Fermi-Dirac Function¶
Complete Fermi-Dirac Integrals¶
The complete Fermi-Dirac integral \(F_j(x)\) is given by,
Note that the Fermi-Dirac integral is sometimes defined without the normalisation factor in other texts.
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gsl_sf_fermi_dirac_m1(x)¶
This routine computes the complete Fermi-Dirac integral with an index of -1. This integral is given by \(F_{-1}(x) = e^x / (1 + e^x)\).
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gsl_sf_fermi_dirac_0(x)¶
This routine computes the complete Fermi-Dirac integral with an index of 0. This integral is given by \(F_0(x) = \ln(1 + e^x)\).
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gsl_sf_fermi_dirac_1(x)¶
This routine computes the complete Fermi-Dirac integral with an index of 1, \(F_1(x) = \int_0^\infty (t /(\exp(t-x)+1)) dt\).
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gsl_sf_fermi_dirac_2(x)¶
This routine computes the complete Fermi-Dirac integral with an index of 2, \(F_2(x) = (1/2) \int_0^\infty (t^2 /(\exp(t-x)+1)) dt\).
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gsl_sf_fermi_dirac_int(j, x)¶
This routine computes the complete Fermi-Dirac integral with an integer index of \(j\), \(F_j(x) = (1/\Gamma(j+1)) \int_0^\infty (t^j /(\exp(t-x)+1)) dt\).
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gsl_sf_fermi_dirac_mhalf(x)¶
This routine computes the complete Fermi-Dirac integral \(F_{-1/2}(x)\).
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gsl_sf_fermi_dirac_half(x)¶
This routine computes the complete Fermi-Dirac integral \(F_{1/2}(x)\).
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gsl_sf_fermi_dirac_3half(x)¶
This routine computes the complete Fermi-Dirac integral \(F_{3/2}(x)\).
Incomplete Fermi-Dirac Integrals¶
The incomplete Fermi-Dirac integral F_j(x,b) is given by,
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gsl_sf_fermi_dirac_inc_0(x, b)¶
This routine computes the incomplete Fermi-Dirac integral with an index of zero, \(F_0(x,b) = \ln(1 + e^{b-x}) - (b-x)\).