DLMF: Untitled Document

§10.74 Methods of Computation

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§10.74(vi) Zeros and Associated Values

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Hankel Transform

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H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §19.12, §19.5.
Encodings:: BibTeX

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J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Notes:: Associated computer program available online
External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

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C. Van Loan (1992) Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics, Vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-285-8, MathReview (S. Hitotumatu), ZentralBlatt
Referenced by:: §3.11(v).
Encodings:: BibTeX

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A. van Wijngaarden (1953) On the coefficients of the modular invariant

J(\tau)

. Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 56, pp. 389–400.

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External Links:: MathReview (J. Lehner), ZentralBlatt
Referenced by:: §23.17(ii).
Encodings:: BibTeX

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M. N. Vrahatis, T. N. Grapsa, O. Ragos, and F. A. Zafiropoulos (1997a) On the localization and computation of zeros of Bessel functions. Z. Angew. Math. Mech. 77 (6), pp. 467–475.

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External Links:: ISSN 0044-2267, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vi).
Encodings:: BibTeX

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… ►Conversely, for a given

\lambda

, the value of

\bigtriangleup(\lambda)

is needed for the computation of

\nu

. For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of

Q(x)

; see Magnus and Winkler (1966, §2.3, pp. 28–36). …

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National Bureau of Standards (1944) Tables of Lagrangian Interpolation Coefficients. Columbia University Press, New York.

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Notes:: Technical Director: Arnold N. Lowan
External Links:: MathReview (W. Feller), ZentralBlatt
Referenced by:: §3.3(ii).
Encodings:: BibTeX

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National Physical Laboratory (1961) Modern Computing Methods. 2nd edition, Notes on Applied Science, No. 16, Her Majesty’s Stationery Office, London.

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External Links:: MathReview (A. S. Householder), ZentralBlatt
Referenced by:: §3.8(iv).
Encodings:: BibTeX

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National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview, ZentralBlatt
Referenced by:: §28.1, §28.26(i), 6th item, Notations S, Notations S.
Encodings:: BibTeX

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G. Nemes (2013a) An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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External Links:: ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt
Referenced by:: §5.11(i), §5.11(i).
Encodings:: BibTeX

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C. J. Noble (2004) Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Comm. 159 (1), pp. 55–62.

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Notes:: Double-precision Fortran-90 code.
External Links:: Document, Code
Referenced by:: §33.23(vi), 8th item.
Encodings:: BibTeX

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GAP (website) The GAP Group, Centre for Interdisciplinary Research in Computational Algebra, University of St. Andrews, United Kingdom.

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Notes:: Groups, Algorithms, Programming: A system for computational discrete algebra with emphasis on computational group theory.
External Links:: GAP
Referenced by:: 2nd item.
Encodings:: BibTeX

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W. Gautschi (1967) Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.

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External Links:: Document, MathReview (E. Frank), ZentralBlatt
Referenced by:: §10.74(iv), §14.32, §3.10(iii), §3.6(iii).
Encodings:: BibTeX

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W. Gautschi (1970) Efficient computation of the complex error function. SIAM J. Numer. Anal. 7 (1), pp. 187–198.

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External Links:: Document, MathReview (P. Barrucand), ZentralBlatt
Referenced by:: §7.22(i), §7.25(iii).
Encodings:: BibTeX

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W. Gautschi (1979b) A computational procedure for incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 466–481.

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External Links:: ISSN 0098-3500, Document, ZentralBlatt
Referenced by:: §8.25(iv), §8.25(v).
Encodings:: BibTeX

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D. Goldberg (1991) What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys 23 (1), pp. 5–48.

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External Links:: ISSN 0360-0300, Document
Referenced by:: §3.1(i).
Encodings:: BibTeX

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§28.34 Methods of Computation

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§28.34(i) Characteristic Exponents

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§28.34(ii) Eigenvalues

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§28.34(iii) Floquet Solutions

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§28.34(iv) Modified Mathieu Functions

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J. C. Lagarias, V. S. Miller, and A. M. Odlyzko (1985) Computing

\pi(x)

: The Meissel-Lehmer method. Math. Comp. 44 (170), pp. 537–560.

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External Links:: ISSN 0025-5718, FullText, MathReview (Kenneth A. Jukes), ZentralBlatt
Referenced by:: §27.18.
Encodings:: BibTeX

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S. Lai and Y. Chiu (1990) Exact computation of the

3

-

j

and

6

-

j

symbols. Comput. Phys. Comm. 61 (3), pp. 350–360.

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Notes:: A Fortran program is included.
External Links:: Document, ZentralBlatt
Referenced by:: §34.15.
Encodings:: BibTeX

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S. Lai and Y. Chiu (1992) Exact computation of the

9

-

j

symbols. Comput. Phys. Comm. 70 (3), pp. 544–556.

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Notes:: Double-precision Fortran
External Links:: ISSN 0010-4655, Document, Code, MathReview
Referenced by:: 7th item.
Encodings:: BibTeX

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J. D. Louck (1958) New recursion relation for the Clebsch-Gordan coefficients. Phys. Rev. (2) 110 (4), pp. 815–816.

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External Links:: Document, MathReview, ZentralBlatt
Referenced by:: §34.3(iii).
Encodings:: BibTeX

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Y. L. Luke (1977b) Algorithms for the Computation of Mathematical Functions. Academic Press, New York.

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External Links:: MathReview (Gh. Adam)
Referenced by:: §16.26.
Encodings:: BibTeX

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… ►Difference equations are simple and attractive for computation. … ►Then computation of

w_{n}

by forward recursion is unstable. … ►

Example 1. Bessel Functions

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Example 2. Weber Function

… ►The results of the computations are displayed in Table 3.6.1. …

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A. B. Olde Daalhuis (1996) Hyperterminants. I. J. Comput. Appl. Math. 76 (1-2), pp. 255–264.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §10.17(v), §10.40(iv), §2.11(v), §9.7(v).
Encodings:: BibTeX

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A. B. Olde Daalhuis (2000) On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

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T. Oliveira e Silva (2006) Computing

\pi(x)

: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

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Notes:: Online fulltext contains postpublication annotations.
External Links:: FullText
Referenced by:: §27.18.
Encodings:: BibTeX

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F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.7(ii), §2.7(ii).
Encodings:: BibTeX

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M. L. Overton (2001) Numerical Computing with IEEE Floating Point Arithmetic. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:: Including one theorem, one rule of thumb, and one hundred and one exercises
External Links:: ISBN 0-89871-482-6, MathReview (Jesse L. Barlow), ZentralBlatt
Referenced by:: §3.1(i).
Encodings:: BibTeX

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26.5.1

C\left(n\right)=\frac{1}{n+1}\genfrac{(}{)}{0.0pt}{}{2n}{n}=\frac{1}{2n+1}% \genfrac{(}{)}{0.0pt}{}{2n+1}{n}=\genfrac{(}{)}{0.0pt}{}{2n}{n}-\genfrac{(}{)}% {0.0pt}{}{2n}{n-1}=\genfrac{(}{)}{0.0pt}{}{2n-1}{n}-\genfrac{(}{)}{0.0pt}{}{2n% -1}{n+1}.

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Defines:: $C\left(\NVar{n}\right)$ : Catalan number
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and $n$ : nonnegative integer
Referenced by:: §26.5(iv)
Permalink:: http://dlmf.nist.gov/26.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(i), §26.5 and Ch.26

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26.5.5

C\left(n+1\right)=\sum_{k=0}^{\left\lfloor n/2\right\rfloor}\genfrac{(}{)}{0.0% pt}{}{n}{2k}2^{n-2k}C\left(k\right).

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Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iii), §26.5 and Ch.26

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computation of coefficients

21—30 of 56 matching pages

21: 10.74 Methods of Computation

§10.74 Methods of Computation

§10.74(vi) Zeros and Associated Values

Hankel Transform

22: Bibliography V

23: 28.29 Definitions and Basic Properties

24: Bibliography N

25: Bibliography G

26: 28.34 Methods of Computation

§28.34 Methods of Computation

§28.34(i) Characteristic Exponents

§28.34(ii) Eigenvalues

§28.34(iii) Floquet Solutions

§28.34(iv) Modified Mathieu Functions

27: Bibliography L

28: 3.6 Linear Difference Equations

Example 1. Bessel Functions

Example 2. Weber Function

29: Bibliography O

30: 26.5 Lattice Paths: Catalan Numbers