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The Darwin-Howie-Whelan equations are commonly used to describe and simulatethe scattering of fast electrons in transmission electron microscopy. They area system of infinitely many envelope functions, derived from the Schr\"odingerequation. However, for the simulation of images only a finite set of envelopefunctions is used, leading to a system of ordinary differential equations inthickness direction of the specimen. We study the mathematical structure ofthis system and provide error estimates to evaluate the accuracy of specialapproximations, like the two-beam and the systematic-row approximation.

On the Darwin-Howie-Whelan equations for the scattering of fast electrons described by the Schr\"odinger equation