An analytic expression for the ionization amplitude of hydrogen by electron impact is found to contain a polynomial by an optimal truncation in an asymptotic series and a convergent series. The ionization amplitude, i.e., the transition matrix element on the energy shell, is decomposed into two parts: the structure-scattering factor and correlation factor, based on an approximation of the projectile plane wave in coplanar asymmetric geometries.The contribution of these factors to the triple differential cross section is evaluated using the method of optional truncation of asymptotic and convergent series.