| @cindex psi function |
| @cindex digamma function |
| @cindex polygamma functions |
| |
| The polygamma functions of order @math{n} are defined by |
| @tex |
| \beforedisplay |
| $$ |
| \psi^{(n)}(x) = \left(d \over dx\right)^n \psi(x) = \left(d \over dx\right)^{n+1} \log(\Gamma(x)) |
| $$ |
| \afterdisplay |
| @end tex |
| @ifinfo |
| |
| @example |
| \psi^@{(n)@}(x) = (d/dx)^n \psi(x) = (d/dx)^@{n+1@} \log(\Gamma(x)) |
| @end example |
| |
| @end ifinfo |
| @noindent |
| where @math{\psi(x) = \Gamma'(x)/\Gamma(x)} is known as the digamma function. |
| These functions are declared in the header file @file{gsl_sf_psi.h}. |
| |
| @menu |
| * Digamma Function:: |
| * Trigamma Function:: |
| * Polygamma Function:: |
| @end menu |
| |
| @node Digamma Function |
| @subsection Digamma Function |
| |
| @deftypefun double gsl_sf_psi_int (int @var{n}) |
| @deftypefunx int gsl_sf_psi_int_e (int @var{n}, gsl_sf_result * @var{result}) |
| These routines compute the digamma function @math{\psi(n)} for positive |
| integer @var{n}. The digamma function is also called the Psi function. |
| @comment Domain: n integer, n > 0 |
| @comment Exceptional Return Values: GSL_EDOM |
| @end deftypefun |
| |
| |
| @deftypefun double gsl_sf_psi (double @var{x}) |
| @deftypefunx int gsl_sf_psi_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the digamma function @math{\psi(x)} for general |
| @math{x}, @math{x \ne 0}. |
| @comment Domain: x != 0.0, -1.0, -2.0, ... |
| @comment Exceptional Return Values: GSL_EDOM, GSL_ELOSS |
| @end deftypefun |
| |
| |
| @deftypefun double gsl_sf_psi_1piy (double @var{y}) |
| @deftypefunx int gsl_sf_psi_1piy_e (double @var{y}, gsl_sf_result * @var{result}) |
| These routines compute the real part of the digamma function on the line |
| @math{1+i y}, @math{\Re[\psi(1 + i y)]}. |
| @comment exceptions: none |
| @comment Exceptional Return Values: none |
| @end deftypefun |
| |
| |
| @node Trigamma Function |
| @subsection Trigamma Function |
| |
| @deftypefun double gsl_sf_psi_1_int (int @var{n}) |
| @deftypefunx int gsl_sf_psi_1_int_e (int @var{n}, gsl_sf_result * @var{result}) |
| These routines compute the Trigamma function @math{\psi'(n)} for |
| positive integer @math{n}. |
| @comment Domain: n integer, n > 0 |
| @comment Exceptional Return Values: GSL_EDOM |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_psi_1 (double @var{x}) |
| @deftypefunx int gsl_sf_psi_1_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the Trigamma function @math{\psi'(x)} for |
| general @math{x}. |
| @comment Domain: x != 0.0, -1.0, -2.0, ... |
| @comment Exceptional Return Values: GSL_EDOM, GSL_ELOSS |
| @end deftypefun |
| |
| @node Polygamma Function |
| @subsection Polygamma Function |
| |
| @deftypefun double gsl_sf_psi_n (int @var{n}, double @var{x}) |
| @deftypefunx int gsl_sf_psi_n_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the polygamma function @c{$\psi^{(n)}(x)$} |
| @math{\psi^@{(n)@}(x)} for |
| @c{$n \ge 0$} |
| @math{n >= 0}, @math{x > 0}. |
| @comment Domain: n >= 0, x > 0.0 |
| @comment Exceptional Return Values: GSL_EDOM |
| @end deftypefun |