Faddeeva function

The Faddeeva function or Kramp function is a scaled complex complementary error function,

${\displaystyle w(z):=e^{-z^{2}}\operatorname {erfc} (-iz)=\operatorname {erfcx} (-iz)=e^{-z^{2}}\left(1+{\frac {2i}{\sqrt {\pi }}}\int _{0}^{z}e^{t^{2}}{\text{d}}t\right).}$

It is related to the Fresnel integral, to Dawson's integral, and to the Voigt function.

The function arises in various physical problems, typically relating to electromagnetic responses in complicated media.

• problems involving small-amplitude waves propagating through Maxwellian plasmas, and in particular appears in the plasma's permittivity from which dispersion relations are derived, hence it is sometimes referred to as the plasma dispersion function (although this name is sometimes used instead for the rescaled function Z(z) = i√π w(z) defined by Fried and Conte, 1961).
• the infrared permittivity functions of amorphous oxides have resonances (due to phonons) that are sometimes too complicated to fit using simple harmonic oscillators. The Brendel–Bormann oscillator model uses an infinite superposition of oscillators having slightly different frequencies, with a Gaussian distribution. The integrated response can be written in terms of the Faddeeva function.
• the Faddeeva function is also used in the analysis of electromagnetic waves of the type used in AM radio. Groundwaves are vertically polarised waves propagating over a lossy ground with finite resistivity and permittivity.
• the Faddeeva function also describes the changes of the neutron cross sections of materials as temperature is varied.