  A comprehensive analysis on the photon self-energy, the fermion self-energy,
and the fermion vertex function is presented at one loop in the context of
quantum electrodynamics (QED) with 1 extra dimension. In 5-dimensional
theories, characterized by an infinite number of Kaluza-Klein fields, one-loop
amplitudes involve discrete as well as continuous sums, $\sum^\infty_{n=1}\int
d^4k$, that could diverge. Using dimensional regularization, we express such
sums as products of gamma and Epstein functions, both defined on the complex
plane, with divergences arising from poles of these functions in the limit as $
D \to 4$. Using the analytical properties of the Epstein function, we show that
the ultraviolet divergences generated by the Kaluza-Klein sums can be
consistently renormalized, which means that the corresponding renormalized
quantities reduce to the usual ones of QED at the limit of a very large
compactification scale $R^{-1}$. The main features of QED at the one-loop level
were studied. We use the mass-dependent $\mu$-scheme to calculate, in QED with
an arbitrary number $n$ of extra dimensions, a beta function fulfilling all
desirable physical requirements. We argue that in this type of theories, with a
large mass spectrum covering a wide energy range, beta functions should not be
calculated by using mass-independent renormalization schemes. We show that the
beta function is finite for any energy $\mu$. In particular, it reduces to the
usual QED result $e^3/12\pi^2$ for $m\ll \mu \ll R^{-1}$ and vanishes for $m\gg
\mu$, with $m$ the usual fermion mass. Throughout the work, the decoupling
nature of all our results obtained from the analytical properties of the
Epstein function is stressed.
