A singular integral is an integral whose integrand has a singularity at one or more points
in the domain of integration. Such integrals
may converge, in which case they are said to exist.
A standard example is the Hilbert transform.
The logarithmic integral with upper endpoint
greater than 1 also has an interior singularity and
is defined by a Cauchy principal value (NIST
DLMF, eqn. 6.2.8).
Such singular integrals are often defined by removing a small neighborhood of the singularity and taking the limit
as this neighborhood shrinks.
See also
Harmonic Analysis,
Hilbert Transform,
Improper Integral,
Riesz
Transform,
Triangular Hilbert Transform,
Trilinear Hilbert Transform
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References
Hackbusch, W. (Ed.). Numerical Techniques for Boundary Element Methods, Proceedings of the Seventh GAMM Seminar
Held at the Christian-Albrechts Universität, Kiel, January 25-27, 1991
Braunschweig, Germany: Vieweg, 1992.Huang, Q. and Cruse, T. A.
"Some Notes on Singular Integral Techniques in Boundary Element Analysis."
Int. J. Numer. Meth. Eng. 36, 2643-2659, 1993.Kutt, H. R.
"The Numerical Evaluation of the Principal Value Integrals by Finite Part Integration."
Numer. Math. 24, 205-210, 1974.NIST Digital Library of
Mathematical Functions. "Exponential and Logarithmic Integrals." §6.2(i).
https://dlmf.nist.gov/6.2.i.
Paulino, G. H. "Singular
Integrals." Fall 2001. http://cee.ce.uiuc.edu/paulino/BEM/handouts/sing.pdfStein,
E. M. Singular
Integrals and Differentiability Properties of Functions. Princeton, NJ: Princeton
University Press, 1971.Tanaka, M.; Sladek, V.; and Sladek, J. "Regularization
Techniques Applied to Boundary Element Method." AMSE Appl. Mech. Rev. 47,
457-499, 1994.Referenced on Wolfram|Alpha
Singular Integral
Cite this as:
Weisstein, Eric W. "Singular Integral."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SingularIntegral.html
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