## File: Complete-Fermi_002dDirac-Integrals.html

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 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151  GNU Scientific Library – Reference Manual: Complete Fermi-Dirac Integrals

7.18.1 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 dt (t^j / (\exp(t-x) + 1))

Note that the Fermi-Dirac integral is sometimes defined without the normalisation factor in other texts.

Function: double gsl_sf_fermi_dirac_m1 (double x)
Function: int gsl_sf_fermi_dirac_m1_e (double x, gsl_sf_result * result)

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).

Function: double gsl_sf_fermi_dirac_0 (double x)
Function: int gsl_sf_fermi_dirac_0_e (double x, gsl_sf_result * result)

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).

Function: double gsl_sf_fermi_dirac_1 (double x)
Function: int gsl_sf_fermi_dirac_1_e (double x, gsl_sf_result * result)

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)).

Function: double gsl_sf_fermi_dirac_2 (double x)
Function: int gsl_sf_fermi_dirac_2_e (double x, gsl_sf_result * result)

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)).

Function: double gsl_sf_fermi_dirac_int (int j, double x)
Function: int gsl_sf_fermi_dirac_int_e (int j, double x, gsl_sf_result * result)

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)).

Function: double gsl_sf_fermi_dirac_mhalf (double x)
Function: int gsl_sf_fermi_dirac_mhalf_e (double x, gsl_sf_result * result)

These routines compute the complete Fermi-Dirac integral F_{-1/2}(x).

Function: double gsl_sf_fermi_dirac_half (double x)
Function: int gsl_sf_fermi_dirac_half_e (double x, gsl_sf_result * result)

These routines compute the complete Fermi-Dirac integral F_{1/2}(x).

Function: double gsl_sf_fermi_dirac_3half (double x)
Function: int gsl_sf_fermi_dirac_3half_e (double x, gsl_sf_result * result)

These routines compute the complete Fermi-Dirac integral F_{3/2}(x).