Rigorous contour integral analysis of generalized Keldysh theory of strong laser photoionization

The Keldysh integral for the photoionization rate of a H atom under a strong laser field is rigorously computed using a residue theorem. For the case of λ = 248 nm, the contribution from the pole at z = 0 in z = exp (iωt) domain is negligibly small compared with other two poles.