A convolution integral representation of the thermal Sunyaev-Zel'dovich effect

Analytical expressions for the non-relativistic and relativistic Sunyaev-Zel'dovich effect (SZE) are derived by means of suitable convolution integrals. The establishment of these expressions is based on the fact that the SZE disturbed spectrum, at high frequencies, possesses the form of a Laplace transform of the single line distortion profile (structure factor). Implications of this description of the SZE related to light scattering in optically thin plasmas are discussed.