# 3.5.1.3.57 Real_polygamma

## Definition:

$psi\_deriv =real\_polygamma(x,k)$ evaluates an approximation to the kth derivative of the psi function $\psi (x)$ by

$\Psi ^k(x)=\frac{d^k}{dx^k}\Psi (x)=\frac{d^k}{dx^k}(\frac d{dx}\log _e\Gamma (x))$

where x is real with x≠0, -1, -2, ... and k=0,1,......6.

For more information please review the s14aec function in the NAG document.

## Parameters:

x (input, double)
The argument x of the function.
Constraint: x must not be 'too close' to a non-positive integer.
k (input, double)
The argument k of the function.
Constraint: 0≤k≤6
$psi\_deri$ (output, double)
Approximation to the kth derivative of the psi function $\psi (x)$.