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Cauchy’s integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order. (English) Zbl 1202.30068
Summary: The modified Riemann-Liouville fractional derivative applies to functions which are fractional differentiable but not differentiable, in such a manner that they cannot be analyzed by means of the Djrbashian fractional derivative. It provides a fractional Taylor series for functions which are infinitely fractional differentiable, and this result suggests to introduce a definition of analytic functions of fractional order. Cauchy’s conditions for fractional differentiability in the complex plane and the Cauchy integral formula are derived for these kinds of functions.
MSC:
30E99Miscellaneous topics of analysis in the complex domain
26A33Fractional derivatives and integrals (real functions)
References:
[1]Jumarie, G.: Stochastic differential equations with fractional Brownian motion input, Internat. J. Systems sci. 24, No. 6, 1113-1132 (1993) · Zbl 0771.60043 · doi:10.1080/00207729308949547
[2]Jumarie, G.: On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Appl. math. Lett. 18, 817-826 (2005) · Zbl 1075.60068 · doi:10.1016/j.aml.2004.09.012
[3]Jumarie, G.: Modified Riemann–Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput. math. Appl. 51, 1367-1376 (2006) · Zbl 1137.65001 · doi:10.1016/j.camwa.2006.02.001
[4]Jumarie, G.: New stochastic fractional models for malthusian growth, the Poissonian birth process and optimal management of populations, Math. comput. Modelling 44, 231-254 (2006) · Zbl 1130.92043 · doi:10.1016/j.mcm.2005.10.003
[5]Jumarie, G.: Fractional partial differential equations and modified Riemann–Liouville derivatives. Method for solution, J. appl. Math. comput. 24, No. 1–2, 31-48 (2007) · Zbl 1145.26302 · doi:10.1007/BF02832299
[6]Jumarie, G.: Lagrangian mechanics of fractional order, Hamilton–Jacobi fractional PDE and Taylor’s series of non differentiable functions, Chaos solitons fractals 32, No. 3, 969-987 (2007) · Zbl 1154.70011 · doi:10.1016/j.chaos.2006.07.053
[7]Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions, Appl. math. Lett. 22, 378-385 (2009) · Zbl 1171.26305 · doi:10.1016/j.aml.2008.06.003
[8]Kolwankar, K. M.; Gangal, A. D.: Holder exponents of irregular signals and local fractional derivatives, Pramana J. Phys. 48, 49-68 (1997)
[9]Kolwankar, K. M.; Gangal, A. D.: Local fractional Fokker–Planck equation, Phys. rev. Lett. 80, 214-217 (1998) · Zbl 0945.82005 · doi:10.1103/PhysRevLett.80.214
[10]Al-Akaidi, M.: Fractal speech processing, (2004)
[11]Anh, V. V.; Leonenko, N. N.: Spectral theory of renormalized fractional random fields, Teor. imovir. Math. stat. 66, 3-14 (2002) · Zbl 1029.60040
[12]Campos, L. M. C.: On a concept of derivative of complex order with applications to special functions, IMA J. Appl. math. 33, 109-133 (1984) · Zbl 0565.30034 · doi:10.1093/imamat/33.2.109
[13]Campos, L. M. C.: Fractional calculus of analytic and branched functions, Recent advances in fractional calculus (1993) · Zbl 0789.30030
[14]Caputo, M.: Linear model of dissipation whose Q is almost frequency dependent II, Geophys. J. R. astron. Soc. 13, 529-539 (1967)
[15], CISM lecture notes 378 (1997)
[16]Djrbashian, M. M.; Nersesian, A. B.: Fractional derivative and the Cauchy problem for differential equations of fractional order, Izv. acad. Nauk arm. SSR 3, No. 1, 3-29 (1968)
[17]Falconer, K.: Techniques in fractal geometry, (1997) · Zbl 0869.28003
[18]Hilfer, R.: Fractional time evolution, Applications of fractional calculus in physics, 87-130 (2000) · Zbl 0994.34050
[19]Huang, F.; Liu, F.: The space–time fractional diffusion equation with Caputo derivatives, J. appl. Math. comput. 19, No. 1–2, 179-190 (2005) · Zbl 1085.35003 · doi:10.1007/BF02935797
[20]Jumarie, G.: Further results on Fokker–Planck equation of fractional order, Chaos solitons fractals 12, 1873-1886 (2001) · Zbl 1046.82515 · doi:10.1016/S0960-0779(00)00152-1
[21]Jumarie, G.: On the representation of fractional Brownian motion as an integral with respect to (dt)α, Appl. math. Lett. 18, 739-748 (2005) · Zbl 1082.60029 · doi:10.1016/j.aml.2004.05.014
[22]Kober, H.: On fractional integrals and derivatives, Q. J. Math. 11, 193-215 (1940) · Zbl 0025.18502
[23]Letnivov, A. V.: Theory of differentiation of fractional order, Sb. math. 3, 1-7 (1868)
[24], Frontiers in mathematical biology (1994)
[25]Liouville, J.: Sur le calcul des différentielles à indices quelconques, J. ecole polytech. 13, 71 (1832)
[26]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1973)
[27]Nishimoto, K.: Fractional calculus, (1989) · Zbl 0707.26009
[28]Oldham, K. B.; Spanier, J.: The fractional calculus. Theory and application of differentiation and integration to arbitrary order, (1974)
[29]Osler, T. J.: Taylor’s series generalized for fractional derivatives and applications, SIAM J. Math. anal. 2, No. 1, 37-47 (1971) · Zbl 0215.12101 · doi:10.1137/0502004
[30]Podlubny, I.: Fractional differential equations, (1999)
[31]Ross, B.: Fractional calculus and its applications, Lecture notes in mathematics 457 (1974)
[32]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives. Theory and applications, (1987)
[33]Shaher, M.; Odibat, Z. M.: Fractional Green function for linear time-fractional inhomogeneous partial differential equations in fluid mechanics, J. appl. Math. comput. 24, No. 1–2, 167-178 (2007) · Zbl 1134.35093 · doi:10.1007/BF02832308