Havriliak-Negami Relaxation

The Havriliak-Negami relaxation is the function

(1)

where is a scale parameter and, conventionally, are shape parameters. It includes the Debye relaxation , the Cole-Cole relaxation , and the Davidson-Cole relaxation as special cases (Havriliak and Negami 1966).

For , it has the normalized Stieltjes transform representation

(2)

where, on setting

(3)

the probability density is

(4)

Here is the principal complex argument. Using a single-argument inverse tangent for without correcting its quadrant can give the wrong sign. The asymptotic behaviors of the probability density are

			(5)
			(6)

(Ribeiro de Almeida et al. 2026).

For the Davidson-Cole case and , the corresponding probability density instead has compact support and is

(7)

At , it degenerates to the delta function .