One-loop QED vertex in any covariant gauge: Its complete analytic form.

The one loop vertex in QED is calculated in arbitrary covariant gauges as an analytic function of its momenta. The vertex is decomposed into a longitudinal part, that is fully responsible for ensuring the Ward and Ward-Takahashi identities are satisfied, and a transverse part. The transverse part is decomposed into 8 independent components each being separately free of kinematic singularities in any covariant gauge in a basis that modifies that proposed by Ball and Chiu. Analytic expressions for all 11 components of the O(α) vertex are given explicitly in terms of elementary functions and one Spence function. These results greatly simplify in particular kinematic regimes.