Fourier cosine and sine transforms

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1: 1.14 Integral Transforms

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§1.14(ii) Fourier Cosine and Sine Transforms

… ►In this subsection we let $F_c(x)={ℱ}_{c}f\left(x\right)$, $F_s(x)={ℱ}_{s}f\left(x\right)$, $G_c(x)={ℱ}_{c}g\left(x\right)$, and $G_s(x)={ℱ}_{s}g\left(x\right)$. ►

Inversion

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Table 1.14.2: Fourier cosine transforms.

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$f(t)$	$\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle {\int }_0^{\mathrm{\infty }}}f(t)\cos \left(xt\right)dt$,	$x>0$
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Table 1.14.3: Fourier sine transforms.

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$f(t)$	$\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle {\int }_0^{\mathrm{\infty }}}f(t)\sin \left(xt\right)dt$,	$x>0$
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2: 15.17 Mathematical Applications

… ►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. …

3: Errata

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There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$ℱ\left(f\right)\left(x\right)$	$ℱf\left(x\right)$
Fourier Cosine	${ℱ}_c\left(f\right)\left(x\right)$	${ℱ}_{c}f\left(x\right)$
Fourier Sine	${ℱ}_s\left(f\right)\left(x\right)$	${ℱ}_{s}f\left(x\right)$
Laplace	$ℒ\left(f\right)\left(s\right)$	$ℒf\left(s\right)$	$ℒ(f(t);s)$
Mellin	$ℳ\left(f\right)\left(s\right)$	$ℳf\left(s\right)$	$ℳ(f;s)$
Hilbert	$\mathscr{H}\left(f\right)\left(s\right)$	$\mathscr{H}f\left(s\right)$	$\mathscr{H}(f;s)$
Stieltjes	$𝒮\left(f\right)\left(s\right)$	$𝒮f\left(s\right)$	$𝒮(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

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4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►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). …

5: 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”.
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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”.
sin^2 +cos^2	the expression ${\sin }^2+{\cos }^2$.
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trigonometric	the word ”trigonometric” or any of the various trigonometric functions such as $\sin$, $\cos$, $\tan$, and $\cot$.
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6: 7.14 Integrals

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Fourier Transform

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Laplace Transforms

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Laplace Transforms

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7.14.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(t\right)dt=\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.27 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(\sqrt{\frac{2t}{\pi }}\right)dt=\frac{{(\sqrt{a^2+1}+a)}^{\frac{1}{2}}}{2a\sqrt{a^2+1}},$$ $\mathrm{\Re }a>0$,

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.29 in modified form

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7: 1.8 Fourier Series

§1.8 Fourier Series

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Uniqueness of Fourier Series

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

8: 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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9: 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)): … ►

Sine and Cosine Functions

… ►Integral representation (1.17.12_1), (1.17.12_2) is the equivalent of the transform pairs, (1.14.9) (1.14.11), (1.14.10) (1.14.12), respectively. … ►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …

10: 30.15 Signal Analysis

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30.15.3 $${\int }_{-\tau }^{\tau }\frac{\sin \sigma (t-s)}{\pi (t-s)}{\varphi }_n(s)ds={\mathrm{\Lambda }}_n{\varphi }_n(t).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $n\ge m$: integer degree, $\tau >0$: parameter, $\sigma >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$

Permalink:

http://dlmf.nist.gov/30.15.E3

Encodings:

pMML, png, TeX

See also:

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

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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 $[-\sigma ,\sigma ]$. … ►

30.15.11 $$\arccos \sqrt{\mathrm{B}}+\arccos \sqrt{\alpha }=\arccos \sqrt{{\mathrm{\Lambda }}_0},$$

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Symbols:

$\arccos z$: arccosine function, $\mathrm{\Lambda }$ and $\mathrm{B}$: maximum bound

Permalink:

http://dlmf.nist.gov/30.15.E11

Encodings:

pMML, png, TeX

See also:

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

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