In the present investigation compressibility effect on two-dimensional steady natural convection boundary layer flow of a viscous fluid past a vertical plate has been studied. The governing equations are transformed into a non-dimensional form and the solutions are obtained by four different methods, namely, a perturbation method for the small values of local compressibility variable, xi, asymptotic solutions for large xi, the direct finite difference method and the implicit finite difference method with Keller-box scheme for all xi. In order to examine the results obtained from the four methods, calculations have been carried out for a wide range of parameters. The results obtained by the above mentioned four methods are compared and found in excellent agreement. Moreover, the effects of various practical values of the relative temperature difference, alpha, and the surface temperature wave amplitude a on the local skin friction, the rate of heat transfer and the isotherms are shown graphically. (C) 2016 Published by Elsevier Ltd.