TY - JOUR A1 - Mickelsson, Jouko A1 - Paycha, Sylvie T1 - The logarithmic residue density of a generalized Laplacian T2 - Journal of the Australian Mathematical Society N2 - We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold definesan invariant polynomial-valued differential form. We express it in terms of a finite sum of residues ofclassical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas providea pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for atwisted Dirac operator on a flat space of any dimension. These correspond to special cases of a moregeneral formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either aCampbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula. KW - residue KW - index KW - Dirac operators Y1 - 2011 UR - https://publishup.uni-potsdam.de/frontdoor/index/index/docId/42517 SN - 0263-6115 SN - 1446-8107 VL - 90 IS - 1 SP - 53 EP - 80 PB - Cambridge Univ. Press CY - Cambridge ER -