Fourier integrals

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31: 22.11 Fourier and Hyperbolic Series

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22.11.13 $${\mathrm{sn}}^2(z,k)=\frac{1}{k^2}\left(1-\frac{E}{K}\right)-\frac{2{\pi }^2}{k^2{K}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^n}{1-q^{2n}}\cos \left(2n\zeta \right).$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§22.11

Permalink:

http://dlmf.nist.gov/22.11.E13

Encodings:

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See also:

Annotations for §22.11 and Ch.22

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32: 29.6 Fourier Series

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29.6.60 $$\sum _{p=1}^{\mathrm{\infty }}{B}_{2p}{D}_{2p}=\frac{4}{\pi }{\int }_0^K{\left({\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(x,k^2)\right)}^{2}dx.$$

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Symbols:

${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer, $p$: nonnegative integer, $x$: real variable, $k$: real parameter, $\nu$: real parameter, $B_{2p+1}$: coefficents and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.6.E60

Encodings:

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See also:

Annotations for §29.6(iv), §29.6 and Ch.29

33: 22.20 Methods of Computation

… ►This formula for $\mathrm{dn}$ becomes unstable near $x=K$. If only the value of $\mathrm{dn}(x,k)$ at $x=K$ is required then the exact value is in the table 22.5.1. … ►If $k,k^{\prime }$ are given with $k^2+k_{}^{\prime }{}_{}{}^2=1$ and , then $K,K^{\prime }$ can be found from … ► $\mathrm{am}(x,k)$ can be computed from its definition (22.16.1) or from its Fourier series (22.16.9). … ►Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. …

34: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►where the integral kernel is given by … ►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). …

35: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
"Fourier transform" and series	both the phrase “Fourier transform” and the word “series”.
Fourier or series	at least one of the words “Fourier” or “series”.
Fourier (transform or series)	at least one of “Fourier transform” or “Fourier series”.
1/(2pi) and "Fourier transform"	both $1/(2\pi )$ and the phrase “Fourier transform”.
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int_$^$ sin	any definite integral of sin
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36: 28.11 Expansions in Series of Mathieu Functions

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28.11.3 $$1=2\sum _{n=0}^{\mathrm{\infty }}{A}_0^{2n}(q){\mathrm{ce}}_{2n}(z,q),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E3

Encodings:

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See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.4 $$\cos 2mz=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m}^{2n}(q){\mathrm{ce}}_{2n}(z,q),$$ $m\ne 0$,

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E4

Encodings:

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See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.5 $$\cos (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m+1}^{2n+1}(q){\mathrm{ce}}_{2n+1}(z,q),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E5

Encodings:

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See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.6 $$\sin (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+1}^{2n+1}(q){\mathrm{se}}_{2n+1}(z,q),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E6

Encodings:

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See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.7 $$\sin (2m+2)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E7

Encodings:

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See also:

Annotations for §28.11, §28.11 and Ch.28

37: Bibliography Z

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Su un integrale definito del prodotto di due funzioni di Struve. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 112 (1-2), pp. 63–81 (Italian).

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External Links:

ISSN 0001-4419, MathReview, ZentralBlatt

Referenced by:

§11.7(ii).

Encodings:

BibTeX

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A fast Fourier-Bessel transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 35 (7), pp. 1128–1133 (Russian).

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Notes:

English translation in Comput. Math. Math. Phys. 35(7), pp. 901–905

External Links:

ISSN 0044-4669, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Symmetric elliptic integrals of the third kind. Math. Comp. 24 (109), pp. 199–214.

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External Links:

FullText, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.21(i), §19.21(iii), §19.22(iii), §19.25(i), §19.26(i).

Encodings:

BibTeX

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38: Bibliography G

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Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.

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Notes:

Includes Algol60 program, maximum accuracy 10S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii), §8.28(vi).

Encodings:

BibTeX

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A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§6.14(iii), §6.17, §6.7(iv).

Encodings:

BibTeX

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Definite integrals of the complete elliptic integral $K$ . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

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A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.

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External Links:

ISSN 0044-2275, Document, MathReview

Referenced by:

§10.71(ii).

Encodings:

BibTeX

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Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

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External Links:

ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt

Referenced by:

§18.28(xi), §18.38(iii).

Encodings:

BibTeX

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39: Bibliography L

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Traité des fonctions elliptiques et des intégrales Eulériennes. Huzard-Courcier, Paris.

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Notes:

vol.1, 1825; vol.2, 1826; three supplements, 1828–1832.

Referenced by:

§19.14(ii), Ch.19.

Encodings:

BibTeX

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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

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Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

BibTeX

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Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

BibTeX

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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

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40: 3.11 Approximation Techniques

… ►In fact, (3.11.11) is the Fourier-series expansion of $f(\cos \theta )$; compare (3.11.6) and §1.8(i). … ►

Example. The Discrete Fourier Transform

… ►is called a discrete Fourier transform pair. ►

The Fast Fourier Transform

►The direct computation of the discrete Fourier transform (3.11.38), that is, of …