Formulation using the dielectric constant

Böttcher and Bordewijk, page 40:

\hat{\epsilon}(\omega) &= \epsilon_\infty + (\epsilon - \epsilon_\infty) \int_0^\infty \frac{g(\tau)}{1 + i \omega \tau} d\tau\\ &\text{with}\\ &\int_0^\infty g(\tau) d\tau = 1

For a g(\tau) = \delta(\tau - \tau_1) it follows (see eqs. 8.182 in B&B):

\hat{\epsilon}(\omega) &= \epsilon_\infty + (\epsilon - \epsilon_\infty) \frac{g(\tau)}{1 + i \omega \tau_1}

For multiple relaxation times in a discrete case (8.187 in B&W):

\hat{\epsilon}(\omega) &= \epsilon_\infty + (\epsilon - \epsilon_\infty) \sum_k \frac{g_k}{1 + i \omega \tau_k}\\ &\text{with}\\ &\sum_k g_k = 1

Transformation to conductivity

Tarasov and Titov substitute \epsilon for \sigma (they cite the electrostatic analogy and the formulation for dielectric materials with losses):

\hat{\sigma}(\omega) &= \sigma_\infty + (\sigma - \sigma_\infty) \int_0^{\infty} \frac{g(\tau)}{1 + i \omega \tau} d\tau\\ &\text{with}:\\ &\int_0^{\infty} g(\tau) d\tau