k-distribution of transmission function and theory of Dirichlet series

Abstract A representation of the transmission function by a series of exponents is discussed in the context of the Dirichlet series theory. A rigorous expression for the distribution function of the absorption coefficient is obtained for a homogeneous medium. In this case it is a formalization of the ordering procedure for absorption coefficients. A rigorous extension of this expression to the case of a nonhomogeneous medium, overlapping spectra and integrals with the source function is also performed. The resulting relations can be used to derive approximate formulas. Thus, such a formula whose accuracy can be evaluated is written for a nonhomogeneous medium and in relation to this formula, the meaning of the correlation of k -distributions is discussed. It is shown that the number of terms in the series of exponents used for the transmission function can be greatly reduced by means of the foregoing results. As a general conclusion, it follows that transmission functions should be represented by series of exponents using the distribution function for absorption coefficients rather than the distribution function density for absorption coefficients.