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Integral Transforms and Special Functions

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An Introduction to Anomalous Diffusion and Relaxation

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Abstract

This chapter provides the essential mathematical tools to be used in the subsequent chapters, and is intended to make the book as self-contained as possible. The first part of the chapter is dedicated to review some useful properties of the integral transforms of Fourier and Laplace, illustrating their applicability with a few examples of physical problems. The second part of the chapter reviews some basic properties of the gamma and related functions. In connection with these special functions, we introduce the definition of the mathematical Mellin transform, with a short discussion on the Mellin-Barnes integral representation, to be used later on to face the problems of fractional diffusion. The last part of the chapter is dedicated to a concise introduction to some special functions of the fractional calculus, like the Mittag-Leffler function, with its generalizations, the Wright function, the Meijer \({\text {G}}\)–function, and the \({\text {H}}-\)function of Fox .

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Authors and Affiliations

  1. Departamento de Física, Universidade Estadual de Maringá, Maringá, Paraná, Brazil

    Luiz Roberto Evangelista

  2. Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa, Paraná, Brazil

    Ervin Kaminski Lenzi

Authors
  1. Luiz Roberto Evangelista
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  2. Ervin Kaminski Lenzi
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Correspondence to Luiz Roberto Evangelista .

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Evangelista, L.R., Lenzi, E.K. (2023). Integral Transforms and Special Functions. In: An Introduction to Anomalous Diffusion and Relaxation. PoliTO Springer Series. Springer, Cham. https://doi.org/10.1007/978-3-031-18150-4_1

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