  Compton scattering between electrons and photons plays a crucial role in
astrophysical plasmas. Many important aspects of this process can be captured
by using the so-called Compton scattering kernel. For isotropic media, exact
analytic expressions (valid at all electron and photon energies) do exist but
are hampered by numerical issues and often are presented in complicated ways.
In this paper, we summarize, simplify and improve existing analytic expressions
for the Compton scattering kernel, with an eye on clarity and physical
understanding. We provide a detailed overview of important properties of the
kernel covering a wide range of energies and highlighting aspects that have not
been appreciated as much previously. We discuss analytic expressions for the
moments of the kernel, comparing various approximations and demonstrating their
precision. We also illustrate the properties of the scattering kernel for
thermal electrons at various temperatures and photon energies, introducing new
analytic approximations valid to high temperatures. The obtained improved
formulae for the kernel and its moments should prove useful in many
astrophysical computations, one of them being the evolution of spectral
distortions of the cosmic microwave background in the early Universe. A novel
code, CSpack, for efficient computations of the Compton scattering kernel and
its properties (in the future also including anisotropies in the initial
electron and photon distributions) is being developed in a series of papers and
will be available within one month.
