fractional derivatives

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21: Bibliography C

… ►

B. W. Char (1980) On Stieltjes’ continued fraction for the gamma function. Math. Comp. 34 (150), pp. 547–551.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §5.10.
Encodings:: BibTeX

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A. D. Chave (1983) Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.

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Notes:: Fortran code.
External Links:: ISSN 0098-3500, Document
Referenced by:: §10.77(ix).
Encodings:: BibTeX

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A. R. Curtis (1964b) Tables of Jacobian Elliptic Functions Whose Arguments are Rational Fractions of the Quarter Period. National Physical Laboratory Mathematical Tables, Vol. 7, Her Majesty’s Stationery Office, London.

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External Links:: ISBN 00-791-1158-0, MathReview (John Todd), ZentralBlatt
Referenced by:: §22.20(vii), §22.21.
Encodings:: BibTeX

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A. Cuyt, V. Petersen, B. Verdonk, H. Waadeland, W. B. Jones, and C. Bonan-Hamada (2007) Handbook of Continued Fractions for Special Functions. Kluwer Academic Publishers Group, Dordrecht.

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Referenced by:: Profile Annie A. M. Cuyt.
Encodings:: BibTeX

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A. Cuyt, V. B. Petersen, B. Verdonk, H. Waadeland, and W. B. Jones (2008) Handbook of Continued Fractions for Special Functions. Springer, New York.

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External Links:: ISBN 978-1-4020-6948-2, MathReview Entry, ZentralBlatt
Referenced by:: §10.10, §10.33, §10.55, §10.74(v), §12.6, §13.17, §13.5, §15.7, §16.4(iv), §17.6(vi), §17.7(ii), §18.13, §3.10(ii), §3.10(iii), §4.25, §4.39, §4.9(i), §4.9(ii), §4.9(iii), §5.10, §5.15, §6.9, §7.18(v), §7.22(i), §7.9, §8.17(v), §8.19(vii), §8.9.
Encodings:: BibTeX

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22: 1.2 Elementary Algebra

… ►

§1.2(iii) Partial Fractions

… ►and

f^{(k)}

is the

k

-th derivative of

f

(§1.4(iii)). …

23: Bibliography J

… ►

L. Jacobsen, W. B. Jones, and H. Waadeland (1986) Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

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External Links:: MathReview (Marietta J. Tretter), ZentralBlatt
Referenced by:: §8.25(iv).
Encodings:: BibTeX

… ►

W. B. Jones and W. J. Thron (1974) Numerical stability in evaluating continued fractions. Math. Comp. 28 (127), pp. 795–810.

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External Links:: FullText, MathReview (N. N. Abdelmalek), ZentralBlatt
Referenced by:: §3.10(iii).
Encodings:: BibTeX

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W. B. Jones and W. J. Thron (1980) Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.

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External Links:: ISBN 0-201-13510-8, MathReview (E. Frank), ZentralBlatt
Referenced by:: §1.12(ii), §1.12(iii), §1.12(iv), §1.12(v), §13.17, §13.17, §13.5, §13.5, §4.25, §5.10.
Encodings:: BibTeX

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W. B. Jones and W. Van Assche (1998) Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:: MathReview (Olav Njåstad), ZentralBlatt
Referenced by:: §5.10, §5.10.
Encodings:: BibTeX

… ►

JTEM (website) Java Tools for Experimental Mathematics

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Notes:: Single and double precision. Contains packages to compute elliptic functions, Riemann theta functions and their derivatives, and Riemann matrices for Riemann surfaces.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

…

24: 18.39 Applications in the Physical Sciences

… ►Here the term

-\frac{\hbar^{2}}{2m}\tfrac{{\partial}^{2}}{{\partial x}^{2}}

represents the quantum kinetic energy of a single particle of mass

m

, and

V(x)

its potential energy. … ►

18.39.8

\mathcal{H}\psi_{n}(x)=\left(-\frac{\hbar^{2}}{2m}\frac{{\mathrm{d}}^{2}}{{% \mathrm{d}x}^{2}}+V(x)\right)\psi_{n}(x)=\epsilon_{n}\psi_{n}(x),

n=0,1,2,\dots,

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: (18.39.17), (18.39.18), §18.39(i), §18.39(i), §18.39(i), §18.39(ii)
Permalink:: http://dlmf.nist.gov/18.39.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

… ►

18.39.38

\mathbf{L}_{p}^{m}(\rho)=\frac{{\mathrm{d}}^{m}}{{\mathrm{d}\rho}^{m}}\mathbf{% L}_{p}^{0}(\rho),

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $m$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.39.E38
Encodings:: pMML, png, TeX
See also:: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

… ►

18.39.39

\mathbf{L}_{p}^{0}(\rho)={\mathrm{e}}^{\rho}\frac{{\mathrm{d}}^{p}}{{\mathrm{d% }\rho}^{p}}(\rho^{p}{\mathrm{e}}^{-\rho}),

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $\mathrm{e}$ : base of natural logarithm
Permalink:: http://dlmf.nist.gov/18.39.E39
Encodings:: pMML, png, TeX
See also:: Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

… ►Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with

Z

protons, involve Laguerre polynomials of fractional index. …

25: 1.10 Functions of a Complex Variable

… ►is analytic in

D

and its derivatives of all orders can be found by differentiating under the sign of integration. … ►

§1.10(x) Infinite Partial Fractions

… ►

Mittag-Leffler’s Expansion

… ►Many properties are a direct consequence of this representation: Taking the

x

-derivative gives us …and hence

\frac{\mathrm{d}}{\mathrm{d}x}C^{(\lambda)}_{n}\left(x\right)=2\lambda C^{(% \lambda+1)}_{n-1}\left(x\right)

, that is (18.9.19). …

26: 2.4 Contour Integrals

… ►If, in addition, the corresponding integrals with

Q

and

F

replaced by their derivatives

Q^{(j)}

and

F^{(j)}

j=1,2,\dots,m

, converge uniformly, then by repeated integrations by parts … ►However, if

p^{\prime}(t_{0})=0

, then

\mu\geq 2

and different branches of some of the fractional powers of

p_{0}

are used for the coefficients

b_{s}

; again see §2.3(iii). … ►with

p, q

and their derivatives evaluated at

t_{0}

. … ►Suppose that on the integration path

\mathscr{P}

there are two simple zeros of

\ifrac{\partial p(\alpha,t)}{\partial t}

that coincide for a certain value

\widehat{\alpha}

\alpha

. … ►with

a

and

b

chosen so that the zeros of

\ifrac{\partial p(\alpha,t)}{\partial t}

correspond to the zeros

w_{1}(\alpha),w_{2}(\alpha)

, say, of the quadratic

w^{2}+2aw+b

. …

27: 25.11 Hurwitz Zeta Function

… ►

§25.11(vi) Derivatives

►

$a$ -Derivative

… ►

$s$ -Derivatives

►In (25.11.18)–(25.11.24) primes on

\zeta

denote derivatives with respect to

s

. … ►Similarly, as

a\to\infty

in the sector

|\operatorname{ph}a|\leq\frac{1}{2}\pi-\delta(<\frac{1}{2}\pi)

, …

28: 3.11 Approximation Techniques

… ►For convergence results for Padé approximants, and the connection with continued fractions and Gaussian quadrature, see Baker and Graves-Morris (1996, §4.7). … ►From the equations

\ifrac{\partial S}{\partial a_{k}}=0

k=0,1,\dots,n

, we derive the normal equations … ►Given

n+1

distinct points

x_{k}

in the real interval

[a,b]

, with (

a=

)

x_{0}<x_{1}<\cdots<x_{n-1}<x_{n}

(

=b

), on each subinterval

[x_{k},x_{k+1}]

k=0,1,\ldots,n-1

, a low-degree polynomial is defined with coefficients determined by, for example, values

f_{k}

and

f_{k}^{\prime}

of a function

f

and its derivative at the nodes

x_{k}

and

x_{k+1}

. …By taking more derivatives into account, the smoothness of the spline will increase. …

29: 21.7 Riemann Surfaces

… ►by setting

\lambda=\tilde{\lambda}/\tilde{\eta}

\mu=\tilde{\mu}/\tilde{\eta}

, and then clearing fractions. … ►

21.7.7

\left(\frac{\partial}{\partial z_{1}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol% {{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right)\Big{|}_{\mathbf{z}=\boldsymbol{{0}}},\dots,\frac{\partial}{\partial z_% {g}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)\Big{|}_{\mathbf{z}=% \boldsymbol{{0}}}\right)\neq\boldsymbol{{0}}.

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Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(ii), §21.7 and Ch.21

… ►

21.7.8

\zeta=\sum_{j=1}^{g}\omega_{j}\frac{\partial}{\partial z_{j}}\theta\genfrac{[}% {]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}% \middle|\boldsymbol{{\Omega}}\right)\Big{|}_{\mathbf{z}=\boldsymbol{{0}}}.

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Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector and $\omega$ : differential
Permalink:: http://dlmf.nist.gov/21.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(ii), §21.7 and Ch.21

…

30: Bibliography K

… ►

D. Karp, A. Savenkova, and S. M. Sitnik (2007) Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.

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External Links:: ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt
Referenced by:: §19.12, §19.5.
Encodings:: BibTeX

… ►

M. K. Kerimov and S. L. Skorokhodov (1985b) Calculation of the complex zeros of Hankel functions and their derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 25 (11), pp. 1628–1643, 1741.

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Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 25(6), pp. 26–36
External Links:: ISSN 0044-4669, Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §10.21(ix), §10.74(vi), 4th item, 5th item.
Encodings:: BibTeX

… ►

M. K. Kerimov and S. L. Skorokhodov (1986) On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).

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Notes:: In Russian. English translation: Soviet Math. Dokl. 33(3), 650–653, (1986).
External Links:: ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §10.21(i), §10.74(vi).
Encodings:: BibTeX

►

M. K. Kerimov and S. L. Skorokhodov (1987) On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.

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Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 27(6), pp. 18–25
External Links:: ISSN 0044-4669, Document, MathReview (Hans-Jürgen Albrand), ZentralBlatt
Referenced by:: §10.21(ix), §10.74(vi), 5th item.
Encodings:: BibTeX

… ►

T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §15.14, §15.14.
Encodings:: BibTeX

…