# Hilbert transform

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##### 1: 1.14 Integral Transforms
###### §1.14(v) HilbertTransform
The Hilbert transform of a real-valued function $f(t)$ is defined in the following equivalent ways: …
###### Fourier Transform
A special case is the rule for Hilbert transforms1.14(v)):
3.5.46 $\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)=\frac{1}{\pi}\pvint_{-% \infty}^{\infty}\frac{f(t)}{t-x}\mathrm{d}t,$ $x\in\mathbb{R}$,
##### 3: Errata
• Changes

• 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.

• Several changes have been made in §1.16(vii) to

1. (i)

make consistent use of the Fourier transform notations $\mathscr{F}\left(f\right)$, $\mathscr{F}\left(\phi\right)$ and $\mathscr{F}\left(u\right)$ where $f$ is a function of one real variable, $\phi$ is a test function of $n$ variables associated with tempered distributions, and $u$ is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

2. (ii)

introduce the partial differential operator $\mathbf{D}$ in (1.16.30);

3. (iii)

clarify the definition (1.16.32) of the partial differential operator $P(\mathbf{D})$; and

4. (iv)

clarify the use of $P(\mathbf{D})$ and $P(\mathbf{x})$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

• An entire new Subsection 1.16(viii) Fourier Transforms of Special Distributions, was contributed by Roderick Wong.

• The validity constraint $|\operatorname{ph}z|<\tfrac{1}{6}\pi$ was added to (9.5.6). Additionally, specific source citations are now given in the metadata for all equations in Chapter 9 Airy and Related Functions.

• The relation between Clebsch-Gordan and $\mathit{3j}$ symbols was clarified, and the sign of $m_{3}$ was changed for readability. The reference Condon and Shortley (1935) for the Clebsch-Gordan coefficients was replaced by Edmonds (1974) and Rotenberg et al. (1959) and the references for $\mathit{3j}$, $\mathit{6j}$, $\mathit{9j}$ symbols were made more precise in §34.1.

• The website’s icons and graphical decorations were upgraded to use SVG, and additional icons and mouse-cursors were employed to improve usability of the interactive figures.

• ##### 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, $\lim_{y\to 0+}\Im\Phi(x+iy)$ 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. …
##### 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 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 $n^{\mathrm{ter}}$ Potenzen (Waringsches Problem). Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 17–36 (German).
• ##### 9: 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).
• ##### 10: Bibliography S
• J. L. Schiff (1999) The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.
• J. D. Secada (1999) Numerical evaluation of the Hankel transform. Comput. Phys. Comm. 116 (2-3), pp. 278–294.
• D. Shanks (1955) Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.
• 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.
• B. D. Sleeman (1978) Multiparameter spectral theory in Hilbert space. Research Notes in Mathematics, Vol. 22, Pitman (Advanced Publishing Program), Boston, Mass.-London.