This textbook provides a comprehensive exploration of special functions and fractional calculus, offering a structured approach through solved and proposed exercises. Covering key mathematical concepts such as Mittag-Leffler functions, Kilbas-Saigo functions, and the Erdélyi-Kober fractional integral, it balances theoretical insights with practical applications. Appendices introduce Barnes G-functions and demonstrate the use of Mathematica for fractional calculus, expanding the book’s accessibility. With an updated index and extensive references, this edition serves as a valuable resource for researchers, graduate students, and professionals in applied mathematics and related fields.

Edmundo Capelas de Oliveira is a Full (retired) Professor in the Department of Applied Mathematics at IMECC, Unicamp, Brazil. His research focuses on fractional calculus and its applications.

Jayme Vaz is a Full Professor in the Department of Applied Mathematics at IMECC, Unicamp, Brazil. His research interests include Mathematical Physics, particularly Clifford algebras, and fractional calculus.