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1: 1.15 Summability Methods
§1.15(vii) Fractional Derivatives
1.15.51 D α f ( x ) = d n d x n I n - α f ( x ) ,
1.15.52 D k I α = D n I α + n - k , k = 1 , 2 , , n .
1.15.53 D α D β = D α + β .
Note that D 1 / 2 D D 3 / 2 . …
2: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. …
3: Errata
  • Subsections 1.15(vi), 1.15(vii), 2.6(iii)

    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 α was more precisely identified as the Riemann-Liouville fractional integral operator of order α , 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
  • E. R. Love (1972b) Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14, pp. 385–410.
  • 5: 30.3 Eigenvalues
    §30.3(iii) Transcendental Equation
    If p is an even nonnegative integer, then the continued-fraction equation …If p = 0 or p = 1 , the finite continued-fraction on the left-hand side of (30.3.5) equals 0; if p > 1 its last denominator is β 0 - λ or β 1 - λ . …
    6: 31.18 Methods of Computation
    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 z ; 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 2830.
    7: 33.23 Methods of Computation
    §33.23(v) Continued Fractions
    §33.8 supplies continued fractions for F / F and H ± / H ± . Combined with the Wronskians (33.2.12), the values of F , G , 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
    2.6.46 I μ f ( x ) = s = 0 n - 1 ( - 1 ) s a s s ! Γ ( μ + 1 ) d s + 1 d x s + 1 ( x μ ( ln x - γ - ψ ( μ + 1 ) ) ) - s = 1 n d s Γ ( μ - s + 1 ) x μ - s + 1 x n δ n ( x ) ,
    9: Bibliography M
  • J. P. McClure and R. Wong (1979) Exact remainders for asymptotic expansions of fractional integrals. J. Inst. Math. Appl. 24 (2), pp. 139–147.
  • K. S. Miller and B. Ross (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.
  • S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.
  • C. Mortici (2011a) A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.
  • C. Mortici (2013a) A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.
  • 10: 10.74 Methods of Computation
    §10.74(v) Continued Fractions
    For applications of the continued-fraction expansions (10.10.1), (10.10.2), (10.33.1), and (10.33.2) to the computation of Bessel functions and modified Bessel functions see Gargantini and Henrici (1967), Amos (1974), Gautschi and Slavik (1978), Tretter and Walster (1980), Thompson and Barnett (1986), and Cuyt et al. (2008). …
    §10.74(vi) Zeros and Associated Values
    Necessary values of the first derivatives of the functions are obtained by the use of (10.6.2), for example. …