Revisiting the Longitudinal Development of Electromagnetic Air Showers: Analytical Improvements to the Greisen Formalism with Zenith Angle Dependence

We present a new analytical approach to the longitudinal development of electromagnetic air showers, offering improvements to the classical Greisen formalism. We introduce a novel profile for the slope function $\lambda_1(s)$ that achieves an agreement less than $0.75\%$ with the original $\lambda_1$ for shower age parameter $s$ between $0.3 < s < 1.4$, where $s$ represents the stage of shower development. Our new formalism provides an improved representation of shower evolution, particularly near and beyond the shower maximum. In addition, we derive a complete expression for the number of particles $N(t)$. Our implementation includes the zenithal angle dependence on the number of particles at the detector level at high altitudes, making it particularly useful for high-altitude observatories. This expression is suitable for implementing air shower simulation tool fitting procedures over a wide range of energies and geometries. Our analysis suggests that the proposed new formalism may provide better agreement with the expected evolution of particle numbers compared to the traditional Greisen formulation.