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

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Defines:

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

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and ${\mathit{I}}^{\alpha }$: 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 $${\mathit{D}}^k{\mathit{I}}^{\alpha }={\mathit{D}}^n{\mathit{I}}^{\alpha +n-k},$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$k$: integer, $n$: nonnegative integer, ${\mathit{I}}^{\alpha }$: fractional integral and ${\mathit{D}}^{\alpha }$: 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 $${\mathit{D}}^{\alpha }{\mathit{D}}^{\beta }={\mathit{D}}^{\alpha +\beta }.$$

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Symbols:

${\mathit{D}}^{\alpha }$: 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 ${\mathit{D}}^{1/2}\mathit{D}\ne {\mathit{D}}^{3/2}$. …

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

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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).

• Changes to §8.18(ii)–§8.11(v): A sentence was added in §8.18(ii) to refer to Nemes and Olde Daalhuis (2016). Originally §8.11(iii) was applicable for real variables $a$ and $x=\lambda a$. It has been extended to allow for complex variables $a$ and $z=\lambda a$ (and we have replaced $x$ with $z$ in the subsection heading and in Equations (8.11.6) and (8.11.7)). Also, we have added two paragraphs after (8.11.9) to replace the original paragraph that appeared there. Furthermore, the interval of validity of (8.11.6) was increased from to the sector , and the interval of validity of (8.11.7) was increased from $\lambda >1$ to the sector $\lambda >1$, $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$. A paragraph with reference to Nemes (2016) has been added in §8.11(v), and the sector of validity for (8.11.12) was increased from $|\mathrm{ph}z|\le \pi -\delta$ to $|\mathrm{ph}z|\le 2\pi -\delta$. Two new Subsections 13.6(vii), 13.18(vi), both entitled Coulomb Functions, were added to note the relationship of the Kummer and Whittaker functions to various forms of the Coulomb functions. A sentence was added in both §13.10(vi) and §13.23(v) noting that certain generalized orthogonality can be expressed in terms of Kummer functions.

• Four of the terms in (14.15.23) were rewritten for improved clarity.

• In §15.6 it was noted that (15.6.8) can be rewritten as a fractional integral.

• In applying changes in Version 1.0.12 to (16.15.3), an editing error was made; it has been corrected.

• In §34.1, the reference for Clebsch-Gordan coefficients, Condon and Shortley (1935), was replaced by Edmonds (1974) and Rotenberg et al. (1959). The references for $\mathit{3}j$, $\mathit{6}j$, $\mathit{9}j$ symbols were made more precise.

• Images in Figures 36.3.1, 36.3.2, 36.3.3, 36.3.4, 36.3.5, 36.3.6, 36.3.7, 36.3.8 and Figures 36.3.13, 36.3.14, 36.3.15, 36.3.16, 36.3.17 were resized for consistency.

• Meta.Numerics (website) was added to the Software Index.

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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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External Links:

Document, MathReview (C. Fox), ZentralBlatt

Referenced by:

§1.15(vi), §1.15(vii).

Encodings:

BibTeX

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5: 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 ${\beta }_1-\lambda$. …

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 28–30.

7: 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. …

8: 2.6 Distributional Methods

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

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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, ${\mathit{I}}^{\mu }$: 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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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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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), 1st Erratum (V1.0.14).

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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10: 10.74 Methods of Computation

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§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. … ► …