Numerical evaluation of the Lamb shift in hydrogen

A reliable and precise theoretical understanding of quantum electrodynamic effects in atoms is of crucial importance for the determination of fundamental constants from precision electromagnetic measurements. The numerical evaluation of self-energy corrections in hydrogen-like ions with small atomic number Z entails severe loss of numerical significance at intermediate stages of the calculation and slow convergence in the sum over intermediate angular momenta. We present an evaluation of the scaled self-energy function F(Zα) at Z=1 with a relative uncertainty of 2×10^-9