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1: 1.14 Integral Transforms
§1.14(v) Hilbert Transform
The Hilbert transform of a real-valued function f ( t ) is defined in the following equivalent ways: …
Inversion
Inequalities
Fourier Transform
2: 3.5 Quadrature
A special case is the rule for Hilbert transforms1.14(v)):
3.5.46 f ( x ) = 1 π - f ( t ) t - x d t , x ,
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: Bibliography F
  • B. D. Fried and S. D. Conte (1961) The Plasma Dispersion Function: The Hilbert Transform of the Gaussian. Academic Press, London-New York.
  • 5: 1.15 Summability Methods
    Moreover, lim y 0 + Φ ( x + i y ) is the Hilbert transform of f ( x ) 1.14(v)). …
    6: 18.38 Mathematical Applications
    Integrable Systems
    It has elegant structures, including N -soliton solutions, Lax pairs, and Bäcklund transformations. …
    Riemann–Hilbert Problems
    Radon Transform
    7: Bibliography W
  • Z. Wang and R. Wong (2006) Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.
  • G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.
  • F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.
  • D. V. Widder (1979) The Airy transform. Amer. Math. Monthly 86 (4), pp. 271–277.
  • D. V. Widder (1941) The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.
  • 8: Bibliography C
  • R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.
  • S. M. Candel (1981) An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.
  • B. C. Carlson (1976) Quadratic transformations of Appell functions. SIAM J. Math. Anal. 7 (2), pp. 291–304.
  • N. B. Christensen (1990) Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.
  • R. Courant and D. Hilbert (1953) Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, N.Y..
  • 9: Bibliography H
  • P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
  • R. A. Handelsman and J. S. Lew (1970) Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.
  • R. A. Handelsman and J. S. Lew (1971) Asymptotic expansion of a class of integral transforms with algebraically dominated kernels. J. Math. Anal. Appl. 35 (2), pp. 405–433.
  • E. W. Hansen (1985) Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.
  • D. Hilbert (1909) Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n ter Potenzen (Waringsches Problem). Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 17–36 (German).
  • 10: Bibliography D
  • B. Davies (1984) Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.
  • S. R. Deans (1983) The Radon Transform and Some of Its Applications. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.
  • L. Debnath and D. Bhatta (2015) Integral transforms and their applications. Third edition, CRC Press, Boca Raton, FL.
  • P. A. Deift (1998) Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, Vol. 3, New York University Courant Institute of Mathematical Sciences, New York.
  • G. Doetsch (1955) Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).