Riemann–Liouville integral

In mathematics, the Riemann–Liouville integral associates with a real function ƒ : R → R another function I^αƒ of the same kind for each value of the parameter α > 0. The integral is a manner of generalization of the repeated antiderivative of ƒ in the sense that for positive integer values of α, I^αƒ is an iterated antiderivative of ƒ of order α. The Riemann–Liouville integral is named for Bernhard Riemann and Joseph Liouville, the latter of whom was the first to consider the possibility of fractional calculus in 1832. The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions. It was generalized to arbitrary dimensions by Marcel Riesz, who introduced the Riesz potential.

The Riemann–Liouville integral is defined by

where Γ is the Gamma function and a is an arbitrary but fixed base point. The integral is well-defined provided ƒ is a locally integrable function, and α is a complex number in the half-plane re(α) > 0. The dependence on the base-point a is often suppressed, and represents a freedom in constant of integration. Clearly I¹ƒ is an antiderivative of ƒ (of first order), and for positive integer values of α, I^αƒ is an antiderivative of order α by Cauchy formula for repeated integration. Another notation, which emphasizes the basepoint, is

This also makes sense if a = −∞, with suitable restrictions on ƒ.

...
Wikipedia