Riemann–Liouville integral

In mathematics, the Riemann–Liouville integral associates with a real function ƒ : R → R another function Iαƒ 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, Iαƒ 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. In mathematics, the Riemann–Liouville integral associates with a real function ƒ : R → R another function Iαƒ 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, Iαƒ 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.