R: Psi (Polygamma) Function

psi {pracma} | R Documentation

## Psi (Polygamma) Function

### Description

Arbitrary order Polygamma function valid in the entire complex plane.

### Usage

    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

    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

[Package pracma version 2.0.4 Index]