
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<blockquote><p>These routines compute the complete Fermi-Dirac integral with an index
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F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)).
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<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
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</p></blockquote></div>
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— Function: double <b>gsl_sf_fermi_dirac_int</b> (<var>int j, double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fint-538"></a></var><br>
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— 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-539"></a></var><br>
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<blockquote><p>These routines compute the complete Fermi-Dirac integral with an integer
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F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1)).
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<!-- Complete integral F_j(x) for integer j -->
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<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
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</p></blockquote></div>
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— Function: double <b>gsl_sf_fermi_dirac_mhalf</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fmhalf-540"></a></var><br>
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— 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-541"></a></var><br>
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<blockquote><p>These routines compute the complete Fermi-Dirac integral
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<!-- {$F_{-1/2}(x)$} -->
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<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
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</p></blockquote></div>
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— Function: double <b>gsl_sf_fermi_dirac_half</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005fhalf-542"></a></var><br>
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— 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-543"></a></var><br>
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<blockquote><p>These routines compute the complete Fermi-Dirac integral
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<!-- {$F_{1/2}(x)$} -->
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<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
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</p></blockquote></div>
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— Function: double <b>gsl_sf_fermi_dirac_3half</b> (<var>double x</var>)<var><a name="index-gsl_005fsf_005ffermi_005fdirac_005f3half-544"></a></var><br>
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— 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-545"></a></var><br>
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<blockquote><p>These routines compute the complete Fermi-Dirac integral
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<!-- {$F_{3/2}(x)$} -->
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<!-- Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW -->
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</p></blockquote></div>
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