.. index::

   single: psi function

   single: digamma function

   single: polygamma functions

The polygamma functions of order :math:`n` are defined by

.. only:: not texinfo

   .. math:: \psi^{(n)}(x) = \left(d \over dx\right)^n \psi(x) = \left(d \over dx\right)^{n+1} \log(\Gamma(x))

.. only:: texinfo

   ::

      \psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+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`.

Digamma Function

----------------

.. function:: double gsl_sf_psi_int (int n)

              int gsl_sf_psi_int_e (int n, gsl_sf_result * result)

   These routines compute the digamma function :math:`\psi(n)` for positive

   integer :data:`n`.  The digamma function is also called the Psi function.

.. Domain: n integer, n > 0

.. Exceptional Return Values: GSL_EDOM

.. function:: double gsl_sf_psi (double x)

              int gsl_sf_psi_e (double x, gsl_sf_result * result)

   These routines compute the digamma function :math:`\psi(x)` for general

   :data:`x`, :math:`x \ne 0`.

.. Domain: x != 0.0, -1.0, -2.0, ...

.. Exceptional Return Values: GSL_EDOM, GSL_ELOSS

.. function:: double gsl_sf_psi_1piy (double y)

              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

   :math:`1 + i y`, :math:`\Re[\psi(1 + i y)]`.

.. exceptions: none

.. Exceptional Return Values: none

Trigamma Function

-----------------

.. function:: double gsl_sf_psi_1_int (int n)

              int gsl_sf_psi_1_int_e (int n, gsl_sf_result * result)

   These routines compute the Trigamma function :math:`\psi'(n)` for

   positive integer :math:`n`.

.. Domain: n integer, n > 0 

.. Exceptional Return Values: GSL_EDOM

.. function:: double gsl_sf_psi_1 (double x)

              int gsl_sf_psi_1_e (double x, gsl_sf_result * result)

   These routines compute the Trigamma function :math:`\psi'(x)` for

   general :data:`x`.

.. Domain: x != 0.0, -1.0, -2.0, ...

.. Exceptional Return Values: GSL_EDOM, GSL_ELOSS

Polygamma Function

------------------

.. function:: double gsl_sf_psi_n (int n, double x)

              int gsl_sf_psi_n_e (int n, double x, gsl_sf_result * result)

   These routines compute the polygamma function :math:`\psi^{(n)}(x)` for

   :math:`n \ge 0`, :math:`x > 0`.

.. Domain: n >= 0, x > 0.0

.. Exceptional Return Values: GSL_EDOM