# Psi (Digamma) Function¶

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.

## Digamma Function¶

gsl_sf_psi_int(n)

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

gsl_sf_psi(x)

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

gsl_sf_psi_1piy(x)

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

## Trigamma Function¶

gsl_sf_psi_1_int(n)

This routine computes the Trigamma function $$\psi'(n)$$ for positive integer $$n$$.

gsl_sf_psi_1(x)

This routine computes the Trigamma function $$\psi'(x)$$ for general $$x$$.

## Polygamma Function¶

gsl_sf_psi_n(n, x)

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