The integral analogue of the Leibniz rule

Author:
Thomas J. Osler

Journal:
Math. Comp. **26** (1972), 903-915

MSC:
Primary 65D25

DOI:
https://doi.org/10.1090/S0025-5718-1972-0314240-4

MathSciNet review:
0314240

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper demonstrates that the classical Leibniz rule for the derivative of the product of two functions

**[1]**A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi,*Tables of Integral Transforms*. Vols. 1, 2, McGraw-Hill, New York, 1954, 1955. MR**15**, 868; MR**16**, 586.**[2]**Y. L. Luke,*The Special Functions and Their Approximations*. Vols. 1, 2, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR**39**#3039; MR**40**#2909.**[3]**T. J. Osler, ``Leibniz rule for fractional derivatives generalized and an application to infinite series,''*SIAM J. Appl. Math.*, v. 18, 1970, pp. 658-674. MR**41**#5562. MR**0260942 (41:5562)****[4]**T. J. Osler, ``The fractional derivative of a composite function,''*SIAM J. Math. Anal.*, v. 1, 1970, pp. 288-293. MR**41**#5563. MR**0260943 (41:5563)****[5]**T. J. Osler, ``Taylor's series generalized for fractional derivatives and applications,''*SIAM J. Math. Anal.*, v. 2, 1971, pp. 37-48. MR**0294612 (45:3682)****[6]**T. J. Osler, ``Fractional derivatives and Leibniz rule,''*Amer. Math. Monthly*, v. 78, 1971, pp. 645-649. MR**1536368****[7]**T. J. Osler, ``A further extension of the Leibniz rule to fractional derivatives and its relation to Parseval's formula,''*SIAM J. Math. Anal.*, v. 3, 1972, pp. 1-16. MR**0323970 (48:2323a)****[8]**T. J. Osler, ``An integral analogue of Taylor's series and its use in computing Fourier transforms,''*Math. Comp.*, v. 26, 1972, pp. 449-460. MR**0306828 (46:5950)****[9]**T. J. Osler, ``A correction to Leibniz rule for fractional derivatives.'' (To appear.) MR**0323971 (48:2323b)****[10]**E. C. Titchmarsh,*The Theory of Functions*, 2nd ed., Oxford Univ. Press, London, 1939.**[11]**E. C. Titchmarsh,*Introduction to the Theory of Fourier Integrals*, 2nd ed., Clarendon Press, Oxford, 1948.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0314240-4

Keywords:
Fractional derivative,
Leibniz rule,
Fourier transforms,
Parseval relation,
special functions

Article copyright:
© Copyright 1972
American Mathematical Society