Analytical Calculation to All Orders inZαof Vacuum Polarization at Short Distances

We have calculated the leading terms of the potential due to vacuum polarization about a point nucleus, $V\left(\mathcal{r}\right)=m[a_{-1\mathrm{}}{\left(m\mathcal{r}\right)}^{-1\mathrm{}}+a_0+a_1(m\mathcal{r})+a_{2\lambda }{\left(m\mathcal{r}\right)}^{2\lambda }]$. Our results for $a_1$ and $a_{2\lambda }$, to all orders in $Z\alpha$, extend the previously known results which included terms only up to ${\left(Z\alpha \right)}^3$. For $a_{-1\mathrm{}}$ we confirm previous results. The new portions of $a_1$ and $a_{2\lambda }$ do not remove present discrepancies in muonic x-ray transition such as $5g\to 4f$ in ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$.