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Fractional derivatives of products of Airy functions. (English) Zbl 1141.33002
In the present paper the author has investigated fractional derivatives of two products of Airy functions first kind and second kind. He has also proved the Wronskian \(W(x)\) of the system of integral of order half and its Hilbert transform \(\overline W(x)= -HW(x)\) are special functions. In the last he has established some integral relations. The work is good.

33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
26A33 Fractional derivatives and integrals
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[1] Ablowitz, M.J.; Clarkson, P.A., Solitons, nonlinear evolution equations and inverse scattering, Lecture notes series, vol. 149, (1991), Cambridge University Press · Zbl 0762.35001
[2] ()
[3] Aspnes, D.E., Electric-field effects on optical absorption near thresholds in solids, Phys. rev., 147, 554-566, (1966)
[4] Aspnes, D.E., Electric-field effects on the dielectric constant of solids, Phys. rev., 153, 972-982, (1967)
[5] Babich, V.M.; Buldyrev, V.S., Short wavelength diffraction theory—asymptotic methods, Springer series on wave phenomena, (1991), Springer Berlin · Zbl 0742.35002
[6] Duoandikoetxea, J., Fourier analysis, Graduate studies in mathematics, vol. 29, (2001), Amer. Math. Soc. Providence, RI
[7] Jones, D.S., High-frequency refraction and diffraction in general media, Philos. trans. R. soc. lond. ser. A math. phys. eng. sci., 255, 1058, 341-387, (1963) · Zbl 0138.44201
[8] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F., Higher transcendental functions, vol. III, (1955), McGraw-Hill New York · Zbl 0064.06302
[9] Fock, V.A., Electromagnetic diffraction and propagation problems, (1965), Pergamon Press Oxford
[10] Kato, T., On the Cauchy problem for the (generalized) korteweg – de Vries equation, (), 93-128
[11] Katznelson, Y., An introduction to harmonic analysis, (1968), Wiley New York · Zbl 0169.17902
[12] Kenig, C.; Ponce, G.; Vega, L., On the (generalized) korteweg – de Vries equation, Duke math. J., 59, 585-610, (1989) · Zbl 0795.35105
[13] Kenig, C.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized korteweg – de Vries equation via the contraction principle, Comm. pure appl. math., 46, 4, 527-620, (1993) · Zbl 0808.35128
[14] Kormilitsin, B.T., Propagation of electromagnetic waves in a medium with permittivity \(\operatorname{\&z.epsiv;}(z) = \operatorname{\&z.epsiv;}_0 + \operatorname{\&z.epsiv;}_1 z\) incited by a filament-type source, Radiotekhnika i elektronika, 11, 6, 1130-1134, (1966), (in Russian)
[15] Landau, L.D.; Lifshitz, E.M., Quantum mechanics. nonrelativistic theory, (1965), Pergamon Press Oxford · Zbl 0178.57901
[16] Laurenzi, B.J., Moment integrals of powers of Airy functions, Z. angew. math. phys., 44, 891-908, (1993) · Zbl 0784.33003
[17] Mainardi, F.; Pagnini, G., The wright functions as solutions of the time-fractional diffusion equation, Appl. math. comput., 141, 51-62, (2003) · Zbl 1053.35008
[18] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[19] Ostrovsky, L.A., Nonlinear internal waves in a rotating Ocean, Oceanology, 18, 181-191, (1978)
[20] Ostrovsky, L.A.; Stepanyants, Yu.A., Nonlinear surface and internal waves in rotating fluids, () · Zbl 0719.76021
[21] Reid, W.H., Integral representations for products of Airy functions, Z. angew. math. phys., 46, 159-170, (1995) · Zbl 0824.33002
[22] Reid, W.H., Integral representations for products of Airy functions. part 2. cubic products, Z. angew. math. phys., 48, 646-655, (1997) · Zbl 0874.33003
[23] Reid, W.H., Integral representations for products of Airy functions. part 3. quartic products, Z. angew. math. phys., 48, 656-664, (1998) · Zbl 0874.33004
[24] Stein, E.M., Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton · Zbl 0207.13501
[25] Titchmarsh, E.C., Introduction to the theory of Fourier integrals, (1962), Oxford Univ. Press Glasgow, New York · Zbl 0115.44301
[26] Vallée, O.; Soares, M., Airy functions and applications to physics, (2004), Imperial College Press London · Zbl 1056.33006
[27] Vallée, O.; Soares, M.; de Izarra, C., An integral representation for the product of Airy functions, Z. angew. math. phys., 48, 156-160, (1997) · Zbl 0879.33002
[28] Varlamov, V.; Liu, Y., Cauchy problem for the Ostrovsky equation, Discrete contin. dyn. syst., 10, 3, 731-753, (2004) · Zbl 1059.35035
[29] Varlamov, V., Oscillatory integrals generated by the Ostrovsky equation, Z. angew. math. phys., 56, 1-29, (2005) · Zbl 1108.35134
[30] V. Varlamov, Semi-integer derivatives of the Airy function and related properties of the Korteweg – de Vries-type equations, Z. Angew. Math. Phys., in press · Zbl 1155.35008
[31] Wolfram Functions Site, http://functions.wolfram.com/BesselAiryStruveFunctions/AiryAi/26
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