Padé-Approximation to the $α^3$ Vacuum-Polarization Function and the Lamb Shift of the Muonic Hydrogen

The contribution of the $\alpha^3$ single electron-loop vacuum-polarization diagrams to the Lamb shift of the muonic hydrogen has been evaluated recently by two methods. One uses the exact parametric representation of the vacuum-polarization function while the other relies on the Pad\'{e} approximation method. High numerical precision of these calculations enables us to examine the accuracy of the Monte-Carlo integration as well as that of the Pad\'{e} method applied to the Lamb shift problem.