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I am trying to determine the integrand in the equivariant Atiyah-Patodi-Singer index theorem by Donnelli ("Eta invariants for G-spaces") for spin manifolds if one only has isolated fixed points. In "Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture" by Gilkey, Leahy and Park they state (equation (1.6.20.a)) without proof or reference, that it is given by $\prod_{j=1}^k\frac{\sqrt{\lambda_j}}{(\lambda_j-1)}$, where $\lambda_j=e^{i\theta_j}$ and $\theta_j$ denotes the rotation angles at a given fixed point. I don't see how to get there and I can't find a reference. Can anyone give an explanation or a reference? Thanks!

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