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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: 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 ν = ± 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: 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 .
    6: 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 ,
    7: 1.8 Fourier Series
    §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 2 π ( f ) ( 2 π n ) . …
    8: 18.17 Integrals
    §18.17(v) Fourier Transforms
    Jacobi
    Ultraspherical
    Legendre
    Hermite
    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 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 ,