The Mittag-Leffler and Gamma Function

The Mittag-Leffler function is a basic function in fractional calculus and is an entire function in the complex plane. Entire functions are generalization of polynomials to infinite degree, do not have singularities in the complex plane, and possess a wealth of beautiful properties. Entire functions are defined by a Taylor series that converges for any finite complex number. In addition to the summation formulation and as a consequence of Weierstrass's factorization theorem, entire functions possess a product form in which all zeros appear.

The study of properties of entire functions has led Mittag-Leffler, around 1900, to the study of his function. The Mittag-Leffler function is a primary example of an entire function, tunable in its order, and is represented by the simple eyeing Taylor series, in which another basic complex function, the Gamma function pops up.

The Mittag-Leffler and Gamma Function mainly targets the mathematical properties of the Mittag-Leffler function in easy-to-understand language, not its applications to fractional analysis nor its numerical evaluation. Since the Gamma function plays a crucial role in the properties of the Mittag-Leffler function, a comprehensive treatment of the Mittag-Leffler function requires the knowledge of the Gamma function. The second part of the book attempts to present a complete study, enriched with historical references.

Piet Van Mieghem, Professor, Delft University of Technology

Piet Van Mieghem is Professor at the Delft University of Technology with a chair in telecommunication networks that he broadened to Network Science. His main research interests lie in the analysis of complex networks such as infrastructural, biological, brain, social networks. He has published five books. He is an IEEE Fellow and received an Advanced ERC grant 2020 for Virus Spread in Networks. He was a visiting scientist at MIT (1992-1993) and a visiting professor at UCLA (2005), Cornell University (2009), Stanford University, and Princeton University (2022).