1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
|
@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/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
@end example
@end ifinfo
@noindent
Note that the Fermi-Dirac integral is sometimes defined without the
normalisation factor in other texts.
@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
|
