New proof of the convergence of the Laplacian series at physical surfaces of the planets

It is proven that the external gravitational potential of a planet may be written as a series in spherical functions - $V(r,\phi,\lambda)=f \sum_{n=0}^{\infty} \frac{1}{r^{n+1}}\int \mu (r’,\phi’, \lambda’)P_n(cos \gamma)r’^{n+2} dr’ d sin \phi’ d\lambda’$, uniformly converging everywhere on and outside the physical surface of the planet, with a normalized nonlinear dispersion of the relief height conforming to the condition $\bar{D}_J<\frac{1+\hat{H}/R}{\beta J \sqrt{2 J+1}}\left( \frac{R-\Delta}{R+\hat{H}} \right)^J$, where $\Delta>0$ is the maximum depth of the relief with respect to the mean sphere radius R; $\hat{Н}$ is the height; the density of the surface masses (lying above the sphere passing through the observation point $r, \phi, \lambda$) is an analytical function of the coordinates.