# sine transform

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##### 1: 1.14 Integral Transforms
###### §1.14(ii) Fourier Cosine and SineTransforms
1.14.10 $\mathscr{F}_{\mkern-2.0mus}\left(f\right)\left(x\right)=\mathscr{F}_{\mkern-2.% 0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)=\sqrt{\frac{2}{\pi}}\int^{\infty}% _{0}f(t)\sin\left(xt\right)\,\mathrm{d}t.$
In this subsection we let $F_{c}(x)=\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)$, $F_{s}(x)=\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)$, $G_{c}(x)=\mathscr{F}_{\mkern-3.0muc}\mskip-1.0mug\mskip 3.0mu\left(x\right)$, and $G_{s}(x)=\mathscr{F}_{\mkern-2.0mus}\mskip-1.0mug\mskip 3.0mu\left(x\right)$.
##### 2: 15.17 Mathematical Applications
Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform1.14(ii)) or as a specialization of a group Fourier transform. …
##### 4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
###### Example 2: Sine and Cosine Transforms, $X=[0,\infty)$
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
• Section 1.14

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.

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

• ##### 6: Bibliography B
• V. Britanak, P. C. Yip, and K. R. Rao (2007) Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations. Elsevier/Academic Press, Amsterdam.
• ##### 7: 19.7 Connection Formulas
###### Imaginary-Argument Transformation
With $\sinh\phi=\tan\psi$, … For two further transformations of this type see Erdélyi et al. (1953b, p. 316). …
##### 10: 2.6 Distributional Methods
2.6.27 $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\frac{\pi}{\sin\left(\pi% \alpha\right)}\sum_{s=0}^{n-1}(-1)^{s}\frac{a_{s}}{z^{s+\alpha}}-\sum_{s=1}^{n% }(s-1)!\frac{c_{s}}{z^{s}}+R_{n}(z),$
2.6.51 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)=(-1)^{s}\pi/\sin\left(\pi% \alpha\right),$