  The theory of fractional calculus has developed in a number of directions
over the years, including: the formulation of multiple different definitions of
fractional differintegration; the extension of various properties of standard
calculus into the fractional scenario; the application of fractional
differintegrals to assorted special functions. Recently, a new variant of
fractional calculus has arisen, namely incomplete fractional calculus. In two
very recent papers, incomplete versions of the Riemann-Liouville and Caputo
fractional differintegrals have been formulated and applied to several
important special functions.
  In the current work, we develop the theory of incomplete fractional calculus
in more depth, investigating further properties of the incomplete
Riemann-Liouville fractional differintegrals and answering some fundamental
questions about these operators. By considering appropriate function spaces, we
formulate rigorously the definitions of incomplete Riemann-Liouville fractional
integration, and justify how this model may be used to analyse a wider class of
functions than classical fractional calculus. By using analytic continuation,
we formulate definitions for incomplete Riemann-Liouville fractional
differentiation, hence extending the incomplete integrals to a fully-fledged
model of fractional calculus.
  We also investigate and analyse these operators further, in order to prove
new properties. These include a Leibniz rule for incomplete differintegrals of
products, and composition properties of incomplete differintegrals with
classical calculus operations. These are natural and expected issues to
investigate in any new model of fractional calculus, and in the incomplete
Riemann-Liouville model the results emerge naturally from the definition
previously proposed.
