Fourier integrals

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21: 1.8 Fourier Series

… ►It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi }ℱ\left(f\right)\left(-2\pi n\right)$. …

22: 22.16 Related Functions

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Integral Representation

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Fourier Series

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Relation to Elliptic Integrals

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Integral Representations

… ►See Figure 22.16.2. …

23: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.16 $${\int }_0^{\mathrm{\infty }}\sin \left(2h\cos y\cosh t\right){\mathrm{Ce}}_{2n}(t,h^2)dt=-\frac{\pi A_0^{2n}(h^2)}{2{\mathrm{ce}}_{2n}(\frac{1}{2}\pi ,h^2)}\left({\mathrm{ce}}_{2n}(y,h^2)\mp \frac{2}{\pi C_{2n}(h^2)}{\mathrm{fe}}_{2n}(y,h^2)\right),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, $\sin z$: sine function, $h$: parameter, $n$: integer, $y$: real variable, $A_m(q)$: Fourier coefficient and $C_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.28.E16

Encodings:

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See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.20 $$\frac{2}{\pi }{\int }_0^{\pi }{𝒞}_{2\mathrm{\ell }}^{(j)}(2hR)\cos \left(2\mathrm{\ell }\varphi \right){\mathrm{ce}}_{2m}(t,h^2)dt={\epsilon }_{\mathrm{\ell }}{(-1)}^{\mathrm{\ell }+m}{A}_{2\mathrm{\ell }}^{2m}(h^2){\mathrm{Mc}}_{2m}^{(j)}(z,h),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function, ${\epsilon }_{\mathrm{\ell }}$: constants and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.26 (in different form and only for $\mathrm{\ell }=0$)

Permalink:

http://dlmf.nist.gov/28.28.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

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28.28.21 $$\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\cos \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{ce}}_{2m+1}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Mc}}_{2m+1}^{(j)}(z,h),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.26 (in different form and only for $\mathrm{\ell }=0$)

Referenced by:

Erratum (V1.0.11) for Equations (28.28.21) and (28.28.22)

Permalink:

http://dlmf.nist.gov/28.28.E21

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally the prefactor and uppper limit of integration were given incorrectly as ${\displaystyle \frac{2}{\pi }}{\int }_0^{\pi }$.

Reported 2015-05-20 by Ruslan Kabasayev

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

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28.28.22 $$\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{se}}_{2m+1}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Ms}}_{2m+1}^{(j)}(z,h),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\sin z$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.27 (in different form and only for $\mathrm{\ell }=0$)

Referenced by:

Erratum (V1.0.11) for Equations (28.28.21) and (28.28.22)

Permalink:

http://dlmf.nist.gov/28.28.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally the prefactor and uppper limit of integration were given incorrectly as ${\displaystyle \frac{2}{\pi }}{\int }_0^{\pi }$.

Reported 2015-05-20 by Ruslan Kabasayev

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

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28.28.23 $$\frac{2}{\pi }{\int }_0^{\pi }{𝒞}_{2\mathrm{\ell }+2}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+2)\varphi \right){\mathrm{se}}_{2m+2}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+2}^{2m+2}(h^2){\mathrm{Ms}}_{2m+2}^{(j)}(z,h).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\sin z$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.28 (in different form and only for $\mathrm{\ell }=0$)

Permalink:

http://dlmf.nist.gov/28.28.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

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24: 10.23 Sums

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10.23.11 $$a_k=\frac{1}{2\pi \mathrm{i}}{\int }_{|t|=c^{\prime }}f(t)O_k\left(t\right)dt,$$ ,

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Defines:

$a_k$ (locally)

Symbols:

$O_n\left(x\right)$: Neumann’s polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: nonnegative integer, $z$: complex variable and $f(t)$

A&S Ref:

9.1.83

Referenced by:

Erratum (V1.1.2) for Equation (10.23.11)

Permalink:

http://dlmf.nist.gov/10.23.E11

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

Originally the contour of integration written incorrectly as $|z|=c^{\prime }$, has been corrected to be $|t|=c^{\prime }$.

Reported 2021-03-22 by Mark Dunster

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

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Fourier–Bessel Expansion

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10.23.18

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $f(t)$

Permalink:

http://dlmf.nist.gov/10.23.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

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10.23.19 $$a_m=\frac{2}{{(J_{\nu +1}\left(j_{\nu ,m}\right))}^2}{\int }_0^{1}tf(t)J_{\nu }\left(j_{\nu ,m}t\right)dt,$$ $\nu \ge -\frac{1}{2}$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $dx$: differential of $x$ , $\int$: integral, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$ , $m$: integer, $\nu$: complex parameter, $a_k$ and $f(t)$

Referenced by:

§10.74(vii)

Permalink:

http://dlmf.nist.gov/10.23.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

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25: 15.17 Mathematical Applications

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§15.17(ii) Conformal Mappings

… ►Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. … ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …

26: 18.2 General Orthogonal Polynomials

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18.2.1 $${\int }_a^b{p}_n(x)p_m(x)w(x)dx=0,$$ $n\ne m$.

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer, $a$: interval endpoint, $b$: interval endpoint and $x$: real variable

A&S Ref:

22.1.1

Referenced by:

§18.2(i), §18.2(iii), §18.2(viii), §18.5(iii)

Permalink:

http://dlmf.nist.gov/18.2.E1

Encodings:

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See also:

Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

… ►More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by $d\mu (x)$, where the measure $\mu$ is the Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ corresponding to a bounded nondecreasing function $\alpha$ on the closure of $(a,b)$ with an infinite number of points of increase, and such that for all $n$. … ►For such a system, functions $f\in L_w^2((a,b))$ and sequences $\{{\lambda }_n\}$ ($n=0,1,2,\mathrm{\dots }$) satisfying can be related to each other in a similar way as was done for Fourier series in (1.8.1) and (1.8.2): ►

18.2.24 $${\lambda }_n=h_n^{-1}{\int }_a^{b}f(x)p_n(x)w(x)dx$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $n$: nonnegative integer, $x$: real variable and $h_n$

Referenced by:

§18.18(i)

Permalink:

http://dlmf.nist.gov/18.2.E24

Encodings:

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See also:

Annotations for §18.2(viii), §18.2 and Ch.18

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18.2.26 $${\mu }_n={\int }_a^b{x}^{n}d\mu (x),$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(x), §18.34(ii)

Permalink:

http://dlmf.nist.gov/18.2.E26

Encodings:

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See also:

Annotations for §18.2(ix), §18.2 and Ch.18

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27: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:

ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

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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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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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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

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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28: Bibliography O

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

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External Links:

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).

Encodings:

BibTeX

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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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Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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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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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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29: 30.15 Signal Analysis

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§30.15(ii) Integral Equation

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§30.15(iii) Fourier Transform

… ►Equations (30.15.4) and (30.15.6) show that the functions ${\varphi }_n$ are $\sigma$-bandlimited, that is, their Fourier transform vanishes outside the interval $[-\sigma ,\sigma ]$. … ►

30.15.7 $${\int }_{-\tau }^{\tau }{\varphi }_k(t){\varphi }_n(t)dt={\mathrm{\Lambda }}_n{\delta }_{k,n},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $n\ge m$: integer degree, $\tau >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$

Permalink:

http://dlmf.nist.gov/30.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(iv), §30.15 and Ch.30

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30.15.8 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\varphi }_k(t){\varphi }_n(t)dt={\delta }_{k,n}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $n\ge m$: integer degree and ${\varphi }_n(t)$: function

Permalink:

http://dlmf.nist.gov/30.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(iv), §30.15 and Ch.30

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30: 18.17 Integrals

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18.17.18 $${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n+1}^{(\lambda )}\left(x\right)\sin \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda +1\right)J_{2n+\lambda +1}\left(y\right)}{(2n+1)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }}.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E18

Encodings:

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See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

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18.17.32 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{x}^{\nu -1-2n}{L}_{2n}^{(\nu -1-2n)}\left(ax\right)\cos \left(xy\right)dx=\frac{{(-1)}^n\mathrm{\Gamma }\left(\nu \right)}{2(2n)!}{y}^{2n}\left({(a+iy)}^{-\nu }+{(a-iy)}^{-\nu }\right),$$ $\nu >2n$, $a>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

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