Let

be real and

be an integrable real function of a real variable. The following is the formula for the fractional differintegral of order

of

, using the Riemann–Liouville definition [1] with zero as the lower limit of integration:

where

denotes the ordinary derivative of order

, where

is a positive integer.

In this Demonstration, the arguments

and

denote the two arguments of the incomplete beta function, while

usually means the order of the differintegral, and

is the degree of a polynomial. For practical purposes, some singularities have been excluded and replaced with their ordinary derivatives; these singularities arise from the plots of the beta function for nonpositive arguments, the derivatives of integer order for the logarithmic function, and order zero for the exponential. Closed-form expressions were chosen for the differintegral to speed up plotting. For the triangle wave function, plot exclusions were introduced for visual purposes, although the derivative does not exist at these discontinuities.

For the exponential function, the lower limit equals negative infinity. In this Demonstration, the original graph of the exponential is compared to the graph of the fractional differintegral of the exponential with

as the lower limit. For example, setting

generates a result that deviates from the exponential function itself. This corresponds to the semi-differentiation of the exponential function, but fails to take into consideration that a fractional differintegral is, in general, nonlocal and depends on a range specific to the function. The author believes that this is very valuable in demonstrating the nonlocality of differintegration operations.

This was a project for Advanced Topics in Mathematics II, 2019–2020, Torrey Pines High School, San Diego, CA.

[1] K. B. Oldham and J. Spanier,

*The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order*, Mineola, NY: Dover Publications Inc., 2006.