# fractional derivatives

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##### 1: 1.15 Summability Methods
###### §1.15(vii) FractionalDerivatives
1.15.51 $D^{\alpha}f(x)=\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}I^{n-\alpha}f(x),$
1.15.52 $D^{k}I^{\alpha}=D^{n}I^{\alpha+n-k},$ $k=1,2,\dots,n$.
1.15.53 $D^{\alpha}D^{\beta}=D^{\alpha+\beta}.$
Note that $D^{1/2}D\not=D^{3/2}$. …
##### 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
• Other Changes

• 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 $0<\lambda<1$ to the sector $0<\lambda<1,|\operatorname{ph}a|\leq\frac{\pi}{2}-\delta$, and the interval of validity of (8.11.7) was increased from $\lambda>1$ to the sector $\lambda>1$, $|\operatorname{ph}a|\leq\frac{3\pi}{2}-\delta$. 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 $|\operatorname{ph}z|\leq\pi-\delta$ to $|\operatorname{ph}z|\leq 2\pi-\delta$. 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 $\mathit{3j}$, $\mathit{6j}$, $\mathit{9j}$ 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.

• ##### 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
30.3.4 $-1<\frac{\mathrm{d}\lambda^{m}_{n}\left(\gamma^{2}\right)}{\mathrm{d}(\gamma^{% 2})}<0.$
###### §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 2830.
##### 7: 33.23 Methods of Computation
###### §33.23(v) Continued Fractions
§33.8 supplies continued fractions for $F_{\ell}'/F_{\ell}$ and ${H^{\pm}_{\ell}}'/{H^{\pm}_{\ell}}$. Combined with the Wronskians (33.2.12), the values of $F_{\ell}$, $G_{\ell}$, 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
2.6.46 $I^{\mu}f(x)=\sum_{s=0}^{n-1}\frac{(-1)^{s}a_{s}}{s!\Gamma\left(\mu+1\right)}% \frac{{\mathrm{d}}^{s+1}}{{\mathrm{d}x}^{s+1}}\left(x^{\mu}\left(\ln x-\gamma-% \psi\left(\mu+1\right)\right)\right)-\sum_{s=1}^{n}\frac{d_{s}}{\Gamma\left(% \mu-s+1\right)}x^{\mu-s}+\frac{1}{x^{n}}\delta_{n}(x),$
##### 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. …