We present a critical analysis of the analytic ansatz wave functions that have been proposed for use in the calculation of triple–differential cross–sections for the electron–impact ionization of atoms. We will first discuss, in detail, the mathematical background to these approximations, and we will give, for the first time, cross–sections for the electron–impact ionization of hydrogen using the most elaborate of these. We will show that they give generally unsatisfactory agreement with experiment. We will argue that this failure comes from an inability to model the crucial region where the three particles are close together. The analysis presented should be of value not only in the case of electron–impact ionization but also for a wide range of Coulombic three–body problems in atomic physics.