Psi (Polygamma) Function

Arbitrary order Polygamma function valid in the entire complex plane.

psi(k, z)

Arguments

k

order of the polygamma function, whole number greater or equal 0.

z

numeric complex number or vector.

Details

Computes the Polygamma function of arbitrary order, and valid in the entire complex plane. The polygamma function is defined as

$$\psi(n, z) = \frac{d^{n+1}}{dz^{n+1}} \log(\Gamma(z))$$

If n is 0 or absent then psi will be the Digamma function. If n=1,2,3,4,5 etc. then psi will be the tri-, tetra-, penta-, hexa-, hepta- etc. gamma function.

Value

Returns a complex number or a vector of complex numbers.

Examples

psi(2) - psi(1)         # 1
#> [1] 1
-psi(1)                 # Eulers constant: 0.57721566490153  [or, -psi(0, 1)]
#> [1] 0.5772157
psi(1, 2)               # pi^2/6 - 1     : 0.64493406684823
#> [1] 0.6449341
psi(10, -11.5-0.577007813568142i)
#> [1] -4.984032e-06+6.22611e-07i
                        # is near a root of the decagamma function