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11: 6.16 Mathematical Applications
§6.16(i) The Gibbs Phenomenon
Consider the Fourier series … Compare Figure 6.16.1. … It occurs with Fourier-series expansions of all piecewise continuous functions. … …
12: 2.10 Sums and Sequences
This identity can be used to find asymptotic approximations for large n when the factor v j changes slowly with j , and u j is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). …
13: 1.14 Integral Transforms
§1.14 Integral Transforms
1.14.7 F ( x ) G ( x ) d x = f ( t ) g ( t ) d t ,
1.14.7_5 F ( x ) G ( x ) ¯ d x = f ( t ) g ( t ) ¯ d t ,
§1.14(viii) Compendia
14: 1.16 Distributions
1.16.29 ( ϕ ) ( 𝐱 ) = ϕ ( 𝐱 ) = 1 ( 2 π ) n / 2 n ϕ ( 𝐭 ) e i 𝐱 𝐭 d 𝐭 ,
1.16.38 ( δ ) , ϕ = δ , ( ϕ ) = δ , 1 2 π ϕ ( t ) e i x t d t = 1 2 π ϕ ( t ) d t = 1 2 π 1 , ϕ , ϕ 𝒯 .
The second to last equality follows from the Fourier integral formula (1.17.8). …
15: Bibliography P
  • A. Pinkus and S. Zafrany (1997) Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.
  • 16: Bibliography H
  • P. Henrici (1986) Applied and Computational Complex Analysis. Vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons Inc.], New York.
  • 17: 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.

  • 18: 15.14 Integrals
    §15.14 Integrals
    Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). … For other integral transforms see Erdélyi et al. (1954b), Prudnikov et al. (1992b, §4.3.43), and also §15.9(ii).
    19: 28.10 Integral Equations
    28.10.1 2 π 0 π / 2 cos ( 2 h cos z cos t ) ce 2 n ( t , h 2 ) d t = A 0 2 n ( h 2 ) ce 2 n ( 1 2 π , h 2 ) ce 2 n ( z , h 2 ) ,
    28.10.2 2 π 0 π / 2 cosh ( 2 h sin z sin t ) ce 2 n ( t , h 2 ) d t = A 0 2 n ( h 2 ) ce 2 n ( 0 , h 2 ) ce 2 n ( z , h 2 ) ,
    28.10.3 2 π 0 π / 2 sin ( 2 h cos z cos t ) ce 2 n + 1 ( t , h 2 ) d t = h A 1 2 n + 1 ( h 2 ) ce 2 n + 1 ( 1 2 π , h 2 ) ce 2 n + 1 ( z , h 2 ) ,
    28.10.4 2 π 0 π / 2 cos z cos t cosh ( 2 h sin z sin t ) ce 2 n + 1 ( t , h 2 ) d t = A 1 2 n + 1 ( h 2 ) 2 ce 2 n + 1 ( 0 , h 2 ) ce 2 n + 1 ( z , h 2 ) ,
    28.10.5 2 π 0 π / 2 sinh ( 2 h sin z sin t ) se 2 n + 1 ( t , h 2 ) d t = h B 1 2 n + 1 ( h 2 ) se 2 n + 1 ( 0 , h 2 ) se 2 n + 1 ( z , h 2 ) ,
    20: 10.74 Methods of Computation
    §10.74(iii) Integral Representations
    The integral representation used is based on (10.32.8). …
    §10.74(vii) Integrals
    Fourier–Bessel Expansion
    Spherical Bessel Transform