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