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 fundamen tal constants from precision electromagnetic measurements. The numerical ev aluation of self-energy corrections in hydrogen-like ions with small atomic number Z entails severe loss of numerical significance at intermediate sta ges of the calculation and slow convergence in the sum over intermediate an gular momenta, We present an evaluation of the scaled self-energy function F(Z alpha) at Z = 1 with a relative uncertainty of 2 x 10(-9).