Fractional Calculus　An Extension of Differential and Integral Calculus Revised Edition

概要：

This book is a monograph on an extension of differential and integral calculus. The theory in this book is built on an operator-based formulation of fractional calculus. An integer power of a linear operator satisfying Leibniz's rule is extended to a complex power and many formulas of differential and integral calculus are extended into a domain of complex powers. Such extensions include, for example, Leibniz's theorem, composition rules of operators, integration by parts, the calculus of variations, the Laplace transform, the Fourier transform, and various kinds of special functions including the Bessel polynomial introduced by Krall and Frink. Faa di Bruno's formula on differentiation of a composite function is also extended to a more general formula. In the last chapter of this book, miscellaneous problems in mathematics, physics, and engineering are illustrated as applications. The reader of this book is assumed to be acquainted with undergraduate-level complex analysis together with a bit of knowledge of function theory, topology, and abstract algebra.

author:
Takahiro INOUE
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 Academic Research Publication, Japan.