The integral analogue of the Leibniz rule

Abstract

This paper demonstrates that the classical Leibniz rule for the derivative of the product of two functions \[ D N u υ = ∑ k = 0 N ( k N ) D N − k u D k υ {D^N}u\upsilon = \sum \limits _{k = 0}^N {(_k^N)} {D^{N - k}}u{D^k}\upsilon \] has the integral analog \[ D α u υ = ∫ − ∞ ∞ ( ω α ) D α − ω u D ω υ d ω . {D^\alpha }u\upsilon = \int _{ - \infty }^\infty {(_\omega ^\alpha )} {D^{\alpha - \omega }}u{D^\omega }\upsilon d\omega .{\text { }} \] The derivatives occurring are “fractional derivatives.” Various generalizations of the integral are given, and their relationship to Parseval’s formula from the theory of Fourier integrals is revealed. Finally, several definite integrals are evaluated using our results.