fractional derivatives

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11: 10.41 Asymptotic Expansions for Large Order

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10.41.4 $$K_{\nu }\left(\nu z\right)\sim {\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{\mathrm{e}}^{-\nu \eta }}{{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k(p)}{{\nu }^k},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.8

Permalink:

http://dlmf.nist.gov/10.41.E4

Encodings:

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See also:

Annotations for §10.41(ii), §10.41 and Ch.10

… ►The expansions (10.41.3)–(10.41.6) also hold uniformly in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ , with the branches of the fractional powers in (10.41.3)–(10.41.8) extended by continuity from the positive real $z$-axis. … ► …

12: 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. … ► …

13: 18.40 Methods of Computation

… ►The problem of moments is simply stated and the early work of Stieltjes, Markov, and Chebyshev on this problem was the origin of the understanding of the importance of both continued fractions and OP’s in many areas of analysis. … ►

Derivative Rule Approach

►An alternate, and highly efficient, approach follows from the derivative rule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that …In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: … ►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. …

14: 10.72 Mathematical Applications

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10.72.1 $$\frac{d^{2}w}{{dz}^2}=\left(u^{2}f(z)+g(z)\right)w,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

Referenced by:

§10.72(i), §10.72(ii), §10.72(iii)

Permalink:

http://dlmf.nist.gov/10.72.E1

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See also:

Annotations for §10.72(i), §10.72 and Ch.10

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Multiple or Fractional Turning Points

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15: Bibliography W

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The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt (Mehdi Hassani)

Referenced by:

§5.4(iii).

Encodings:

BibTeX

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Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc., New York.

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Notes:

Reprinted by Chelsea Publishing, New York, 1973 and AMS Chelsea Publishing, Providence, RI, 2000.

External Links:

ISBN 0-8218-2106-7, MathReview (R. C. Buck), ZentralBlatt

Referenced by:

§3.10(iii), §4.25, §4.9(i), §4.9(ii), Ch.4, §5.10.

Encodings:

BibTeX

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Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.

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Notes:

Includes Fortran code in the appendices to compute the functions to an accuracy between ${10}^{-2}$ and ${10}^{-5}$.

External Links:

ISSN 0022-4073, Document

Referenced by:

§7.21, §7.21.

Encodings:

BibTeX

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16: 8.19 Generalized Exponential Integral

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§8.19(v) Recurrence Relation and Derivatives

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8.19.13 $$\frac{d}{dz}{E}_p\left(z\right)=-E_{p-1}\left(z\right),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable and $p$: parameter

A&S Ref:

5.1.36 (Extended to general values of $p$.)

Referenced by:

§8.19(x)

Permalink:

http://dlmf.nist.gov/8.19.E13

Encodings:

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See also:

Annotations for §8.19(v), §8.19 and Ch.8

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8.19.14 $$\frac{d}{dz}({\mathrm{e}}^z{E}_p\left(z\right))={\mathrm{e}}^z{E}_p\left(z\right)\left(1+\frac{p-1}{z}\right)-\frac{1}{z}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.19.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(v), §8.19 and Ch.8

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$p$-Derivatives

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§8.19(vii) Continued Fraction

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17: 5.15 Polygamma Functions

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5.15.6 $${\psi }^{(n)}\left(1-z\right)+{(-1)}^{n-1}{\psi }^{(n)}\left(z\right)={(-1)}^n\pi \frac{d^n}{{dz}^n}\cot \left(\pi z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\psi \left(z\right)$: psi (or digamma) function, $n$: nonnegative integer and $z$: complex variable

A&S Ref:

6.4.7

Permalink:

http://dlmf.nist.gov/5.15.E6

Encodings:

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See also:

Annotations for §5.15 and Ch.5

… ►For $B_{2k}$ see §24.2(i). ►For continued fractions for ${\psi }^{\prime }\left(z\right)$ and ${\psi }^{\prime \prime }\left(z\right)$ see Cuyt et al. (2008, pp. 231–238).

18: Bibliography R

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A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

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Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

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High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.

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Referenced by:

3rd item.

Encodings:

BibTeX

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

Encodings:

BibTeX

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19: 10.43 Integrals

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§10.43(iii) Fractional Integrals

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10.43.15 $${\mathrm{Ki}}_{-n}\left(x\right)={(-1)}^n\frac{d^n}{{dx}^n}{K}_0\left(x\right),$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $n$: integer and $x$: real variable

A&S Ref:

11.2.9

Permalink:

http://dlmf.nist.gov/10.43.E15

Encodings:

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See also:

Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

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§10.43(iv) Integrals over the Interval ($0,\mathrm{\infty }$)

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On the interval , $x^{-1}g(x)$ is continuously differentiable and each of $xg(x)$ and $xd(x^{-1}g(x))/dx$ is absolutely integrable.

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20: 3.3 Interpolation

… ►with the derivative … ►For Hermite interpolation, trigonometric interpolation, spline interpolation, rational interpolation (by using continued fractions), interpolation based on Chebyshev points, and bivariate interpolation, see Bulirsch and Rutishauser (1968), Davis (1975, pp. 27–31), and Mason and Handscomb (2003, Chapter 6). …