Voigt profile

(Centered) Voigt

Probability density function Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6. The black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.
Cumulative distribution function
Parameters	${\displaystyle \gamma ,\sigma >0}$
Support	${\displaystyle x\in (-\infty ,\infty )}$
PDF	${\displaystyle {\frac {\Re [w(z)]}{\sigma {\sqrt {2\pi }}}},~~~z={\frac {x+i\gamma }{\sigma {\sqrt {2}}}}}$
CDF	(complicated - see text)
Mean	(not defined)
Median	${\displaystyle 0}$
Mode	${\displaystyle 0}$
Variance	(not defined)
Skewness	(not defined)
Ex. kurtosis	(not defined)
MGF	(not defined)
CF	${\displaystyle e^{-\gamma |t|-\sigma ^{2}t^{2}/2}}$

In spectroscopy, the Voigt profile (named after Woldemar Voigt) is a line profile resulting 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 computational expense of the convolution operation, the Voigt profile is often approximated using a pseudo-Voigt profile.

All normalized line profiles can be considered to be probability distributions. The Gaussian profile has a Gaussian, or normal, distribution and a Lorentzian profile has a Lorentz, or Cauchy, distribution. Without loss of generality, we can consider only centered profiles, which peak at zero. The Voigt profile is then a convolution of a Lorentz profile and a Gaussian profile:

where x is the shift from the line center, ${\displaystyle G(x;\sigma )}$ is the centered Gaussian profile:

and ${\displaystyle L(x;\gamma )}$ is the centered Lorentzian profile: