Calculation of the vacuum-polarization potential

The lowest-order vacuum-polarization potential, known as Uehling potential, is expanded for a spherical charge distribution in a convergent form valid for all distances. The accuracy of this expansion is carefully examined at different distances. The ratios of the vacuum-polarization potentials of orders $\alpha \left(Z\alpha \right)$, ${\alpha }^2\left(Z\alpha \right)$, $\alpha {\left(Z\alpha \right)}^3$, $\alpha {\left(Z\alpha \right)}^5$, and $\alpha {\left(Z\alpha \right)}^7$ to the Coulomb potential for a point nucleus are also calculated and presented in figures for $r\le 3.5{\lambda }_e$. Several simple fitting curves are suggested.