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Diffstat (limited to 'gsl-1.9/doc/specfunc-zeta.texi')
| -rw-r--r-- | gsl-1.9/doc/specfunc-zeta.texi | 98 |
1 files changed, 98 insertions, 0 deletions
diff --git a/gsl-1.9/doc/specfunc-zeta.texi b/gsl-1.9/doc/specfunc-zeta.texi new file mode 100644 index 0000000..a773212 --- /dev/null +++ b/gsl-1.9/doc/specfunc-zeta.texi @@ -0,0 +1,98 @@ +@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 + |