sine transform

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1: 1.14 Integral Transforms

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§1.14(ii) Fourier Cosine and Sine Transforms

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1.14.10 $${ℱ}_s\left(f\right)\left(x\right)={ℱ}_{s}f\left(x\right)=\sqrt{\frac{2}{\pi }}{\int }_0^{\mathrm{\infty }}f(t)\sin \left(xt\right)dt.$$

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Defines:

${ℱ}_s\left(f\right)\left(s\right)$: Fourier sine transform

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $F_s(x)$: sine transformation of $f(t)$

Referenced by:

§1.17(ii), §1.18(vi), §1.18(vi)

Permalink:

http://dlmf.nist.gov/1.14.E10

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

This equation defines the general notation ${ℱ}_s\left(f\right)\left(x\right)$ or ${ℱ}_{s}f\left(x\right)$. Previously it defined the local notation $F_s(x)$ used in this subsection.

See also:

Annotations for §1.14(ii), §1.14 and Ch.1

►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 1.14.3: 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: 6.14 Integrals

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§6.14(i) Laplace Transforms

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4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Example 2: Sine and Cosine Transforms, $X=[0,\mathrm{\infty })$

►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: 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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6: Bibliography B

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Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations. Elsevier/Academic Press, Amsterdam.

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External Links:

ISBN 978-0-12-373624-6; 0-12-373624-2, MathReview (Gerd Teschke)

Referenced by:

§18.3.

Encodings:

BibTeX

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7: 19.7 Connection Formulas

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Reciprocal-Modulus Transformation

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Imaginary-Modulus Transformation

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Imaginary-Argument Transformation

►With $\sinh \varphi =\tan \psi$, … ►For two further transformations of this type see Erdélyi et al. (1953b, p. 316). …

8: 7.14 Integrals

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7.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}S\left(t\right)dt=\frac{1}{a}\mathrm{g}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $S\left(z\right)$: 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:

7.4.28 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E6

Encodings:

pMML, png, TeX

See also:

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

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7.14.8 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}S\left(\sqrt{\frac{2t}{\pi }}\right)dt=\frac{{(\sqrt{a^2+1}-a)}^{\frac{1}{2}}}{2a\sqrt{a^2+1}},$$ $\mathrm{\Re }a>0$.

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Symbols:

$S\left(z\right)$: 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:

7.4.30 in modified form

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E8

Encodings:

pMML, png, TeX

See also:

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

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9: 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”.
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Table 2: Wildcard Examples

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Query	What it stands for
sin?	sin, sine, sinh, sinc, …
si$	si, sin, sine, sinh, sinc, sinInt, similar, …
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10: 2.6 Distributional Methods

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2.6.27 $$𝒮f\left(z\right)=\frac{\pi }{\sin \left(\pi \alpha \right)}\sum _{s=0}^{n-1}{(-1)}^s\frac{a_s}{z^{s+\alpha }}-\sum _{s=1}^n(s-1)!\frac{c_s}{z^s}+R_n(z),$$

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Symbols:

$𝒮\left(f\right)\left(s\right)$: Stieltjes transform, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sin z$: sine function, $R_n(z)$: remainder, $c\ne 0$: real, $f(t)$: locally integrable function, $a_n$: coefficients and $n$: nonnegative integer

Referenced by:

§2.6(ii)

Permalink:

http://dlmf.nist.gov/2.6.E27

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Stieltjes transform was changed to $𝒮f\left(z\right)$ from $𝒮(f;z)$.

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

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2.6.51 $$ℳf\left(s\right)={(-1)}^s\pi /\sin \left(\pi \alpha \right),$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $f(t)$: locally integrable function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.6.E51

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §2.6(iii), §2.6(iii), §2.6 and Ch.2

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