An improved formula for the relativistic corrections to the kinematical   Sunyaev-Zeldovich effect for clusters of galaxies
  We improve the calculation of Nozawa, Itoh, and Kohyama (1998) to provide a formula for relativistic corrections to the thermal and kinematical Sunyaev-Zeldovich effects that is accurate to fourth order in theta_e = kT_e/m_ec^2, T_e and m_e being the electron temperature and electron mass, respectively. We also carry out a direct numerical integration of the Boltzmann collision term and confirm the excellent accuracy of the analytic formula. This formula will be useful for the analysis of the observational data of the forthcoming experiments of the kinematical Sunyaev-Zeldovich effect for clusters of galaxies. 
Satoshi Nozawa, Naoki Itoh, Yasuhiko Suda, and Yoichi Ohhata