In recent articles [K. Mishima et al., Phys. Rev. A, 66, 033401 (2002); S. D. Chao, Phys. Rev. A, 72, 053414 (2005)] it was proposed to use the residue theorem for the exact calculation of the transition amplitude describing strong-field ionization of atomic systems within Keldysh theory. This should avoid the necessity of applying the method of steepest descent (saddle-point approximation). Comparing the results of both approaches for atomic hydrogen a difference by a factor of 2 was found for the $1s$ and an even more drastic deviation for the $2s$ state. Thus it was concluded that the use of the saddle-point approximation is problematic. In this work the deviations are explained and it is shown that the previous conclusion is based on an unjustified neglect of an important contribution occurring in the application of the residue theorem. Furthermore, the applicability of the method of steepest descent for the ionization of Rydberg states is discussed and an improvement of the standard result is suggested that successfully removes the otherwise drastic failure for large principal quantum numbers.

• Received 6 October 2006

DOI:https://doi.org/10.1103/PhysRevA.75.033403