(untitled) (Comprehensive Journal of Science, 2026)
This paper provides an extensive analytical investigation into the Riemann-Liouville R-L) fractional integral, a core operator in fractional calculus. By generalizing the) classical n-fold integration process to an arbitrary real order α > 0, the R-L operator enables the modeling of systems with long-range memory and non-local interactions. This research provides a rigorous derivation from Cauchy’s repeated integration formula and explores fundamental, non-classical properties, including linearity, the semigroup property, and its action on diverse function classes. We present detailed mathematical examples demonstrating the emergence of special functions like the Mittag-Leffler function. Furthermore, a comparative analysis between Riemann-Liouville and Caputo formulations is conducted, followed by an exploration of Laplace transforms in the fractional domain. This comprehensive study aims to bridge the gap between abstract mathematical theory and applied computational modeling, providing a robust framework for understanding fractional-order dynamics .