The integral analogue of the Leibniz rule
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- by Thomas J. Osler PDF
- Math. Comp. 26 (1972), 903-915 Request permission
Abstract:
This paper demonstrates that the classical Leibniz rule for the derivative of the product of two functions \[ {D^N}u\upsilon = \sum \limits _{k = 0}^N {(_k^N)} {D^{N - k}}u{D^k}\upsilon \] has the integral analog \[ {D^\alpha }u\upsilon = \int _{ - \infty }^\infty {(_\omega ^\alpha )} {D^{\alpha - \omega }}u{D^\omega }\upsilon d\omega .{\text { }}\] 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.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Math. Comp. 26 (1972), 903-915
- MSC: Primary 65D25
- DOI: https://doi.org/10.1090/S0025-5718-1972-0314240-4
- MathSciNet review: 0314240