The exact expression of the Voigt profile function

We solve in exact and closed-form a classical problem of mathematical physics of great interest in spectroscopy: the convolution of a Gaussian and a Lorentzian distribution that define the Voigt profile function, V(x). The solution is based in three steps: a power series development following the integral expression for V(x), the ordinary differential equation (ODE) satisfied by that expansion, and the corresponding solution of the ODE. This work converts in obsolete all graphical, numerical and semi-analytical approximations published previously. All results are clearly expressed in terms of the complementary error function Φc(a)=1-erf(a), where a=ln2γL/γG is, basically, the relation between Lorentzian and Gaussian widths.