The proof

As t —► oo (In.ll) still holds in case ΟΧ φ 0, with Tr replaced by b-Tr^ (the limit is independent of v because the limiting operator is trace class). Similarly Getzler’s scaling argument carries over to this setting to give (In. 17), uniformly in x. Finally a similar scaling argument applies to (In.26) (as shown by Bismut and Freed [18], [19]). This allows (In.24) to be inte­ grated over (0, oo) giving the limiting formula:

(In.27)