Laser experiments are widely used to investigate excitation of Rayleigh-Taylor modes, which are of great importance for astrophysical applications. Measured growth rates are normally compared with either the sharp interface or the smooth gradient model. In the present paper an analytical solution is obtained that is valid for arbitrary density gradient scale L. It is a further development of the Mikaelian & Lindl model. New explicit presentation omega(k) is found which describes all discrete modes at all transverse wavenumbers k with one parametric expression. A critical value of kL is shown to exist when two independent solutions for the fastest growing main mode become degenerate, in this case the growth rate is calculated exactly. The focus is on astrophysical applications when boundary conditions are at infinity. The case of rigid walls is also considered to study the interrelation with the Chandrasekhar model. Results are supposed to be used for nonlinear RT treatment to analyze mixing in supernovae and other RT-driven objects.