This book is the third revised edition of author's book entitled "Fractional Calculus -An extension of differential and integral calculus-" which is a monograph on the theory of fractional calculus. The theory in this book is built on an operator-based formulation and a wide range of contents has been studied under this formulation. Many fundamental formulas regarding differential and integral operators in classical calculus have been extended into those regarding real- or complex-powers of such a kind of operators. The feature of this book is that almost all fundamental subjects in fractional calculus have been studied from a new viewpoint. For example, this book covers Leibniz's theorem, composition rules of operators, integration by parts, the calculus of variations, the Laplace and Fourier transforms, and various kinds of special functions including the Bessel polynomial introduced by Krall and Frink, an extension of Faà di Bruno's formula, and applications to miscellaneous problems in mathematics, physics, and engineering. This edition also covers the formula valid for both linear and nonlinear semi p-operators, the relation to Riemann's zeta function and to Sato's hyperfunction, and the integral representations on raising powers to powers of a linear semi p-operator. The reader of this book is assumed to be acquainted with undergraduate-level complex analysis together with a bit of knowledge of functional analysis, topology, and abstract algebra.

Takahiro INOUE is a professor emeritus, Kumamoto University, a Doctor of Engineering, and a Fellow of IEICE(The Institute of Electronics, Information and Communication Engineers). He is the author of Chapters 21 to 24 of "Electronic Circuit Handbook" published in 2006 by Asakura Publishing Co., Japan, the main author of the text book titled "Analog Electronic Circuits Learning by Examples" published in 2009 by Morikita Publishing Co., Japan, and the author of "A unified theory of generalized differentiation and integration," "Fractional calculus," and "The advanced theory of fractional calculus" published in 2016, 2018, and 2021, respectively, by BookWay, Japan.