The Mittag-Leffler functions \(E_\alpha(z)\) is named after the great Swedish mathematician Gösta Magnus Mittag-Leffler (1846–1927) who defined it by a power series \[ E_\alpha(z)= \sum^\infty_{n=0} z^k/\Gamma(k\alpha+ 1),\quad \alpha\in\mathbb{C},\quad \text{Re\,}\alpha>0, \] and studied properties in 1902–1905 in five subsequent notes in connection with his summation method for divergent series. During the first half of the twentieth century the Mittag-Leffler function remained almost unknown to the majority of scientists. In the 1960’s it was recognized to belong to a more general class of higher transcendental functions, known as the Fox \(H\)-functions.
Successful applications of the Mittag-Leffler function and its generalizations, and their direct involvement in problems of physics, biology, chemistry, engineering and other applied sciences in recent decades has made it better known. The real importance of this function was recognized only when its special role in fractional calculus was discovered. Because the fractional calculus has attracted wide interest in different areas of applied sciences, the authors of the present book call the classical Mittag-Leffler function, as the Queen function of fractional calculus, and they consider all the related functions as her court. The present volume offers a self-contained, comprehensive treatment of Mittag-Leffler functions, ranging from elementary matters to the latest research results.
In addition to the first part, which contains the general theory if five chapters, the authors devote three chapters to the applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave problems phenomena, as well as stochastics.
Six parts, as Appendices, as well as an extended Bibliography conclude this well written book. A special gift to the readers is offered by the authors at the end of each chapter, with a special section on historical and bibliographical notes. A collection of interesting exercises and problems are given in addition, with the aim to attract students or anyone interested in the subject.
The book is intended for a broad audience, comprising graduate students, university instructors and the scientists in fields of pure and applied mathematics, as well as researchers in applied sciences.