Analytical Theory

Most of analytical calculations of in the EPC theory are related to the number of particles at a given depth performed in the framework of the straightforward model. Going to deal with the electron–photon cascade theory, we slightly change designations of basic cascade functions, characterizing differential (with respect to observed energy E) spectra particles in cascades initiated by single-particle sources, and make replacements \( G_{a\rightarrow e}(t,E_0\rightarrow E)\mapsto P(t, E|E_0),\quad G_{a\rightarrow \gamma }(t,E_0\rightarrow E)\mapsto \varGamma (t, E|E_0).\) Information about the primary particles of the cascade will be stored in the initial conditions of the equations for these functions: \( P(0,E)=\delta (E-E_0),\quad \varGamma (0,E)=0 \) means that the cascade is initiated by primary electron with energy \(E_0\) , whereas conditions \( P(0,E)=0,\quad \varGamma (0,E)=\delta (E-E_0)\) say, that the primary particle is a photon.