# Derivatives of the DiGamma function: Psi

### 5.8.17 Derivatives of the DiGamma function: Psi

The DiGamma function is the derivative of the logarithm of the Γ function (see Section 5.8.13),

ψ(z) =

d
dz

ln(Γ(z)) =

Γ′(z)
Γ(z)

This function is used to evaluated sums of rational functions having poles at integers.

The Psi function computes the DiGamma function and its derivatives.

- Psi takes one mandatory argument and one optional argument:
  - a, a real number.
  - Optionally, n, a non-negative integer.
- Psi(a) returns the value of the DiGamma function ψ(a).
- Psi(a,n) returns the nth derivative of the DiGamma function at x=a.

#### Examples.

- Input:
  Psi(3)
  Output:

  3
  2

  −γ

- Input:
  evalf(Psi(3))
  Output:

  0.922784335098

- Input:
  Psi(3,1)
  Output:

  π 2
  6

  −

  5
  4