psi

Psi (polygamma) function

Syntax

`Y = psi(X)Y = psi(k,X)`

Description

`Y = psi(X)` evaluates the ψ function for each element of array `X`. `X` must be real and nonnegative. The ψ function, also known as the digamma function, is the logarithmic derivative of the gamma function

$\begin{array}{c}\psi \left(x\right)=\text{digamma}\left(x\right)\\ =\frac{d\left(\mathrm{log}\left(\Gamma \left(x\right)\right)\right)}{dx}\\ =\frac{d\left(\Gamma \left(x\right)\right)/dx}{\Gamma \left(x\right)}\end{array}$

`Y = psi(k,X)` evaluates the `k`th derivative of ψ at the elements of `X`. `psi(0,X)` is the digamma function, `psi(1,X)` is the trigamma function, `psi(2,X)` is the tetragamma function, etc.

Examples

Example 1

Use the `psi` function to calculate Euler's constant, γ.

```format long -psi(1) ans = 0.57721566490153 -psi(0,1) ans = 0.57721566490153```

Example 2

The trigamma function of 2, `psi(1,2)`, is the same as (π2/6) – 1.

```format long psi(1,2) ans = 0.64493406684823 pi^2/6 - 1 ans = 0.64493406684823```

References

[1] Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, Sections 6.3 and 6.4.