Self-energy correction to the hyperfine splitting and the electrongfactor in hydrogenlike ions

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The hyperfine structure (hfs) and the $g$ factor of a bound electron are caused by external magnetic fields. For the hfs, the magnetic field is due to the nuclear spin. A uniform-in-space and constant-in-time magnetic field is used to probe the bound-electron $g$ factor. The self-energy corrections to these effects are more difficult to evaluate than those to the Lamb shift. Here, we describe a numerical approach for both effects in the notoriously problematic regime of hydrogenlike bound systems with low nuclear charge numbers. The calculation is nonperturbative in the binding Coulomb field. Accurate numerical values for the remainder functions are provided for $2P$ states and for $\mathit{nS}$ states with $n=1,2,3$.