  In honour of Detlef D\"urr, we report on a mathematical rigorous computation
of the electric vacuum polarisation current and extract the well-known
expression for the second order perturbation. Intermediate steps in the
presented calculation demonstrate, to the knowledge of the authors for the
first time, mathematical rigorous versions of the combined dimensional and
Pauli-Villars regularisation schemes. These are employed as computational tools
to infer convenient integral representations during the computation. The said
second order expression is determined up to a remaining degree of freedom of a
real number -- without ill-defined terms from start to end.
