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 theta domain and the unique contour in the z = exp (i theta) domain (theta = omega t). Particularly in z = exp (i theta) domain we find that one of the poles exists at z = 0 which corresponds to a large enclosed circle in the u = sin 0 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 lambda = 248 nm. (C) 2007 Elsevier B.V. All rights reserved.