[en] The self-energy operator for an electron in an external Coulomb potential is investigated analytically using a mass eigenfunction expansion concept reported earlier. Contour integration techniques in the complex m² plane are used to combine bound state and continuum contributions into a single integral. The result is a relatively simple integral representation for the mass operator. Only terms ignoring the ''shift correction'' are considered in this preliminary study. A transformation to a basis of relativistic Coulomb Sturmian functions exhibits the Zα dependence of the integrand in a strikingly simple way. The entire investigation is set in the framework of the ''scalar formalism'' for quantum electrodynamics investigated earlier by a number of authors and based on the ''second-order'' Dirac equation, {Pix(1+iσ)xPi+m²}Phi = 0, where Phi is a 2 x 1 Pauli spinor