RIEMANN–LIOUVILLE FRACTIONAL OPERATORS OF LOGARITHMIC FUNCTION WITH CONVERGENCE ANALYSIS

Abstract

Fractional calculus extends classical calculus to non-integer orders, yet the systematic study of fractional operators applied to logarithmic functions remains limited. This study derives and analyzes Riemann-Liouville fractional operators applied to ln(x) for orders 0<alpha<=1 . Using Taylor series expansion around x=1 , explicit infinite series formulations involving gamma function ratios and fractional powers of (x-1) are obtained. Rigorous convergence analysis using the ratio test and boundary point examination reveals an important asymmetry: the fractional integral converges on (0,2], whereas the fractional derivative converges on (0,2). This difference arises from distinct growth behaviors of the associated gamma functions. Matlab simulations validate theoretical results and demonstrate limiting behaviors: as alpha approaches 0, both operators converge to ln(x); as alpha approaches 1, they converge to xln(x)-x and 1/x respectively, confirming consistency with classical calculus.