| @cindex Fermi-Dirac function |
| |
| The functions described in this section are declared in the header file |
| @file{gsl_sf_fermi_dirac.h}. |
| |
| @menu |
| * Complete Fermi-Dirac Integrals:: |
| * Incomplete Fermi-Dirac Integrals:: |
| @end menu |
| |
| @node Complete Fermi-Dirac Integrals |
| @subsection Complete Fermi-Dirac Integrals |
| @cindex complete Fermi-Dirac integrals |
| @cindex Fj(x), Fermi-Dirac integral |
| The complete Fermi-Dirac integral @math{F_j(x)} is given by, |
| @tex |
| \beforedisplay |
| $$ |
| F_j(x) := {1\over\Gamma(j+1)} \int_0^\infty dt {t^j \over (\exp(t-x) + 1)} |
| $$ |
| \afterdisplay |
| @end tex |
| @ifinfo |
| |
| @example |
| F_j(x) := (1/r\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1)) |
| @end example |
| @end ifinfo |
| |
| @deftypefun double gsl_sf_fermi_dirac_m1 (double @var{x}) |
| @deftypefunx int gsl_sf_fermi_dirac_m1_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the complete Fermi-Dirac integral with an index of @math{-1}. |
| This integral is given by |
| @c{$F_{-1}(x) = e^x / (1 + e^x)$} |
| @math{F_@{-1@}(x) = e^x / (1 + e^x)}. |
| @comment Exceptional Return Values: GSL_EUNDRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_fermi_dirac_0 (double @var{x}) |
| @deftypefunx int gsl_sf_fermi_dirac_0_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the complete Fermi-Dirac integral with an index of @math{0}. |
| This integral is given by @math{F_0(x) = \ln(1 + e^x)}. |
| @comment Exceptional Return Values: GSL_EUNDRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_fermi_dirac_1 (double @var{x}) |
| @deftypefunx int gsl_sf_fermi_dirac_1_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the complete Fermi-Dirac integral with an index of @math{1}, |
| @math{F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1))}. |
| @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_fermi_dirac_2 (double @var{x}) |
| @deftypefunx int gsl_sf_fermi_dirac_2_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the complete Fermi-Dirac integral with an index |
| of @math{2}, |
| @math{F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1))}. |
| @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_fermi_dirac_int (int @var{j}, double @var{x}) |
| @deftypefunx int gsl_sf_fermi_dirac_int_e (int @var{j}, double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the complete Fermi-Dirac integral with an integer |
| index of @math{j}, |
| @math{F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1))}. |
| @comment Complete integral F_j(x) for integer j |
| @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_fermi_dirac_mhalf (double @var{x}) |
| @deftypefunx int gsl_sf_fermi_dirac_mhalf_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the complete Fermi-Dirac integral |
| @c{$F_{-1/2}(x)$} |
| @math{F_@{-1/2@}(x)}. |
| @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_fermi_dirac_half (double @var{x}) |
| @deftypefunx int gsl_sf_fermi_dirac_half_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the complete Fermi-Dirac integral |
| @c{$F_{1/2}(x)$} |
| @math{F_@{1/2@}(x)}. |
| @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_fermi_dirac_3half (double @var{x}) |
| @deftypefunx int gsl_sf_fermi_dirac_3half_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the complete Fermi-Dirac integral |
| @c{$F_{3/2}(x)$} |
| @math{F_@{3/2@}(x)}. |
| @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW |
| @end deftypefun |
| |
| |
| @node Incomplete Fermi-Dirac Integrals |
| @subsection Incomplete Fermi-Dirac Integrals |
| @cindex incomplete Fermi-Dirac integral |
| @cindex Fj(x,b), incomplete Fermi-Dirac integral |
| The incomplete Fermi-Dirac integral @math{F_j(x,b)} is given by, |
| @tex |
| \beforedisplay |
| $$ |
| F_j(x,b) := {1\over\Gamma(j+1)} \int_b^\infty dt {t^j \over (\exp(t-x) + 1)} |
| $$ |
| \afterdisplay |
| @end tex |
| @ifinfo |
| |
| @example |
| F_j(x,b) := (1/\Gamma(j+1)) \int_b^\infty dt (t^j / (\Exp(t-x) + 1)) |
| @end example |
| @end ifinfo |
| |
| @deftypefun double gsl_sf_fermi_dirac_inc_0 (double @var{x}, double @var{b}) |
| @deftypefunx int gsl_sf_fermi_dirac_inc_0_e (double @var{x}, double @var{b}, gsl_sf_result * @var{result}) |
| These routines compute the incomplete Fermi-Dirac integral with an index |
| of zero, |
| @c{$F_0(x,b) = \ln(1 + e^{b-x}) - (b-x)$} |
| @math{F_0(x,b) = \ln(1 + e^@{b-x@}) - (b-x)}. |
| @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EDOM |
| @end deftypefun |
| |