The weyl fractional calculus

The purpose of this paper is to give a simple development of the Weyl fractional transform W−v, and to establish some of its elementary properties. In particular, we shall define the Weyl fractional derivative Wv and show that $$W^{\alpha + \beta } = W^\alpha W^\beta $$ for all α and β, positive, negative, or zero. We also indicate briefly its relationship to the Hadamard finite part of an improper integral.