Singular Integral

A singular integral is an integral whose integrand reaches an infinite value at one or more points in the domain of integration. Even so, such integrals can converge, in which case, they are said to exist. (If they do not converge, they are said not to exist.) The most commonly encountered example of a singular integral is the Hilbert transform. (However, note that the logarithmic integral is not singular, since it converges in the classical Riemann sense.)

In general, singular integrals can be defined by eliminating a small space including the singularity, and then taking the limit as this small space disappears.

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.

Paulino, G. H. "Singular Integrals." Fall 2001. http://cee.ce.uiuc.edu/paulino/BEM/handouts/sing.pdf

Stein, 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