The exponent parameter , which takes a value between 0 and 1, allows to describe different spectral shapes. When , the Cole-Cole model reduces to the Debye model. When , the relaxation is stretched, i.e. it extends over a wider range on a logarithmic scale than Debye relaxation.
The separation of the complex dielectric constant ε (ω) was reported in the original paper by Cole and Cole as follows:
Upon introduction of hyperbolic functions, the above expressions reduce to:
These equations reduce to the Debye expression when .
Cole-Cole relaxation constitutes a special case of Havriliak-Negami relaxation when the symmetry parameter (β) is equal to 1 - that is, when the relaxation peaks are symmetric. Another special case of Havriliak-Negami relaxation (β<1, α=1) is known as Cole-Davidson relaxation, for an abridged and updated review of anomalous dielectric relaxation in dissored systems see Kalmykov.