By using the propagator expansion method applied to an electon–ion plasma near thermal equilibrium, a closed‐form solution is found for the high‐frequency, collisional dielectric function in the electrostatic approximation to the first order in the plasma parameter when the Balescu–Lenard collision operator [Phys. Fluids3, 52 (1960); Ann. Phys. (N.Y.)3, 390 (1960)] is used to describe electron–electron and electron–ion collisions. The Balescu–Lenard dielectric function is shown to be an entire function of the complex frequency variable &ohgr;. Since an exact solution for the collisional propagator for the Balescu–Lenard problem is probably impossible, these results illustrate the usefulness of the propagator expansion method as a way of obtaining the dielectric function for collisional plasmas. A comparison is made between the Balescu–Lenard result for the plasma conductivity as the wave vectork → 0 and the Guernsey result, obtained by Oberman, Ron, and Dawson [Phys. Fluids5, 1514 (1962)]. By solving the Balescu–Lenard dispersion relation in the long wavelength approximation, a formula is obtained for the total damping rate for Langmuir waves &Ggr;k, which is the sum of the collisionless (Landau) part &ggr;Lkand the collisional part &ggr;&ngr;k. A numerical solution of the Balescu–Lenard dispersion relation has also been performed, and the analytical and numerical results for the damping rates are compared at long wavelengths. Comparisons of the Balescu–Lenard damping rate to the quantum mechanical result obtained by Dubois, Gilinsky, and Kivelson [Phys. Rev. Lett.8, 419 (1962)] and to other results are also made.