Fourier integrals

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1: 1.15 Summability Methods

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§1.15(v) Summability of Fourier Integrals

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Poisson Kernel

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Fejér Kernel

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2: 2.3 Integrals of a Real Variable

… ►For the Fourier integral … ►Then …

3: 7.14 Integrals

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Fourier Transform

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4: 1.17 Integral and Series Representations of the Dirac Delta

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§1.17(ii) Integral Representations

►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): … ►

1.17.12 $$\delta \left(x-a\right)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}(x-a)t}dt.$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $\int$: integral

Referenced by:

§1.16(viii)

Permalink:

http://dlmf.nist.gov/1.17.E12

Encodings:

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See also:

Annotations for §1.17(ii), §1.17 and Ch.1

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1.17.17 $$\frac{1}{2\pi }\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}ka}\left({\int }_{-\pi }^{\pi }\varphi (x){\mathrm{e}}^{\mathrm{i}kx}dx\right)=\varphi (a),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: integer and $\varphi (x)$: continuous function

Permalink:

http://dlmf.nist.gov/1.17.E17

Encodings:

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See also:

Annotations for §1.17(iii), §1.17 and Ch.1

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5: Peter L. Walker

… ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …

6: 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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External Links:

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

Referenced by:

§22.11.

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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7: 13.10 Integrals

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§13.10(iv) Fourier Transforms

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8: 13.23 Integrals

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§13.23(ii) Fourier Transforms

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9: Bibliography C

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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.

External Links:

ZentralBlatt

Referenced by:

§6.16(i).

Encodings:

BibTeX

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A method for numerical calculation of Fourier integrals. Numer. Math. 8 (5), pp. 422–436.

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External Links:

ISSN 0029-599X, Document, MathReview (M. P. S. Madan), ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

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10: Bibliography S

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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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