A Method to Extend the Riemann-Liouville Fractional Integral

Abstract

In this paper, we examine the restricted orders of the Riemann-Liouville Fractional Integral. Specifically, the Riemann-Liouville Fractional Integral exists solely for orders α ∣ R(α) ∈ R+, also known as the positive complex half-plane. The Riemann-Liouville Fractional Integral is fundamental to the field of fractional calculus, which aims to generalize the normal derivative and integral to fractional and complex orders. Extending the input space of the Riemann-Liouville Fractional Integral’s orders to α ∈ C as opposed to the positive complex half-plane allows for the completion of the integral operator in fractional calculus. In this paper, we propose a method to forcefully extend the input orders of the Riemann-Liouville Fractional Integral through the use of formal calculations. That is, we initially find only the general form of the integral. We then use analytic continuation on any singularities in the general form, which results in the form being defined for all orders α ∈ C. We conclude this paper by applying our method on common functions, such as logarithms, trigonometric functions, and rational functions and going over the philosophy of formal calculations and mathematical definitions.

Keywords: fractional calculus, formal calculations, analytic continuation