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 \(n \geq 0, x > 0\).