Abstract
We study the fractional integral (fI) and fractional derivative (fD), attained by the analytic continuation (AC) of Liouville’s fI (LfI) and Riemann-Liouville fI (RLfI). On the AC of RLfI, we give a detailed summary of Lavoie et al’s review. The ACs of RLfI are expressed by means of contour integrals. Two of them use the Cauchy contour, and one is using the Pochhammer contour. In this case, the latter is AC of all the others for the functions treated. For the AC of LfI, one can find studies in Campos’ papers and in Nishimoto’s books, where the AC is using only the Cauchy contour. Here we present also an AC using a modified Pochhammer’s contour. In this case, we see that any of these two ACs is not the AC of the other for all the functions treated. This fact leads to difficulties, if a careful study taking care of the domains of existence of each AC is not adopted. By taking account of this fact, we resolve the difficulties which occur in Nishimoto’s formalism.
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References
P.L. Butzer and R.J. Nessel, Fourier Analysis and Application, Vol. I: One-Dimensional Theory. Birkhäuser Verlag, Basel (1971).
L.M.B.C. Campos, On a Concept of Derivative of Complex Order with Applications to Special Functions. IMA J. Appl. Math. 33 (1984), 109–133.
L.M.B.C. Campos, Rules of derivation with complex order for analytic and branched functions. Portugaliae Mathematica 43 (1986), 347–376.
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).
J.L. Lavoie, R. Tremblay and T.J. Osler, Fundamental properties of fractional derivatives via Pochhammer integrals. In: Fractional Calculus and its Applications (Ed. B. Ross), Lecture Notes in Mathematics 457 (1975), 327–356.
J.L. Lavoie, T.J. Osler and R. Tremblay, Fractional derivatives and special functions. SIAM Review 18 (1976), 240–268.
J. Liouville, Mémoire sur quelque Question de Géométrie et de Mécanique, et sur un genre de calcul pour résoudre ces questions. J. de L’Ecole Polyt. 13(1832), 1–69.
J. Lützen, Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics. Springer-Verlag, New York (1990).
K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley, New York (1993).
T. Morita and K. Sato, Solution of fractional differential equation in terms of distribution theory, Interdiscipl. Inform. Sc. 12 (2006), 71–83.
K. Nishimoto, Fractional Calculus, I. Descartes Press, Koriyama (1989).
K. Nishimoto, An Essence of Nishimoto’s Fractional Calculus. Descartes Press, Koriyama (1991).
M.D. Ortigueira, Fractional Calculus for Scientists and Engineers. Springer, Dordrecht, Heidelberg etc. (2011); DOI: 10.1007/978-94-007-0747-7.
I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).
A.M. Robert, Nonstandard Analysis. Dover, Mineola (2003).
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach Sc. Publ., Amsterdam (1993).
K. Sato, Fractional calculus (Definitions and developments) — Connection with fractals (in Japanese). In: The 11th Symposium on History of Mathematics, Reports of Institute of Mathematics and Computer Sciences, Tsuda College 22 (2001), 22–51.
H. Weyl, In: Hermann Weyl Gesammelte Abhandlung, Band I. Springer-Verlag, Berlin (1968), 663–669.
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis. Cambridge Univ. Press, Cambridge (1935).
D.V. Widder, The Laplace Transform. Princeton Univ. Press, Princeton (1941).
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Morita, T., Sato, Ki. Liouville and Riemann-Liouville fractional derivatives via contour integrals. fcaa 16, 630–653 (2013). https://doi.org/10.2478/s13540-013-0040-9
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DOI: https://doi.org/10.2478/s13540-013-0040-9