# Fermi-Dirac Function¶

## 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 $$F_{3/2}(x)$$.

## Incomplete Fermi-Dirac Integrals¶

The incomplete Fermi-Dirac integral F_j(x,b) is given by,

$F_j(x,b) := (1/\Gamma(j+1)) \int_b^\infty (t^j / (\exp(t-x) + 1)) dt$
gsl_sf_fermi_dirac_inc_0(x, b)

This routine computes the incomplete Fermi-Dirac integral with an index of zero, $$F_0(x,b) = \ln(1 + e^{b-x}) - (b-x)$$.