f36b880f-3ce7-4adb-8831-5913b24acc7820210218102051576wseamdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS1109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40512720202720201910.37394/23206.2020.19http://wseas.org/wseas/cms.action?id=23185The Pioneers of the Mittag-Leffler Functions in Dielectrical and Mechanical Relaxation ProcessesFrancescoMainardiUniversity Bologna and INFN, Department of Physics and Astronomy, Bologna, ITALYArmandoConsiglioUniversitat Wurzburg, Institut fur Theoretische Physik und Astrophysik, Würzburg, GERMANYWe start with a short survey of the basic properties of the Mittag-Leffler functions Then we focus on the key role of these functions to explain the after-effects and relaxation phenomena occurring in dielectrics and in viscoelastic bodies. For this purpose we recall the main aspects that were formerly discussed by two pioneers in the years 1930’s-1940’s whom we have identified with Harold T. 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