


Psi (Digamma) Function
======================

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

.. math::
  \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.

.. index::
  psi function
  polygamma function

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

.. function:: gsl_sf_psi_int(n)

 This routine computes the :index:`digamma function` :math:`\psi(n)` for positive
 integer :math:`n`. The digamma function is also called the Psi function.

.. function:: gsl_sf_psi(x)

 This routine computes the digamma function :math:`\psi(x)` for general
 :math:`x, x \ne 0`.

.. function:: gsl_sf_psi_1piy(x)

 This routine computes the real part of the digamma function on
 the line :math:`1+i y, \operatorname{Re}[\psi(1 + i y)]`.

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

.. function:: gsl_sf_psi_1_int(n)

 This routine computes the :index:`Trigamma function` :math:`\psi'(n)` for
 positive integer :math:`n`.

.. function:: gsl_sf_psi_1(x)

 This routine computes the Trigamma function :math:`\psi'(x)` for general :math:`x`.

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

.. function:: gsl_sf_psi_n(n, x)

 This routine computes the polygamma function :math:`\psi^{(n)}(x)` for
 :math:`n \geq 0, x > 0`.
