Vacuum Polarization in Mesonic Atoms

The effects of vacuum polarization on the energy levels of $\pi$ and $\mu$ mesonic atoms are computed to the lowest order for states $1s(Z<~12)$, $2p(Z<~30)$, $3d(Z<~82)$, $4f(Z<~82)$. If one ignores the finite size of the nucleus, these results may be written explicitly in a closed form. The effect of the finite size of the nucleus is estimated and shown to be ${\left(Z\alpha \right)}^2$.