Fermi-Dirac Function
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Complete Fermi-Dirac Integrals
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F_j(x)   := 1/Gamma[j+1] Integral[ t^j /(Exp[t-x] + 1), (t,0,Infinity)]

/* Complete integral F_(-1)(x) = e^x / (1 + e^x) *
 -- Function: int gsl_sf_fermi_dirac_m1_impl (double x, gsl_sf_result * result)
 -- Function: int gsl_sf_fermi_dirac_m1_e (double x, gsl_sf_result * result)
     Exceptional Return Values: GSL_EUNDRFLW

/* Complete integral F_0(x) = ln(1 + e^x) *
 -- Function: int gsl_sf_fermi_dirac_0_impl (double x, gsl_sf_result * result)
 -- Function: int gsl_sf_fermi_dirac_0_e (double x, gsl_sf_result * result)
     Exceptional Return Values: GSL_EUNDRFLW

/* Complete integral F_1(x) *
 -- Function: int gsl_sf_fermi_dirac_1_impl (double x, gsl_sf_result * result)
 -- Function: int gsl_sf_fermi_dirac_1_e (double x, gsl_sf_result * result)
     Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW

/* Complete integral F_2(x) *
 -- Function: int gsl_sf_fermi_dirac_2_impl (double x, gsl_sf_result * result)
 -- Function: int gsl_sf_fermi_dirac_2_e (double x, gsl_sf_result * result)
     Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW

/* Complete integral F_j(x) * for integer j *
 -- Function: int gsl_sf_fermi_dirac_int_impl (int j, double x, gsl_sf_result * result)
 -- Function: int gsl_sf_fermi_dirac_int_e (int j, double x, gsl_sf_result * result)
     Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW

/* Complete integral F_(-1/2)(x) *
 -- Function: int gsl_sf_fermi_dirac_mhalf_impl (double x, gsl_sf_result * result)
 -- Function: int gsl_sf_fermi_dirac_mhalf_e (double x, gsl_sf_result * result)
     Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW

/* Complete integral F_(1/2)(x) *
 -- Function: int gsl_sf_fermi_dirac_half_impl (double x, gsl_sf_result * result)
 -- Function: int gsl_sf_fermi_dirac_half_e (double x, gsl_sf_result * result)
     Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW

/* Complete integral F_(3/2)(x) *
 -- Function: int gsl_sf_fermi_dirac_3half_impl (double x, gsl_sf_result * result)
 -- Function: int gsl_sf_fermi_dirac_3half_e (double x, gsl_sf_result * result)
     Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW

Incomplete Fermi-Dirac Integrals:
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F_j(x,b) := 1/Gamma[j+1] Integral[ t^j /(Exp[t-x] + 1), (t,b,Infinity)]

/* Incomplete integral F_0(x,b) = ln(1 + e^(b-x)) - (b-x) *
 -- Function: int gsl_sf_fermi_dirac_inc_0_impl (double x, double b, gsl_sf_result * result)
 -- Function: int gsl_sf_fermi_dirac_inc_0_e (double x, double b, gsl_sf_result * result)
     Exceptional Return Values: GSL_EUNDRFLW, GSL_EDOM