| @cindex Zeta functions |
| |
| The Riemann zeta function is defined in Abramowitz & Stegun, Section |
| 23.2. The functions described in this section are declared in the |
| header file @file{gsl_sf_zeta.h}. |
| |
| @menu |
| * Riemann Zeta Function:: |
| * Riemann Zeta Function Minus One:: |
| * Hurwitz Zeta Function:: |
| * Eta Function:: |
| @end menu |
| |
| @node Riemann Zeta Function |
| @subsection Riemann Zeta Function |
| |
| The Riemann zeta function is defined by the infinite sum |
| @c{$\zeta(s) = \sum_{k=1}^\infty k^{-s}$} |
| @math{\zeta(s) = \sum_@{k=1@}^\infty k^@{-s@}}. |
| |
| @deftypefun double gsl_sf_zeta_int (int @var{n}) |
| @deftypefunx int gsl_sf_zeta_int_e (int @var{n}, gsl_sf_result * @var{result}) |
| These routines compute the Riemann zeta function @math{\zeta(n)} |
| for integer @var{n}, |
| @math{n \ne 1}. |
| @comment Domain: n integer, n != 1 |
| @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_zeta (double @var{s}) |
| @deftypefunx int gsl_sf_zeta_e (double @var{s}, gsl_sf_result * @var{result}) |
| These routines compute the Riemann zeta function @math{\zeta(s)} |
| for arbitrary @var{s}, |
| @math{s \ne 1}. |
| @comment Domain: s != 1.0 |
| @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW |
| @end deftypefun |
| |
| |
| @node Riemann Zeta Function Minus One |
| @subsection Riemann Zeta Function Minus One |
| |
| For large positive argument, the Riemann zeta function approaches one. |
| In this region the fractional part is interesting, and therefore we |
| need a function to evaluate it explicitly. |
| |
| @deftypefun double gsl_sf_zetam1_int (int @var{n}) |
| @deftypefunx int gsl_sf_zetam1_int_e (int @var{n}, gsl_sf_result * @var{result}) |
| These routines compute @math{\zeta(n) - 1} for integer @var{n}, |
| @math{n \ne 1}. |
| @comment Domain: n integer, n != 1 |
| @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_zetam1 (double @var{s}) |
| @deftypefunx int gsl_sf_zetam1_e (double @var{s}, gsl_sf_result * @var{result}) |
| These routines compute @math{\zeta(s) - 1} for arbitrary @var{s}, |
| @math{s \ne 1}. |
| @comment Domain: s != 1.0 |
| @comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW |
| @end deftypefun |
| |
| |
| @node Hurwitz Zeta Function |
| @subsection Hurwitz Zeta Function |
| |
| The Hurwitz zeta function is defined by |
| @c{$\zeta(s,q) = \sum_0^\infty (k+q)^{-s}$} |
| @math{\zeta(s,q) = \sum_0^\infty (k+q)^@{-s@}}. |
| |
| @deftypefun double gsl_sf_hzeta (double @var{s}, double @var{q}) |
| @deftypefunx int gsl_sf_hzeta_e (double @var{s}, double @var{q}, gsl_sf_result * @var{result}) |
| These routines compute the Hurwitz zeta function @math{\zeta(s,q)} for |
| @math{s > 1}, @math{q > 0}. |
| @comment Domain: s > 1.0, q > 0.0 |
| @comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW, GSL_EOVRFLW |
| @end deftypefun |
| |
| |
| @node Eta Function |
| @subsection Eta Function |
| |
| The eta function is defined by |
| @c{$\eta(s) = (1-2^{1-s}) \zeta(s)$} |
| @math{\eta(s) = (1-2^@{1-s@}) \zeta(s)}. |
| |
| @deftypefun double gsl_sf_eta_int (int @var{n}) |
| @deftypefunx int gsl_sf_eta_int_e (int @var{n}, gsl_sf_result * @var{result}) |
| These routines compute the eta function @math{\eta(n)} for integer @var{n}. |
| @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_eta (double @var{s}) |
| @deftypefunx int gsl_sf_eta_e (double @var{s}, gsl_sf_result * @var{result}) |
| These routines compute the eta function @math{\eta(s)} for arbitrary @var{s}. |
| @comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW |
| @end deftypefun |
| |