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

**[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]**Thomas J. Osler,*Leibniz rule for fractional derivatives generalized and an application to infinite series*, SIAM J. Appl. Math.**18**(1970), 658–674. MR**0260942****[4]**Thomas J. Osler,*The fractional derivative of a composite function*, SIAM J. Math. Anal.**1**(1970), 288–293. MR**0260943****[5]**Thomas J. Osler,*Taylor’s series generalized for fractional derivatives and applications*, SIAM J. Math. Anal.**2**(1971), 37–48. MR**0294612****[6]**T. J. Osler,*Mathematical Notes: Fractional Derivatives and Leibniz Rule*, Amer. Math. Monthly**78**(1971), no. 6, 645–649. MR**1536368**, 10.2307/2316573**[7]**Thomas J. Osler,*A further extension of the Leibniz rule to fractional derivatives and its relation to Parseval’s formula*, SIAM J. Math. Anal.**3**(1972), 1–16. MR**0323970****[8]**Thomas J. Osler,*An integral analogue of Taylor’s series and its use in computing Fourier transforms*, Math. Comp.**26**(1972), 449–460. MR**0306828**, 10.1090/S0025-5718-1972-0306828-1**[9]**Thomas J. Osler,*A correction to Leibniz rule for fractional derivatives*, SIAM J. Math. Anal.**4**(1973), 456–459. MR**0323971****[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