This Book is useful for Mathematical Analysis student for the point of view in higher studies in Fractional Calculus and Special functions with applications in Mathematical Physics. This text is geared toward advanced graduate students. Not only does it explain the theory underlying the properties of the generalized operator, but it also illustrates the wide variety of fields to which these ideas may be applied. Rather than an exhaustive treatment, it represents an introduction that will appeal to a broad spectrum of students. Accordingly, the mathematics is kept as simple as possible. The first part treatment deals principally with the general properties of differintegral operators and special functions. The last three chapters are mainly oriented toward the applications of these properties to mathematical and other problems.