The advanced theory of fractional calculus, Third edition

概要：

Fractional calculus is an extension of differentiation and integration to fractional order. The birth of fractional calculus was in almost the same times as differential and integral calculus. Therefore, the study of fractional calculus has a long history over three centuries. This book is a monograph on fractional calculus. The feature of this book is that fractional calculus is fully treated in the form of an operator theory in a function space. A differential operator is extended to a general operator satisfying Leibniz's rule. By an integral representation of integer powers of this operator, the power of this operator is extended to complex powers. Thanks to this operator-theoretic treatment, fractional calculus is formulated from a broader viewpoint than previous fractional calculus. The obtained results are general, so that they will be useful in many fields of applications. This book covers many topics such as an axiomatic formulation of a function space to which the extended general operator is applied, an integral representation of a complex power of the operator, the linearity and continuity of a complex power of the operator, generalized Leibniz's theorem, composition rules between complex powers of the operator, generalized fractional integration by parts, the basics of fractional calculus of variations, fractional differ-integration of composite functions, the Laplace and Fourier transforms in fractional calculus, an extended function space composed of distributions and its coset space, and special functions of complex order. In addition, miscellaneous problems in mathematics and physics are also illustrated as applications.