  We use the $\zeta$-function regularization and an integral representation of
the complex power of a pseudo differential operator, to give an unambiguous
definition of the determinant of the Dirac operator. We bring this definition
to a workable form by making use of an asymmetric Wigner representation. The
expression so obtained is amenable to several treatments of which we consider
in detail two, the inverse mass expansion and the gradient expansion, with
concrete examples. We obtain explicit closed expressions for the corresponding
Seeley-DeWitt coefficients to all orders. The determinant is shown to be vector
gauge invariant and to posses the correct axial and scale anomalies. The main
virtue of our approach is that it is conceptually simple and systematic and can
be extended naturally to more general problems (bosonic operators,
gravitational fields, etc). In particular, it avoids defining the real and
imaginary parts of the effective action separately. In addition, it does not
reduce the problem to a bosonic one to apply heat kernel nor performs further
analytical rotations of the fields to make the Dirac operator Hermitian. We
illustrate the flexibility of the method by studying some interesting cases.
