# Fourier cosine and sine transforms

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##### 1: 1.14 Integral Transforms
###### §1.14(ii) FourierCosine and SineTransforms
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. …
##### 3: 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.

• ##### 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). …
##### 7: 1.8 Fourier Series
###### Uniqueness of Fourier Series
It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi}\mathscr{F}\left(f\right)\left(-2\pi n\right)$. …
##### 9: 1.17 Integral and Series Representations of the Dirac Delta
###### §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
30.15.3 $\int_{-\tau}^{\tau}\frac{\sin\sigma(t-s)}{\pi(t-s)}\phi_{n}(s)\,\mathrm{d}s=% \Lambda_{n}\phi_{n}(t).$
###### §30.15(iii) FourierTransform
Equations (30.15.4) and (30.15.6) show that the functions $\phi_{n}$ are $\sigma$-bandlimited, that is, their Fourier transform vanishes outside the interval $[-\sigma,\sigma]$. …
30.15.11 $\operatorname{arccos}\sqrt{\mathrm{B}}+\operatorname{arccos}\sqrt{\alpha}=% \operatorname{arccos}\sqrt{\Lambda_{0}},$