We present a procedure of differential renormalization at the one loop level
which avoids introducing unnecessary renormalization constants and
automatically preserves abelian gauge invariance. The amplitudes are expressed
in terms of a basis of singular functions. The local terms appearing in the
renormalization of these functions are determined by requiring consistency with
the propagator equation. Previous results in abelian theories, with and without
supersymmetry, are discussed in this context.Comment: 13 pages, LaTeX. Some equations corrected and a reference added.
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