@cindex psi function@cindex digamma function@cindex polygamma functionsThe polygamma functions of order @math{m} defined by@c{$\psi^{(m)}(x) = (d/dx)^m \psi(x) = (d/dx)^{m+1} \log(\Gamma(x))$}@math{\psi^@{(m)@}(x) = (d/dx)^m \psi(x) = (d/dx)^@{m+1@} \log(\Gamma(x))},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 positiveinteger @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@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)} forpositive integer @math{n}.@comment Domain: n integer, n > 0 @comment Exceptional Return Values: GSL_EDOM@end deftypefun@node Polygamma Function@subsection Polygamma Function@deftypefun double gsl_sf_psi_n (int @var{m}, double @var{x})@deftypefunx int gsl_sf_psi_n_e (int @var{m}, double @var{x}, gsl_sf_result * @var{result})These routines compute the polygamma function @c{$\psi^{(m)}(x)$}@math{\psi^@{(m)@}(x)} for@c{$c \ge 0$}@math{m >= 0}, @math{x > 0}.  @comment Domain: m >= 0, x > 0.0@comment Exceptional Return Values: GSL_EDOM@end deftypefun