The polygamma functions of order n are defined by

     \psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+1} \log(\Gamma(x))

where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function. These functions are declared in the header file gsl_sf_psi.h.

Digamma Function
----------------

 -- Function: double gsl_sf_psi_int (int N)
 -- Function: int gsl_sf_psi_int_e (int N, gsl_sf_result * RESULT)
     These routines compute the digamma function \psi(n) for positive integer N.  The digamma function is also called the Psi function.

 -- Function: double gsl_sf_psi (double X)
 -- Function: int gsl_sf_psi_e (double X, gsl_sf_result * RESULT)
     These routines compute the digamma function \psi(x) for general x, x \ne 0.

 -- Function: double gsl_sf_psi_1piy (double Y)
 -- Function: int gsl_sf_psi_1piy_e (double Y, gsl_sf_result * RESULT)
     These routines compute the real part of the digamma function on the line 1+i y, \Re[\psi(1 + i y)].

Trigamma Function
-----------------

 -- Function: double gsl_sf_psi_1_int (int N)
 -- Function: int gsl_sf_psi_1_int_e (int N, gsl_sf_result * RESULT)
     These routines compute the Trigamma function \psi'(n) for positive integer n.

 -- Function: double gsl_sf_psi_1 (double X)
 -- Function: int gsl_sf_psi_1_e (double X, gsl_sf_result * RESULT)
     These routines compute the Trigamma function \psi'(x) for general x.

Polygamma Function
------------------

 -- Function: double gsl_sf_psi_n (int N, double X)
 -- Function: int gsl_sf_psi_n_e (int N, double X, gsl_sf_result * RESULT)
     These routines compute the polygamma function $\psi^(n)(x)$ \psi^{(n)}(x) for $n \ge 0$ n >= 0, x > 0.