fractional derivatives

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1: 1.15 Summability Methods

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§1.15(vii) Fractional Derivatives

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1.15.51 $${𝐷}^{\alpha }f(x)=\frac{d^n}{{dx}^n}{𝐼}^{n-\alpha }f(x),$$

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Defines:

${𝐷}^{\alpha }$: fractional derivative (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

§1.15(vii)

Permalink:

http://dlmf.nist.gov/1.15.E51

Encodings:

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See also:

Annotations for §1.15(vii), §1.15 and Ch.1

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1.15.52 $${𝐷}^k{𝐼}^{\alpha }={𝐷}^n{𝐼}^{\alpha +n-k},$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$k$: integer, $n$: nonnegative integer, ${𝐼}^{\alpha }$: fractional integral and ${𝐷}^{\alpha }$: fractional derivative

Referenced by:

§1.15(vii)

Permalink:

http://dlmf.nist.gov/1.15.E52

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vii), §1.15 and Ch.1

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1.15.53 $${𝐷}^{\alpha }{𝐷}^{\beta }={𝐷}^{\alpha +\beta }.$$

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Symbols:

${𝐷}^{\alpha }$: fractional derivative

Permalink:

http://dlmf.nist.gov/1.15.E53

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vii), §1.15 and Ch.1

►Note that ${𝐷}^{1/2}𝐷\ne {𝐷}^{3/2}$. …

2: 12.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. …

3: 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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External Links:

Document, MathReview (C. Fox), ZentralBlatt

Referenced by:

§1.15(vi), §1.15(vii).

Encodings:

BibTeX

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4: Errata

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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 $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$, 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).

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5: 18.17 Integrals

… ►Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively. … ►Formulas (18.17.14) and (18.17.15) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (13.6.19), special cases of (13.3.18) and (13.3.20), respectively. …

6: 30.3 Eigenvalues

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30.3.4

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.3(ii), §30.3 and Ch.30

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§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 ${\beta }_0-\lambda$ or ${\beta }_1-\lambda$. …

7: 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 28–30.

8: 33.23 Methods of Computation

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§33.23(v) Continued Fractions

►§33.8 supplies continued fractions for $F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{\pm }$. Combined with the Wronskians (33.2.12), the values of $F_{\mathrm{\ell }}$, $G_{\mathrm{\ell }}$, 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. …

9: 2.6 Distributional Methods

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2.6.46 $${𝐼}^{\mu }f(x)=\sum _{s=0}^{n-1}\frac{{(-1)}^s{a}_s}{s!\mathrm{\Gamma }\left(\mu +1\right)}\frac{d^{s+1}}{{dx}^{s+1}}\left(x^{\mu }\left(\ln x-\gamma -\psi \left(\mu +1\right)\right)\right)-\sum _{s=1}^n\frac{d_s}{\mathrm{\Gamma }\left(\mu -s+1\right)}{x}^{\mu -s}+\frac{1}{x^n}{\delta }_n(x),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $d_s$: coefficients, $\mu$: order, ${𝐼}^{\mu }$: fractional integral , $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients and ${\delta }_n(x)$: sum

Permalink:

http://dlmf.nist.gov/2.6.E46

Encodings:

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See also:

Annotations for §2.6(iii), §2.6 and Ch.2

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10: 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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External Links:

ISSN 0020-2932, Document, MathReview, ZentralBlatt

Referenced by:

§2.6(iii).

Encodings:

BibTeX

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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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External Links:

ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).

Encodings:

BibTeX

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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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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

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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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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Paul Levrie), ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

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