  We show that the residue density of the logarithm of a generalised Laplacian
on a closed manifold defines an invariant polynomial valued differential form.
We express it in terms of a finite sum of residues of classical
pseudodifferential symbols. In the case of the square of a Dirac operator,
these formulae provide a pedestrian proof of the Atiyah-Singer formula for a
pure Dirac operator in dimension $4$ and for a twisted Dirac operator on a flat
space of any dimension. These correspond to special cases of a more general
formula by S. Scott and D. Zagier announced in \cite{Sc2} and to appear in
\cite{Sc3}. In our approach, which is of perturbative nature, we use either a
Campbell-Hausdorff formula derived by Okikiolu or a non commutative Taylor type
formula.
