Riemann-Liouville Fractional Derivative

For a function , noninteger order , and positive integer with , the left-sided Riemann-Liouville fractional derivative with lower terminal is defined by

(1)

where is the Riemann-Liouville operator of fractional integration. Thus, the Riemann-Liouville fractional derivative performs fractional integration before ordinary differentiation. For integer order , it is defined to be the ordinary th derivative. The special case is a semiderivative.

For lower terminal , the Riemann-Liouville fractional derivative of a power function is

(2)

where is the gamma function. This formula holds for , , and values for which the expression exists. In particular, for a positive noninteger , the Riemann-Liouville fractional derivative of the constant function is

(3)

This is generally nonzero, in contrast to the Caputo fractional derivative of a constant, which is zero.

The fractional derivative of the E_t-function satisfies

(4)

for and .

Riemann-Liouville fractional integrals satisfy the semigroup property

(5)

for . The analogous function composition rule for fractional derivatives does not hold without additional hypotheses; in general,

(6)

Boundary terms or suitable boundary conditions on determine when a derivative composition law is valid.

For lower terminal , the Riemann-Liouville fractional derivative is implemented in the Wolfram Language as FractionalD[f, {x, alpha}].