  This work provides an in-depth mathematical description of the response
functions that are used for spatial and spectral analysis of X-ray data. The
use of such functions is well-known to anyone familiar with the analysis of
X-ray data where they may be identified with the quantities contained in the
Ancillary Response File (ARF), the Redistribution Matrix File (RMF), and the
Exposure Map. Starting from first-principles, explicit mathematical expressions
for these functions, for both imaging and dispersive modes, are arrived at in
terms of the underlying instrumental characteristics of the telescope including
the effects of pointing motion. The response functions are presented in the
context of integral equations relating the expected detector count rate to the
source spectrum incident upon the telescope. Their application to the analysis
of several source distributions is considered. These include multiple, possibly
overlapping, and spectrally distinct point sources, as well as extended
sources. Assumptions and limitations behind the usage of these functions, as
well as their practical computation are addressed.
