  We use a gauge-invariant regularization procedure, called ``split dimensional
regularization'', to evaluate the quark self-energy $\Sigma (p)$ and
quark-quark-gluon vertex function $\Lambda_\mu (p^\prime,p)$ in the Coulomb
gauge, $\vec{\bigtriangledown}\cdot\vec{A}^a = 0$. The technique of split
dimensional regularization was designed to regulate Coulomb-gauge Feynman
integrals in non-Abelian theories. The technique which is based on two complex
regulating parameters, $\omega$ and $\sigma$, is shown to generate a
well-defined set of Coulomb-gauge integrals. A major component of this project
deals with the evaluation of four-propagator and five-propagator Coulomb
integrals, some of which are nonlocal. It is further argued that the standard
one-loop BRST identity relating $\Sigma$ and $\Lambda_\mu$, should by rights be
replaced by a more general BRST identity which contains two additional
contributions from ghost vertex diagrams. Despite the appearance of nonlocal
Coulomb integrals, both $\Sigma$ and $\Lambda_\mu$ are local functions which
satisfy the appropriate BRST identity. Application of split dimensional
regularization to two-loop energy integrals is briefly discussed.
