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.