Derivatives of the DiGamma function

### 7.3.17 Derivatives of the DiGamma function

The DiGamma function is the derivative of the logarithm of the Γ function (see Section 7.3.13),

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

Psi(3)

3
2
| −γ

evalf(Psi(3))

0.922784335098 |  |  |  |  |  |  |  |  |  |

Psi(3,1)

π 2
6
| − |
5
4