Brendel–Bormann oscillator model
The Brendel–Bormann oscillator model is a mathematical formula for the frequency dependence of the complex-valued relative permittivity, sometimes referred to as the dielectric function. The model has been used to fit to the complex refractive index of materials with absorption lineshapes exhibiting non-Lorentzian broadening, such as metals and amorphous insulators, across broad spectral ranges, typically near-ultraviolet, visible, and infrared frequencies. The dispersion relation bears the names of R. Brendel and D. Bormann, who derived the model in 1992, despite first being applied to optical constants in the literature by Andrei M. Efimov  and E. G. Makarova in 1983. Around that time, several other researchers also independently discovered the model. The Brendel-Bormann oscillator model is aphysical because it does not satisfy the Kramers–Kronig relations. The model is non-causal, due to a singularity at zero frequency, and non-Hermitian. These drawbacks inspired J. Orosco and C. F. M. Coimbra to develop a similar, causal oscillator model. Mathematical Formulation. The general form of an oscillator model is given by where The Brendel-Bormann oscillator is related to the Lorentzian oscillator formula_7 and Gaussian oscillator formula_8, given by where The Brendel-Bormann oscillator  formula_19 is obtained from the convolution of the two aforementioned oscillators in the manner of which yields where The square root in the definition of formula_24 must be taken such that its imaginary component is positive. This is achieved by: