The integer on the left, which is just the index on x sL2(X\ +S 8> E) for small s > 0, is sometimes called the extended L2 index.

The absence of the local index theorem in the weighted case means that the form of the integrand, AS(s), in (In.33) cannot be so easily computed. However it is a polynomial in s. The removal of the non-constant terms allows the general formula to be recast as