The Voigt profile (named after Woldemar Voigt) is a probability distribution given by a convolution of a Cauchy-Lorentz distribution and a Gaussian distribution. It is often used in analyzing data from spectroscopy or diffraction. Definition Without loss of generality, we can consider only centered profiles, which peak at zero. The Voigt profile is then where x is the shift from the line center, is the centered Gaussian profile: and is the centered Lorentzian profile: The defining integral can be evaluated as: where Re[w(z)] is the real part of the Faddeeva function evaluated for In the limiting cases of and then simplifies to and , respectively. History and applications In spectroscopy, a Voigt profile results from the convolution of two broadening mechanisms, one of which alone would produce a Gaussian profile (usually, as a result of the Doppler broadening), and the other would produce a Lorentzian profile. Voigt profiles are common in many branches of spectroscopy and diffraction. Due to the expense of computing the Faddeeva function, the Voigt profile is sometimes approximated using a pseudo-Voigt profile. Properties The Voigt profile is normalized: since it is a convolution of normalized profiles. The Lorentzian profile has no moments (other than the zeroth), and so the moment-generating function for the Cauchy distribution is not defined. It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. The characteristic function for the (centered) Voigt profile will then be the product of the two: Since normal distributions and Cauchy distributions are stable distributions, they are each closed under convolution (up to change of scale), and it follows that the Voigt distributions are also closed under convolution. Cumulative distribution function Using the above definition for z , the cumulative distribution function (CDF) can be found as follows: Substituting the definition of the Faddeeva function (scaled complex error function) yields for the indefinite integral: which may be solved to yield where is a hypergeometric function. In order for the function to approach zero as x approaches negative infinity (as the CDF must do), an integration constant of 1/2 must be added. This gives for the CDF of Voigt: The uncentered Voigt profile If the Gaussian profile is centered at and the Lorentzian profile is centered at , the convolution is centered at and the characteristic