Handbook of Differential Entropy

Differential Entropies for Probability Distributions 91

Derivation of the differential entropy for the gamma

distribution

The Gamma function defined by

Γ(x) =

∞

Z

0

t

x−1

e

−t

dt

has already been used for evaluating an integral. A graph and properties of

this function can be found in the Handbook of Mathematical Functions by

Abramowitz and Stegun [1], pp. 255 ff. The derivative of this function can be

found by differentiating with respect to x inside the integral sign to obtain:

Γ

0

(x) =

∞

Z

0

t

x−1

ln(t)e

−t

dt

Then the Psi or Digamma function is given by

Ψ(x) =

Γ

0

(x)

Γ(x)

=

d

dx

(ln (Γ(x))) .

A graph and properties for this function can also be found in Abramowitz and

Stegun [1], ...