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Fourier cosine and sine transforms

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1: 1.14 Integral Transforms
§1.14(ii) Fourier Cosine and Sine Transforms
In this subsection we let F c ( x ) = c f ( x ) , F s ( x ) = s f ( x ) , G c ( x ) = c g ( x ) , and G s ( x ) = s g ( x ) .
Inversion
Table 1.14.2: Fourier cosine transforms.
f ( t ) 2 π 0 f ( t ) cos ( x t ) d t ,

x > 0

Table 1.14.3: Fourier sine transforms.
f ( t ) 2 π 0 f ( t ) sin ( x t ) d t ,

x > 0

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.

    Transform New Abbreviated Old
    Notation Notation Notation
    Fourier ( f ) ( x ) f ( x )
    Fourier Cosine c ( f ) ( x ) c f ( x )
    Fourier Sine s ( f ) ( x ) s f ( x )
    Laplace ( f ) ( s ) f ( s ) ( f ( t ) ; s )
    Mellin ( f ) ( s ) f ( s ) ( f ; s )
    Hilbert ( f ) ( s ) f ( s ) ( f ; s )
    Stieltjes 𝒮 ( f ) ( s ) 𝒮 f ( s ) 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.

  • 4: Guide to Searching the DLMF
    Table 1: Query Examples
    Query

    Matching records contain

    "Fourier transform" and series

    both the phrase “Fourier transform” and the word “series”.

    Fourier (transform or series)

    at least one of “Fourier transform” or “Fourier series”.

    1/(2pi) and "Fourier transform"

    both 1 / ( 2 π ) and the phrase “Fourier transform”.

    sin^2 +cos^2

    the expression sin 2 + cos 2 .

    trigonometric

    the word ”trigonometric” or any of the various trigonometric functions such as sin , cos , tan , and cot .

    5: 7.14 Integrals
    Fourier Transform
    Laplace Transforms
    Laplace Transforms
    7.14.5 0 e - a t C ( t ) d t = 1 a f ( a π ) , a > 0 ,
    7.14.7 0 e - a t C ( 2 t π ) d t = ( a 2 + 1 + a ) 1 2 2 a a 2 + 1 , a > 0 ,
    6: 1.8 Fourier Series
    §1.8 Fourier Series
    Uniqueness of Fourier Series
    §1.8(ii) Convergence
    7: 18.17 Integrals
    §18.17(v) Fourier Transforms
    Jacobi
    Ultraspherical
    Legendre
    Hermite
    8: 30.15 Signal Analysis
    30.15.3 - τ τ sin σ ( t - s ) π ( t - s ) ϕ n ( s ) d s = Λ n ϕ n ( t ) .
    §30.15(iii) Fourier Transform
    Equations (30.15.4) and (30.15.6) show that the functions ϕ n are σ -bandlimited, that is, their Fourier transform vanishes outside the interval [ - σ , σ ] . …
    30.15.11 arccos B + arccos α = arccos Λ 0 ,
    9: 28.2 Definitions and Basic Properties
    With ζ = sin 2 z we obtain the algebraic form of Mathieu’s equation …With ζ = cos z we obtain another algebraic form: … The following three transformations cos ( π ν ) 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.