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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The integral analogue of the Leibniz rule


Author: Thomas J. Osler
Journal: Math. Comp. 26 (1972), 903-915
MSC: Primary 65D25
MathSciNet review: 0314240
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Abstract: This paper demonstrates that the classical Leibniz rule for the derivative of the product of two functions

$\displaystyle {D^N}u\upsilon = \sum\limits_{k = 0}^N {(_k^N)} {D^{N - k}}u{D^k}\upsilon $

has the integral analog

$\displaystyle {D^\alpha }u\upsilon = \int_{ - \infty }^\infty {(_\omega ^\alpha )} {D^{\alpha - \omega }}u{D^\omega }\upsilon d\omega .{\text{ }}$

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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1972-0314240-4
PII: S 0025-5718(1972)0314240-4
Keywords: Fractional derivative, Leibniz rule, Fourier transforms, Parseval relation, special functions
Article copyright: © Copyright 1972 American Mathematical Society