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31: 7.14 Integrals

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Fourier Transform

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32: 10.74 Methods of Computation

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Fourier–Bessel Expansion

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33: 18.17 Integrals

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§18.17(v) Fourier Transforms

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Jacobi

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Ultraspherical

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Legendre

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Hermite

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34: 20.2 Definitions and Periodic Properties

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§20.2(i) Fourier Series

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35: 20.14 Methods of Computation

… ►The Fourier series of §20.2(i) usually converge rapidly because of the factors $q^{{(n+\frac{1}{2})}^2}$ or $q^{n^2}$, and provide a convenient way of calculating values of ${\theta }_j\left(z\right|\tau )$. …

36: 23.8 Trigonometric Series and Products

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§23.8(i) Fourier Series

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37: 24.19 Methods of Computation

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A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

38: 28.22 Connection Formulas

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28.22.5 $$g_{e,2m}(h)={(-1)}^m\sqrt{\frac{2}{\pi }}\frac{{\mathrm{ce}}_{2m}(\frac{1}{2}\pi ,h^2)}{A_0^{2m}(h^2)},$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.6 $$g_{e,2m+1}(h)={(-1)}^{m+1}\sqrt{\frac{2}{\pi }}\frac{{\mathrm{ce}}_{2m+1}^{\prime }(\frac{1}{2}\pi ,h^2)}{h{A}_1^{2m+1}(h^2)},$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.7 $$g_{o,2m+1}(h)={(-1)}^m\sqrt{\frac{2}{\pi }}\frac{{\mathrm{se}}_{2m+1}(\frac{1}{2}\pi ,h^2)}{h{B}_1^{2m+1}(h^2)},$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.8 $$g_{o,2m+2}(h)={(-1)}^{m+1}\sqrt{\frac{2}{\pi }}\frac{{\mathrm{se}}_{2m+2}^{\prime }(\frac{1}{2}\pi ,h^2)}{h^2{B}_2^{2m+2}(h^2)},$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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39: Bibliography S

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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 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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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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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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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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40: Bibliography V

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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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Fourier series representation of Ferrers function $𝖯$ .

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External Links:

FullText

Referenced by:

(14.13.1).

Encodings:

BibTeX

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