The statistical nature of the second order corrections to the thermal   SZE
  This paper shows that the accepted expressions for the second order corrections in the parameter $z$ to the thermal Sunyaev-Zel'dovich effect can be accurately reproduced by a simple convolution integral approach. This representation allows to separate the second order SZE corrections into two type of components. One associated to a single line broadening, directly related to the even derivative terms present in the distortion intensity curve, while the other is related to a frequency shift, which is in turn related to the first derivative term. 
A. Sandoval-Villalbazo and L.S. Garcia-Colin