fractional derivatives
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1: 1.15 Summability Methods
2: 12.1 Special Notation
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►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values.
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3: Errata
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Subsections 1.15(vi), 1.15(vii), 2.6(iii)
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A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order was more precisely identified as the Riemann-Liouville fractional integral operator of order , and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).
4: Bibliography L
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Two index laws for fractional integrals and derivatives.
J. Austral. Math. Soc. 14, pp. 385–410.
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5: 30.3 Eigenvalues
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30.3.4
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§30.3(iii) Transcendental Equation
►If is an even nonnegative integer, then the continued-fraction equation …If or , the finite continued-fraction on the left-hand side of (30.3.5) equals 0; if its last denominator is or . …6: 31.18 Methods of Computation
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►Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of ; see Laĭ (1994) and Lay et al. (1998).
…The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.
7: 33.23 Methods of Computation
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§33.23(v) Continued Fractions
►§33.8 supplies continued fractions for and . Combined with the Wronskians (33.2.12), the values of , , and their derivatives can be extracted. … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. …8: 2.6 Distributional Methods
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2.6.46
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9: Bibliography M
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Exact remainders for asymptotic expansions of fractional integrals.
J. Inst. Math. Appl. 24 (2), pp. 139–147.
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An Introduction to the Fractional Calculus and Fractional Differential Equations.
A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.
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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions.
Ramanujan J. 6 (1), pp. 7–149.
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A new Stirling series as continued fraction.
Numer. Algorithms 56 (1), pp. 17–26.
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A continued fraction approximation of the gamma function.
J. Math. Anal. Appl. 402 (2), pp. 405–410.
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10: 10.74 Methods of Computation
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