In this paper, the complex variable method of N. I. Muskhelishvili is applied to the problem of bending for small deflections of a thin, isotropic, homogeneous, clamped plate with transverse load. The functional equation involved in Muskhelishvili’s method is solved by means of series expansions. The necessary conformal mapping functions are found from the Schwarz-Christoffel formula or expansion of elliptic functions. For uniformly loaded square and rectangular plates, the central (maximum) deflection obtained by the method of Muskhelishvili is compared to the corresponding deflections obtained by other methods. Deflections for the square plate are given for three and five terms, respectively, of the series expansion of the conformal mapping function in order to estimate convergence properties of the method of solution. The solution for a uniformly loaded clamped plate of equilateral triangular planform is also discussed. Central deflection for this case is given.