Fourier-series expansions

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11: 29.20 Methods of Computation

… ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. …The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …

12: Bibliography S

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FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

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Notes:

Single-precision Fortran code.

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Document, Code, ZentralBlatt

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10th item.

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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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ZentralBlatt

Referenced by:

§12.13(i).

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

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A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

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ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(iv).

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An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§17.3(i).

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13: 28.34 Methods of Computation

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Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$.

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Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.

14: Bibliography O

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Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

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ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).

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Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

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MathReview, ZentralBlatt

Referenced by:

§1.8(v), §22.11, §4.27.

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Summing one- and two-dimensional series related to the Euler series. J. Comput. Appl. Math. 98 (2), pp. 245–271.

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ISSN 0377-0427, Document, MathReview (Boris Petrovich Osilenker), ZentralBlatt

Referenced by:

§25.8.

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

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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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ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii), §2.7(ii).

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15: Bibliography P

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On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables. Proc. Roy. Soc. London Ser. A 457 (2016), pp. 2835–2853.

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ISSN 1364-5021, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.40(iv), §2.11(v).

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A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:

Extended-precision Fortran, maximum accuracy 12D.

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Document, Code, ZentralBlatt

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1st item.

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Chebyshev series approximations for the zeros of the Bessel functions. J. Comput. Phys. 53 (1), pp. 188–192.

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ISSN 0021-9991, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.76(ii).

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Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

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Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.

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ISSN 0098-3500, Document, GAMS (module 498; package TOMS), ZentralBlatt

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1st item, 4th item.

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16: 14.13 Trigonometric Expansions

§14.13 Trigonometric Expansions

… ►These Fourier series converge absolutely when . … ►For other trigonometric expansions see Erdélyi et al. (1953a, pp. 146–147).

17: Bibliography T

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Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

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ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§22.11.

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The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

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ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§8.12.

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Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts, Vol. 43, Cambridge University Press, Cambridge.

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ISBN 0-521-45718-1, MathReview (Stefan Kühnlein), ZentralBlatt

Referenced by:

§5.16.

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Introduction to the Theory of Fourier Integrals. Third edition, Chelsea Publishing Co., New York.

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Notes:

Original edition was published in 1937 by Oxford University Press. Expanded bibliography and some improvements were made for the second and third editions.

External Links:

ISBN 0-8284-0324-4, MathReview, ZentralBlatt

Referenced by:

§1.14(i), §1.14(ii), §1.14(iv), §1.14(v), §1.14(vii), §1.8(iv), §10.22(v), §10.32(iii), §5.13.

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Fourier Series. Prentice-Hall Inc., Englewood Cliffs, N.J..

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Notes:

Translated from the Russian by Richard A. Silverman. Second English edition published by Dover Publications Inc., New York, 1976.

External Links:

MathReview (R. P. Boas, Jr.), ZentralBlatt

Referenced by:

§1.8(i), §1.8(ii).

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18: 20.2 Definitions and Periodic Properties

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§20.2(i) Fourier Series

… ►Corresponding expansions for ${\theta }_j^{\prime }\left(z\right|\tau )$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$. …

19: Bibliography C

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An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.

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ISSN 0010-4655, Document, MathReview

Referenced by:

§10.74(vii).

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Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.

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Notes:

Reprinted by Dover, New York, 1950.

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Referenced by:

§6.16(i).

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Macdonald’s evaluation conjectures and difference Fourier transform. Invent. Math. 122 (1), pp. 119–145.

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Notes:

Erratum: see same journal 125(1996), no. 2, p. 391.

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ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§17.14.

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Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.

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ISSN 1815-0659, Document, MathReview (Heinrich Begehr), ZentralBlatt

Referenced by:

§14.28(ii), §14.28(ii).

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An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19 (90), pp. 297–301.

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FullText, MathReview, ZentralBlatt

Referenced by:

§3.11(v).

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