  Hard X-Ray bremsstrahlung continuum spectra, such as from solar flares, are
commonly described in terms of power-law fits, either to the photon spectra
themselves or to the electron spectra responsible for them. In applications
various approximate relations between electron and photon spectral indices are
often used for energies both above and below electron low-energy cutoffs. We
examine the form of the exact relationships in various situations, and for
various cross-sections, showing that empirical relations sometimes used can be
highly misleading and consider how to improve fitting procedures. We obtain
expressions for photon spectra from single, double and truncated power-law
electron spectra for a variety of cross-sections and for the thin and thick
target models and simple analytic expressions for the Bethe-Heitler cases. We
show that above a low-energy cutoff the Kramers and Bethe-Heitler results match
reasonably well with results for exact cross-sections up to energies around 100
keV; that below the low-energy cutoff, Kramers and other constant spectral
index forms commonly used are very poor approximations to accurate results; but
that our analytical forms are a very good match. Analytical forms of the
Bethe-Heitler photon spectra from general power-law electron spectra are an
excellent match to exact results for both thin and thick targets and they
enable much faster spectral fitting than evaluation of the full spectral
integrations.
