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The vertex function is analyzed using covariant spectral regularizationwithout encountering any divergence, either UV or IR. The mathematics ofcovariant spectral regularization for covariant matrix valued measures with oneLorentz index on open subsets of Minkowski space is described. This is thenapplied to the case of the vertex function and expressions for the densitiesassociated with the vertex function in the t channel and the s channel withrespect to Lebesgue measure on Minkowski space are obtained. These densitiesare well defined, non-divergent and analytic over their domains of definitionand are obtained without using renormalization or needing to consider finalstate radiation. The limit of the expression for the vertex function in the tchannel at low energy and low momenta is computed resulting in the classicalresult for the leading order (LO) contribution to the anomalous magnetic momentof the electron. Also the density for the vertex function in the s channel isused to compute the LO vertex correction contribution to the high energy limitof the cross section for the process $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$.

UV and IR divergence-free calculation of the vertex function at arbitrary values of its arguments