# 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 $F_{c}(x)=\mathscr{F}_{\mkern-3.0mu c}\mskip-1.0mu f\mskip 3.0mu \left(x\right)$, $F_{s}(x)=\mathscr{F}_{\mkern-2.0mu s}\mskip-1.0mu f\mskip 3.0mu \left(x\right)$, $G_{c}(x)=\mathscr{F}_{\mkern-3.0mu c}\mskip-1.0mu g\mskip 3.0mu \left(x\right)$, and $G_{s}(x)=\mathscr{F}_{\mkern-2.0mu s}\mskip-1.0mu g\mskip 3.0mu \left(x\right)$.
##### 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.

##### 5: 7.14 Integrals
###### Laplace Transforms
7.14.7 $\int_{0}^{\infty}e^{-at}C\left(\sqrt{\frac{2t}{\pi}}\right)\mathrm{d}t=\frac{(% \sqrt{a^{2}+1}+a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}},$ $\Re a>0$,
##### 8: 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 $\operatorname{arccos}\sqrt{\mathrm{B}}+\operatorname{arccos}\sqrt{\alpha}=% \operatorname{arccos}\sqrt{\Lambda_{0}},$
##### 9: 28.2 Definitions and Basic Properties
With $\zeta={\sin}^{2}z$ we obtain the algebraic form of Mathieu’s equation …With $\zeta=\cos z$ we obtain another algebraic form: … The following three transformations $\cos\left(\pi\nu\right)$ is an entire function of $a,q^{2}$. … The Fourier series of a Floquet solution …
##### 10: Bibliography S
• O. A. Sharafeddin, H. F. Bowen, D. J. Kouri, and D. K. Hoffman (1992) Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.
• D. V. Slavić (1974) Complements to asymptotic development of sine cosine integrals, and auxiliary functions. Univ. Beograd. Publ. Elecktrotehn. Fak., Ser. Mat. Fiz. 461–497, pp. 185–191.
• I. A. Stegun and R. Zucker (1976) Automatic computing methods for special functions. III. The sine, cosine, exponential integrals, and related functions. J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 291–311.
• R. S. Strichartz (1994) A Guide to Distribution Theory and Fourier Transforms. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL.
• S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.