On the Distinguished Role of the MittagLeffler and Wright Functions in Fractional Calculus
Professor Francesco Mainardi

Abstract
Fractional calculus, in allowing integrals and derivatives of any positive real order (the term "fractional" is kept only for historical reasons), can be considered a branch of mathematical analysis which deals with integrodi erential equations where the integrals are of convolution type and exhibit (weakly singular) kernels of powerlaw type. As a matter of fact fractional calculus can be considered a laboratory for special functions and integral transforms. Indeed many problems dealt with fractional calculus can be solved by using Laplace and Fourier transforms and lead to analytical solutions expressed in terms of transcendental functions of MittagLeffler and Wright type. In this plenary lecture we discuss some interesting problems in order to single out the role of these functions. The problems include anomalous relaxation and diffusion and also intermediate phenomena.
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