  There is presently a great interest in studying static and dynamic properties
of highly charged ions that can be produced in large particle accelerators,
like that at GSI in Darmstadt. To perform corresponding theoretical
calculations with great accuracy requires a highly developed machinery of
computational methods that have not until recently been available. In order to
combine many-body perturbation theory with quantum electrodynamics, the
calculations have generally to be performed in the Coulomb gauge, where
applications have not been so developed as in, for instance, the Feynman gauge.
Formulas for the free-electron self energy and vertex correction have been
given without derivation by Adkins (Phys. Rev. D27, 1814 (1983); Phys. Rev.
D34, 2489 (1986)). In the present paper the formulas of Adkins are verified
with detailed derivations.
