Abstract
For a class of functions defined on the real line, a fractional derivative and integral is defined which is an entire function of exponential type of the order. For simplicity, these operations will be called simply fractional differentiation. Properties of this derivative and its relation to other theories is studied.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. C. Gaer and L. A. Rubel, The fractional Derivative via Entire functions, J. Math. Anal. and Appl., 34, (1971), 289–301.
H. Bateman, Higher Trancendental Functions, Vol. 1, Bateman Manuscript Project (A. Erdelyi, Ed.), McGraw-Hill, New York 1953.
R. P. Boas, Jr., Entire Functions, Academic Press, New York 1954.
R. C. Buck, A class of entire functions, Duke Math. J., 13 (1946), pp. 541–559.
P. Civin, Inequalities for trigonometric integrals, Duke Math. J., 8(1941), pp. 656–665.
R. C. Buck, Interpolation series, Trans. Amer. Math. Soc. 64, (1948), 283–298.
P. Dienes, The Taylor Series, Dover Publications Inc., New York 1957.
L. E. Euler, Commentaires de Saint Petersbourg, Vol. V. 55, pp. 1730–1731.
M. Gaer, Fractional Derivatives and Entire Functions, Ph.D. thesis, University of Illinois, 1968.
J. Hadamard, Essai sur l'etude des fonctions donnees par leur developpement de Taylor, J. de Math. (4), 8(1892), pp. 101–186.
G. H. Hardy, Divergent Series, Clarendon Press, Oxford 1949.
G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals, II, Math. Zeit., 34(1931–32), pp. 403–439.
T. J. Osler, Leibniz rule for fractional derivatives generalized and an application to infinite series, SIAM J. Appl. Math. 18 (1970), 658–674.
T. J. Osler, The fractional derivative of a composite function, SIAM J. Math. Anal. 1, (1970), 288–293.
J. Liouville, Memoire: sur le calcul des differentielles a indices quelconques, J. de l'Ecole Polytechnique, 13 (1832), pp. 71–162.
P. Malliavin and L. A. Rubel, On small entire functions of exponential type with given zeros, Bull. Soc. Math. France, 89(1961), pp. 175–206.
P. Montel, Sur les polynomes d'approximation, Bull. Soc. Math. France, 46(1918), pp. 151–192.
E. L. Post, Generalized differentiation, Trans. Amer. Math. Soc., 32(1930), pp. 723–781.
B. Riemann, Versuch einer allgemeinen Auffassung der Integration und Differentiation, The Collected Works of Bernhard Riemann, 2nd Ed., (H. Weber, Ed.), Dover Publications Inc., New York, 1953.
L. A. Rubel, Necessary and sufficient conditions for Carlson's theorem on entire functions, Trans. Amer. Math. Soc., 83 (1956), pp. 417–429.
W. E. Sewell, Generalized derivatives and approximation by polynomials, Trans. Amer. Math. Soc., 41(1937), pp. 84–123.
E. C. Titchmarsh, The Theory of Functions, 2nd Ed., Oxford University Press, London 1960.
H. Weyl, Bemerkungen zum Begriff des Differentialquotienten gebrochner Ordnung, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 62 (1917), pp. 296–302.
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1965.
M. C. Gaer, Newton interpolation of fractional derivatives and iterates of functions and operators, J. Math. Anal. and Appl. (To appear).
Editor information
Rights and permissions
Copyright information
© 1975 Springer-Verlag
About this chapter
Cite this chapter
Gaer, M.C., Rubel, L.A. (1975). The fractional derivative and entire functions. In: Ross, B. (eds) Fractional Calculus and Its Applications. Lecture Notes in Mathematics, vol 457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067104
Download citation
DOI: https://doi.org/10.1007/BFb0067104
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-07161-7
Online ISBN: 978-3-540-69975-0
eBook Packages: Springer Book Archive