The integral analogue of the Leibniz rule
Author:
Thomas J. Osler
Journal:
Math. Comp. 26 (1972), 903915
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 has the integral analog 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.
 [1]
A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi, Tables of Integral Transforms. Vols. 1, 2, McGrawHill, 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
(41 #5562)
 [4]
Thomas
J. Osler, The fractional derivative of a composite function,
SIAM J. Math. Anal. 1 (1970), 288–293. MR 0260943
(41 #5563)
 [5]
Thomas
J. Osler, Taylor’s series generalized for fractional
derivatives and applications, SIAM J. Math. Anal. 2
(1971), 37–48. MR 0294612
(45 #3682)
 [6]
T.
J. Osler, Mathematical Notes: Fractional Derivatives and Leibniz
Rule, Amer. Math. Monthly 78 (1971), no. 6,
645–649. MR
1536368, http://dx.doi.org/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
(48 #2323a)
 [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
(46 #5950), http://dx.doi.org/10.1090/S00255718197203068281
 [9]
Thomas
J. Osler, A correction to Leibniz rule for fractional
derivatives, SIAM J. Math. Anal. 4 (1973),
456–459. 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.
 [1]
 A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi, Tables of Integral Transforms. Vols. 1, 2, McGrawHill, 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. 658674. 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. 288293. 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. 3748. MR 0294612 (45:3682)
 [6]
 T. J. Osler, ``Fractional derivatives and Leibniz rule,'' Amer. Math. Monthly, v. 78, 1971, pp. 645649. 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. 116. 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. 449460. 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:
http://dx.doi.org/10.1090/S00255718197203142404
PII:
S 00255718(1972)03142404
Keywords:
Fractional derivative,
Leibniz rule,
Fourier transforms,
Parseval relation,
special functions
Article copyright:
© Copyright 1972
American Mathematical Society
