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Title: A Unified Theory of Generalized Differentiation and Integration (Third Edition): A NEW APPROACH TO FRACTIONAL CALCULUS

Author: Takahiro INOUE

Category: General Natural Science
Number of pages: 124
Size: B5

Notice: The e-book is available on Amazon Kindle Store.

Language: English

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Book Summary

The subject of this book is the fractional calculus which treats differentiation and integration to fractional order. This book presents a new approach to the fractional calculus being based on linear differentiator-like operators to arbitrary order. In this book, many theorems, corollaries, and lemmas are derived to reveal the properties of fractional differentiation and integration. Various special functions of arbitrary order are derived from this theory. Applications are also given to the solutions of basic differintegral equations, diffusion equations, Schroedinger's equations, electric circuits, and electromagnetic wave propagation problems.

Author Profile

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 has published 175 journal papers and 108 proceedings papers of international conferences. 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" and "Fractional Calculus" published in 2016 and 2018, respectively, by BookWay, Japan.

From the Author

In this book, a differentiation operator in fractional calculus is generalized to a more general operator that I call semi p-operator which may be linear or nonlinear. Then, we consider a function sequence called the lambda-sequence generated by an operator-based equation p^n f = lambda_n f. From the generating function of the lambda-sequence, its integral representation in a complex region is derived. Like the Gamma function which is an extension of a factorial, we can extend an index n of the lambda_n to an arbitrary complex value z via its integral representation. Thereby we can define a linear semi p-operator to arbitrary order. The classical fractional calculus can be derived as a special case. Various special functions of arbitrary order follow from lambda_z by choosing different pairs of p and f. Applications of the theory are given in the last chapter with special emphasis on electromagnetic wave problems. The author believes that the presented theory will provide fractional calculus and special function theory with a new approach.