Fourier–Bessel expansion

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1: 10.74 Methods of Computation

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Fourier–Bessel Expansion

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2: 10.23 Sums

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Fourier–Bessel Expansion

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3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►This may be compared to (1.17.21), the resulting Fourier, or eigenfunction, expansion … … ►The Fourier cosine and sine transform pairs (1.14.9) & (1.14.11) and (1.14.10) & (1.14.12) can be easily obtained from (1.18.57) as for

\nu=\pm\frac{1}{2}

the Bessel functions reduce to the trigonometric functions, see (10.16.1). … ►For

f(x)

even in

x

this yields the Fourier cosine transform pair (1.14.9) & (1.14.11), and for

f(x)

odd the Fourier sine transform pair (1.14.10) & (1.14.12). … …

4: 28.34 Methods of Computation

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(b)

Use of asymptotic expansions and approximations for large $q$ (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

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(b)

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

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(d)

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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(a)

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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(c)

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

5: Bibliography O

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F. Oberhettinger (1973) Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §1.8(v), §22.11, §4.27.
Encodings:: BibTeX

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A. B. Olde Daalhuis (1994) Asymptotic expansions for

q

-gamma,

q

-exponential, and

q

-Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.

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External Links:: ISSN 0022-247X, Document, MathReview (P. K. Banerji), ZentralBlatt
Referenced by:: §5.18(ii).
Encodings:: BibTeX

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F. W. J. Olver (1951) A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Cambridge Philos. Soc. 47, pp. 699–712.

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External Links:: Document, MathReview (J. Todd), ZentralBlatt
Referenced by:: §10.21(vii).
Encodings:: BibTeX

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F. W. J. Olver (1952) Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.

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External Links:: Document, MathReview (F. Oberhettinger), ZentralBlatt
Referenced by:: §10.19(iii), §10.21(vii).
Encodings:: BibTeX

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F. W. J. Olver (1954) The asymptotic expansion of Bessel functions of large order. Philos. Trans. Roy. Soc. London. Ser. A. 247, pp. 328–368.

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External Links:: FullText, MathReview (N. D. Kazarinoff), ZentralBlatt
Referenced by:: §10.20(ii), §10.20(ii), §10.21(ix), §10.21(viii), 2nd item, (9.9.10), (9.9.11), (9.9.12), (9.9.13), (9.9.14), (9.9.15), (9.9.16), (9.9.17), (9.9.18), (9.9.19), (9.9.20), (9.9.21), (9.9.5), (9.9.6), (9.9.7), (9.9.8), (9.9.9), §9.9(iii), §9.9(iv).
Encodings:: BibTeX

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6: Bibliography S

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O. A. Sharafeddin, H. F. Bowen, D. J. Kouri, and D. K. Hoffman (1992) Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

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External Links:: ISSN 0021-9991, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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A. Sidi (2010) A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

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External Links:: ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §2.3(iv).
Encodings:: BibTeX

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D. Slepian and H. O. Pollak (1961) Prolate spheroidal wave functions, Fourier analysis and uncertainty. I. Bell System Tech. J. 40, pp. 43–63.

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External Links:: MathReview (I. Marx), ZentralBlatt
Referenced by:: §30.15(v).
Encodings:: BibTeX

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R. S. Strichartz (1994) A Guide to Distribution Theory and Fourier Transforms. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL.

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External Links:: ISBN 0-8493-8273-4, MathReview (J. Horváth), ZentralBlatt
Referenced by:: §18.17(v).
Encodings:: BibTeX

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S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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External Links:: ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt
Referenced by:: §17.3(i).
Encodings:: BibTeX

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7: Bibliography C

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S. M. Candel (1981) An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.

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External Links:: ISSN 0010-4655, Document, MathReview
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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H. S. Carslaw (1930) 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.
External Links:: ZentralBlatt
Referenced by:: §6.16(i).
Encodings:: BibTeX

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I. Cherednik (1995) 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.
External Links:: ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §17.14.
Encodings:: BibTeX

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H. S. Cohl (2013a) 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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External Links:: ISSN 1815-0659, Document, MathReview (Heinrich Begehr), ZentralBlatt
Referenced by:: §14.28(ii), §14.28(ii).
Encodings:: BibTeX

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J. P. Coleman and A. J. Monaghan (1983) Chebyshev expansions for the Bessel function

J_{n}(z)

in the complex plane. Math. Comp. 40 (161), pp. 343–366.

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External Links:: ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

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8: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

… ►Also, with

I_{n}

and

K_{n}

denoting the modified Bessel functions (§10.25(ii)), and again with

s=0,1,2,\dots

, … ►

28.24.10

\varepsilon_{s}\operatorname{Ke}_{2m}\left(z,h\right)=\sum_{\ell=0}^{\infty}% \frac{A_{2\ell}^{2m}(h^{2})}{A_{2s}^{2m}(h^{2})}\left(I_{\ell-s}\left(he^{-z}% \right)K_{\ell+s}\left(he^{z}\right)+I_{\ell+s}\left(he^{-z}\right)K_{\ell-s}% \left(he^{z}\right)\right),

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{Ke}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable, $\varepsilon_{s}$ : constants and $A_{m}(q)$ : Fourier coefficient
A&S Ref:: 20.8.9
Permalink:: http://dlmf.nist.gov/28.24.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §28.24 and Ch.28

… ►The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the

z

-plane. ►For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).

9: Bibliography W

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G. Wolf (2008) On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.

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External Links:: ISSN 1044-677X, FullText
Referenced by:: §28.4(vii).
Encodings:: BibTeX

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R. Wong and J. F. Lin (1978) Asymptotic expansions of Fourier transforms of functions with logarithmic singularities. J. Math. Anal. Appl. 64 (1), pp. 173–180.

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External Links:: Document, MathReview (B. L. Gupta), ZentralBlatt
Referenced by:: §2.3(iv).
Encodings:: BibTeX

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R. Wong and J.-M. Zhang (1997) Asymptotic expansions of the generalized Bessel polynomials. J. Comput. Appl. Math. 85 (1), pp. 87–112.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §18.34(iii).
Encodings:: BibTeX

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R. Wong (1973b) On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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External Links:: Document, MathReview (L. Berg), ZentralBlatt
Referenced by:: §8.11(v).
Encodings:: BibTeX

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E. M. Wright (1935) The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

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10: Bibliography V

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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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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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B. Ph. van Milligen and A. López Fraguas (1994) Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:: Document
Referenced by:: §14.31(i).
Encodings:: BibTeX

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H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:: ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt
Referenced by:: §31.15(iii).
Encodings:: BibTeX

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H. Volkmer (2021) Fourier series representation of Ferrers function

{\sf P}

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External Links:: FullText
Referenced by:: (14.13.1).
Encodings:: BibTeX

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