We present a nonperturbative numerical evaluation of the one-photon electron self-energy for hydrogenlike ions with low nuclear charge numbers Z = 1 to 5. Our calculation for the 1S state has a numerical uncertainty of 0.8 Hz for hydrogen and 13 Hz for singly ionized helium. Resummation and convergence acceleration techniques that reduce the computer time by about 3 orders of magnitude were employed in the calculation. The numerical results are compared to results based on known terms in the expansion of the self-energy in powers of Zα.
U. D. Jentschura et al., "Calculation of the Electron Self-Energy for Low Nuclear Charge," Physical Review Letters, vol. 82, no. 1, pp. 53-56, American Physical Society (APS), Jan 1999.
The definitive version is available at https://doi.org/10.1103/PhysRevLett.82.53
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