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