3 The Riemann zeta function is defined in Abramowitz & Stegun, Section
4 23.2. The functions described in this section are declared in the
5 header file @file{gsl_sf_zeta.h}.
8 * Riemann Zeta Function::
9 * Riemann Zeta Function Minus One::
10 * Hurwitz Zeta Function::
14 @node Riemann Zeta Function
15 @subsection Riemann Zeta Function
17 The Riemann zeta function is defined by the infinite sum
18 @c{$\zeta(s) = \sum_{k=1}^\infty k^{-s}$}
19 @math{\zeta(s) = \sum_@{k=1@}^\infty k^@{-s@}}.
21 @deftypefun double gsl_sf_zeta_int (int @var{n})
22 @deftypefunx int gsl_sf_zeta_int_e (int @var{n}, gsl_sf_result * @var{result})
23 These routines compute the Riemann zeta function @math{\zeta(n)}
26 @comment Domain: n integer, n != 1
27 @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
30 @deftypefun double gsl_sf_zeta (double @var{s})
31 @deftypefunx int gsl_sf_zeta_e (double @var{s}, gsl_sf_result * @var{result})
32 These routines compute the Riemann zeta function @math{\zeta(s)}
33 for arbitrary @var{s},
35 @comment Domain: s != 1.0
36 @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
40 @node Riemann Zeta Function Minus One
41 @subsection Riemann Zeta Function Minus One
43 For large positive argument, the Riemann zeta function approaches one.
44 In this region the fractional part is interesting, and therefore we
45 need a function to evaluate it explicitly.
47 @deftypefun double gsl_sf_zetam1_int (int @var{n})
48 @deftypefunx int gsl_sf_zetam1_int_e (int @var{n}, gsl_sf_result * @var{result})
49 These routines compute @math{\zeta(n) - 1} for integer @var{n},
51 @comment Domain: n integer, n != 1
52 @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
55 @deftypefun double gsl_sf_zetam1 (double @var{s})
56 @deftypefunx int gsl_sf_zetam1_e (double @var{s}, gsl_sf_result * @var{result})
57 These routines compute @math{\zeta(s) - 1} for arbitrary @var{s},
59 @comment Domain: s != 1.0
60 @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
64 @node Hurwitz Zeta Function
65 @subsection Hurwitz Zeta Function
67 The Hurwitz zeta function is defined by
68 @c{$\zeta(s,q) = \sum_0^\infty (k+q)^{-s}$}
69 @math{\zeta(s,q) = \sum_0^\infty (k+q)^@{-s@}}.
71 @deftypefun double gsl_sf_hzeta (double @var{s}, double @var{q})
72 @deftypefunx int gsl_sf_hzeta_e (double @var{s}, double @var{q}, gsl_sf_result * @var{result})
73 These routines compute the Hurwitz zeta function @math{\zeta(s,q)} for
74 @math{s > 1}, @math{q > 0}.
75 @comment Domain: s > 1.0, q > 0.0
76 @comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW, GSL_EOVRFLW
81 @subsection Eta Function
83 The eta function is defined by
84 @c{$\eta(s) = (1-2^{1-s}) \zeta(s)$}
85 @math{\eta(s) = (1-2^@{1-s@}) \zeta(s)}.
87 @deftypefun double gsl_sf_eta_int (int @var{n})
88 @deftypefunx int gsl_sf_eta_int_e (int @var{n}, gsl_sf_result * @var{result})
89 These routines compute the eta function @math{\eta(n)} for integer @var{n}.
90 @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
93 @deftypefun double gsl_sf_eta (double @var{s})
94 @deftypefunx int gsl_sf_eta_e (double @var{s}, gsl_sf_result * @var{result})
95 These routines compute the eta function @math{\eta(s)} for arbitrary @var{s}.
96 @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW