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

Based on the residue theorem we present a rigorous and detailed mathematical treatment for computing a contour integral appearing in the Keldysh theory of the photoionization rate for the hydrogen atom. We consider two different contours in the u = sin θ domain and the unique contour in the z = exp (iθ) domain (θ = ωt). Particularly in z = exp (iθ) domain we find that one of the poles exists at z = 0 which corresponds to a large enclosed circle in the u = sin θ domain. Consistent results are obtained for both domains. We numerically show that the contribution from z = 0 is negligibly small compared with that from poles for λ = 248 nm.