  Compton scattering is the dominant opacity source in hot neutron stars,
accretion disks around black holes and hot coronae. We collected here a set of
numerical expressions of the Compton scattering redistribution functions for
unpolarized radiation (RF), which are more exact than the widely used
Kompaneets equation. The principal aim of this paper is presentation of the RF
by Guilbert (1981) which is corrected for the computational errors in the
original paper. This corrected RF was used in the series of papers on model
atmosphere computations of hot neutron stars. We have also organized four
existing algorithms for the RF computations into a unified form ready to use in
radiative transfer and model atmosphere codes. The exact method by Nagirner and
Poutanen (1993) was numerically compared to all other algorithms in a very wide
spectral range from hard X-rays to radio waves. Sample computations of the
Compton scattering redistribution functions in thermal plasma were done for
temperatures corresponding to the atmospheres of bursting neutron stars and hot
intergalactic medium. Our formulae are also useful to the study Compton
scattering of unpolarised microwave background radiation in hot intra-cluster
gas and the Sunyaev-Zeldovich effect. We conclude, that the formulae by
Guilbert (1981) and the exact quantum mechanical formulae yield practically the
same redistribution functions for gas temperatures relevant to the atmospheres
of X-ray bursting neutron stars, $T \le 10^8$ K.
