Faltung Formulation of Atmospheric Hadron Cascade

Abstract

The delay integro-differential atmospheric hadron cascade equation in real space is treated by Mellin transform. Faltung integral is introduced to obtain an exact representation of the transformed equation in closed form. In this Faltung formulation, the elasticity distribution function u (η) is the primary input function. This Faltung representation removes the function , which arises from the approximated evaluation of the double energy integral in real space where is just the Mellin transform of u (η), and eliminates some free parameters. The exact flux transform equation is solved by the method of characteristic. Since elasticity distribution transform \(\tilde{u}(s)\) and particle production transform \(\tilde{v}(s)\) appear in the exponentials of the flux transform, simple singularities in \(\tilde{u}(s)\) and \(\tilde{v}(s)\) correspond to essential singularities for the flux transform, and the hadron flux in real space is obtained in terms of the simple and essential residues.