fractional%20derivatives

(0.002 seconds)

9 matching pages

1: Bibliography R

… ►

Yu. L. Ratis and P. Fernández de Córdoba (1993) A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 18D.
External Links:: ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt
Referenced by:: 6th item.
Encodings:: BibTeX

… ►

J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

►

M. Razaz and J. L. Schonfelder (1980) High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.

ⓘ

Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

W. P. Reinhardt (2018) Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

ⓘ

Notes:: arXiv:1812.02196
Referenced by:: §18.40(ii).
Encodings:: BibTeX

… ►

M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt
Referenced by:: §10.23(ii).
Encodings:: BibTeX

…

2: 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. … ►

18.39.38

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

ⓘ

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}),

ⓘ

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

… ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►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. …

3: 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. … ►Results of low (

~{}2

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

20

. … ►

Derivative Rule Approach

… ►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. …

4: Bibliography W

… ►

P. L. Walker (2007) The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview, ZentralBlatt (Mehdi Hassani)
Referenced by:: §5.4(iii).
Encodings:: BibTeX

… ►

H. S. Wall (1948) Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc., New York.

ⓘ

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

… ►

R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

… ►

R. J. Wells (1999) Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.

ⓘ

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

…

5: 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.

ⓘ

External Links:: ISSN 0020-2932, Document, MathReview, ZentralBlatt
Referenced by:: §2.6(iii).
Encodings:: BibTeX

… ►

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.

ⓘ

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

… ►

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.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

… ►

C. Mortici (2011a) A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.

ⓘ

External Links:: ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §5.10.
Encodings:: BibTeX

… ►

C. Mortici (2013a) A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.

ⓘ

External Links:: ISSN 0022-247X, Document, FullText, MathReview (Paul Levrie), ZentralBlatt
Referenced by:: §5.10.
Encodings:: BibTeX

…

6: 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.

ⓘ

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

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

ⓘ

External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

ⓘ

External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
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.

ⓘ

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

…

7: 20.11 Generalizations and Analogs

… ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

8: Bibliography C

… ►

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

ⓘ

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

… ►

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

ⓘ

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

… ►

R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

ⓘ

External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

… ►

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.

ⓘ

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

►

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.

ⓘ

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

…

9: Bibliography L

… ►

P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

… ►

D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

… ►

W. J. Lentz (1976) Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics 15 (3), pp. 668–671.

ⓘ

Referenced by:: §3.10(iii), §3.10(iii), Erratum (V1.0.24) for Paragraph Steed’s Algorithm (in §3.10(iii)).
Encodings:: BibTeX

… ►

L. Lorentzen and H. Waadeland (1992) Continued Fractions with Applications. Studies in Computational Mathematics, North-Holland Publishing Co., Amsterdam.

ⓘ

External Links:: ISBN 0-444-89265-6, MathReview (Olav Njåstad), ZentralBlatt
Referenced by:: §1.12(ii), §1.12(iv), §1.12(v), §15.7, §18.13, §3.10(ii), §3.10(iii), §3.11(iv), §4.25, §4.39, §4.9(i), §4.9(ii), §5.10, §6.9, §7.18(v), §7.9.
Encodings:: BibTeX

… ►

E. R. Love (1972b) Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14, pp. 385–410.

ⓘ

External Links:: Document, MathReview (C. Fox), ZentralBlatt
Referenced by:: §1.15(vi), §1.15(vii).
Encodings:: BibTeX

…