We will generalize the Atiyah-Singer index theorem to the case of manifolds with boundary/cylindrical ends, where the operator near the boundary is of the form d/dt + A, with A non-degenerate self-adjoint. The formula for the index is the same as in the classical case, except we must add an extra term which only depends on A. This term will turn out to be a spectral invariant, i.e. it will be some kind of zeta-function defined in terms of the spectrum of A.