Abstract A general solution for the Helmholtz differential equations is obtained in the complex domain and applied to the nonlinear, free, bending vibrations of plates. The analysis is based on the decoupled nonlinear von Karman field equations by Berger assumption for the large deformations of plates. The decoupled differential equation in terms of the deflection function is a fourth order Helmholtz differential equation. Its solution, called the dynamic deflection function, is obtained in the complex domain by means of newly defined first and second kind and modified Bessel functions. The dynamic deflection function can be applied to any plates having any shape and any boundary condition under any arbitrary dynamic loads. For plates with smooth boundary, the parameters of the dynamic deflection function are determined from the boundary conditions of the plates and the initial conditions of the vibrations. The analyses of plates with piece-wise smooth boundaries are obtained on the mapped planes. The nonlinear, free vibration of circular plates are investigated by the dynamic deflection function. The effect of stretching on the natural circular frequencies are illustrated.