# psi: Psi (Polygamma) Function

Description Usage Arguments Details Value Examples

### Description

Arbitrary order Polygamma function valid in the entire complex plane.

### Usage

 1 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

ψ(n, z) = \frac{d^{n+1}}{dz^{n+1}} \log(Γ(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

 1 2 3 4 5 psi(2) - psi(1) # 1 -psi(1) # Eulers constant: 0.57721566490153 [or, -psi(0, 1)] psi(1, 2) # pi^2/6 - 1 : 0.64493406684823 psi(10, -11.5-0.577007813568142i) # is near a root of the decagamma function 

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