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In recent articles [Mishima et al., Phys. Rev. A, 66, 033401(2002); Chao,Phys. Rev. A, 72, 053414 (2005)] it was proposed to use the residue theorem forthe exact calculation of the transition amplitude describing strong-fieldionization of atomic systems within Keldysh theory. This should avoid thenecessity to apply the method of steepest descent (saddle-point approximation).Comparing the results of both approaches for atomic hydrogen a difference by afactor of 2 was found for the 1s, and an even more drastic deviation for the 2sstate. Thus it was concluded that the use of the saddle-point approximation isproblematic. In this work the deviations are explained and it is shown that theprevious conclusion is based on an unjustified neglect of an importantcontribution occurring in the application of the residue theorem. Furthermore,the applicability of the method of steepest descent for the ionization ofRydberg states is discussed and an improvement of the standard result issuggested that successfully removes the otherwise drastic failure for largeprincipal quantum numbers.Comment: 7 pages, 3 figures, 1 tabl

Exact Keldysh theory of strong-field ionization: Residue method versus saddle-point approximation