  In high-energy hadron-hadron collisions, the dependence of the total
cross-section ($\sigma_{tot}$) with the energy still constitutes an open
problem for QCD. Phenomenological analyses usually relies on analytic
parameterizations provided by the Regge-Gribov formalism and fits to the
experimental data. In this framework, the singularities of the scattering
amplitude in the complex angular momentum plane determine the asymptotic
behavior of $\sigma_{tot}$ in terms of the energy. Usual applications connect
simple and triple pole singularities with asymptotic power and
logarithmic-squared functions of the energy, respectively. More restrict
applications have considered as a leading component for $\sigma_{tot}$ an
empirical function consisting of a logarithmic raised to a real exponent, which
is treated as a free fit parameter. With this function, data reductions lead to
good descriptions of the experimental data and real (not integer) values of the
exponent. In this paper, making use of two independent formalisms (fractional
calculus and Mellin transform), we first show that the singularity associated
with this empirical function is a branch point and then we explore the
mathematical consequences of the result and possible physical interpretations.
After reviewing the determination of the singularities from asymptotic forms of
interest through the Mellin transform, we demonstrate that the same analytic
results can be obtained by means of the Caputo fractional derivative, leading,
therefore, to fractional calculus interpretations. Besides correlating Mellin
transform, non-local fractional derivatives and exploring the generalization
from integer to real exponents and derivative orders, this fractional calculus
result may provide insights for physical interpretations on the asymptotic rise
of $\sigma_{tot}$.
