finite%20Fourier%20series

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11—20 of 380 matching pages

11: 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.

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

… ►

K. V. Leung and S. S. Ghaderpanah (1979) An application of the finite element approximation method to find the complex zeros of the modified Bessel function

K_{n}(z)

. Math. Comp. 33 (148), pp. 1299–1306.

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External Links:: ISSN 0025-5718, FullText, MathReview (R. P. Boas, Jr.), ZentralBlatt
Referenced by:: §10.74(vi), 2nd item.
Encodings:: BibTeX

… ►

M. J. Lighthill (1958) An Introduction to Fourier Analysis and Generalised Functions. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York.

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External Links:: ISBN 0521091284, MathReview (G. Temple), ZentralBlatt
Referenced by:: §1.16(ix), §1.16(v).
Encodings:: BibTeX

… ►

J. N. Lyness (1971) Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.

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External Links:: FullText, MathReview (J. Bryant), ZentralBlatt
Referenced by:: §2.3(iv).
Encodings:: BibTeX

…

12: 34.6 Definition: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

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Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

… ►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

14: 6.16 Mathematical Applications

… ►Consider the Fourier series … ►Compare Figure 6.16.1. … ►It occurs with Fourier-series expansions of all piecewise continuous functions. … … ►

See accompanying text — Figure 6.16.2: The logarithmic integral $\operatorname{li}\left(x\right)$ , together with vertical bars indicating the value of $\pi(x)$ for $x=10,20,\dots,1000$ . Magnify

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
3	3	20	627	37	21637
…

16: 15.17 Mathematical Applications

… ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

17: 3.8 Nonlinear Equations

… ►For the computation of zeros of orthogonal polynomials as eigenvalues of finite tridiagonal matrices (§3.5(vi)), see Gil et al. (2007a, pp. 205–207). For the computation of zeros of Bessel functions, Coulomb functions, and conical functions as eigenvalues of finite parts of infinite tridiagonal matrices, see Grad and Zakrajšek (1973), Ikebe (1975), Ikebe et al. (1991), Ball (2000), and Gil et al. (2007a, pp. 205–213). … ►

3.8.15

p(x)=(x-1)(x-2)\cdots(x-20)

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Permalink:: http://dlmf.nist.gov/3.8.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►Consider

x=20

and

j=19

. We have

p^{\mspace{1.0mu}\prime}(20)=19!

and

a_{19}=1+2+\dots+20=210

. …

18: 20.14 Methods of Computation

… ►The Fourier series of §20.2(i) usually converge rapidly because of the factors

q^{(n+\frac{1}{2})^{2}}

q^{n^{2}}

, and provide a convenient way of calculating values of

\theta_{j}\left(z\middle|\tau\right)

. … ►Hence the first term of the series (20.2.3) for

\theta_{3}\left(z\tau^{\prime}\middle|\tau^{\prime}\right)

suffices for most purposes. In theory, starting from any value of

\tau

, a finite number of applications of the transformations

\tau\to\tau+1

and

\tau\to-1/\tau

will result in a value of

\tau

with

\Im\tau\geq\sqrt{3}/2

; see §23.18. …

19: 15.15 Sums

… ►For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975). …

20: Bibliography M

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

… ►

D. W. Matula and P. Kornerup (1980) Foundations of Finite Precision Rational Arithmetic. In Fundamentals of Numerical Computation (Computer-oriented Numerical Analysis), G. Alefeld and R. D. Grigorieff (Eds.), Comput. Suppl., Vol. 2, Vienna, pp. 85–111.

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External Links:: MathReview (G. Blanch), ZentralBlatt
Referenced by:: §3.1(iii).
Encodings:: BibTeX

… ►

Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

… ►

L. M. Milne-Thomson (1933) The Calculus of Finite Differences. Macmillan and Co. Ltd., London.

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Notes:: Reprinted in 1951 by Macmillan, and in 1981 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI.
External Links:: ZentralBlatt
Referenced by:: §24.16(i), §24.17(i), §24.2(i), §24.2(ii), §24.4(i), §24.4(ii), §24.4(iii), §24.4(v), §26.1, §26.1, Notations B, Notations B.
Encodings:: BibTeX

… ►

D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

…

finite%20Fourier%20series

11—20 of 380 matching pages

11: Bibliography L

12: 34.6 Definition: $\mathit{9j}$ Symbol

13: 1.8 Fourier Series

§1.8 Fourier Series

Uniqueness of Fourier Series

§1.8(ii) Convergence

14: 6.16 Mathematical Applications

15: 26.2 Basic Definitions

16: 15.17 Mathematical Applications

17: 3.8 Nonlinear Equations

18: 20.14 Methods of Computation

19: 15.15 Sums

20: Bibliography M

finite%20Fourier%20series

11—20 of 380 matching pages

11: Bibliography L

12: 34.6 Definition: 9 ⁢ j Symbol

13: 1.8 Fourier Series

§1.8 Fourier Series

Uniqueness of Fourier Series

§1.8(ii) Convergence

14: 6.16 Mathematical Applications

15: 26.2 Basic Definitions

16: 15.17 Mathematical Applications

17: 3.8 Nonlinear Equations

18: 20.14 Methods of Computation

19: 15.15 Sums

20: Bibliography M

12: 34.6 Definition: $\mathit{9j}$ Symbol