# Singular Integral

## singular integral

[′siŋ·gyə·lər ¦int·ə·grəl əv ə ‚dif·ə′ren·chəl i′kwā·zhən]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Singular Integral

(1) A method of representing functions. A singular integral is an integral of the form which, when certain conditions on ∫ are fulfilled, converges to its generating function f(x) as n → ∞. The function Kn is called the kernel of the singular integral. For example, and are the singular integrals of Dirichlet and Vallée Poussin, respectively. The systematic investigation of singular integrals was begun by H. Lebesgue in 1909. Singular integrals attracted attention in connection with the problem of representing and approximating functions of various classes by simpler functions, such as smooth functions and polynomials.

(2) An improper integral. [23–1231–]

References in periodicals archive ?
Multilinear Singular Integral Forms of Christ-Journe Type
The construction of (C, [C.sup.m])-fast/quasifast solvers of periodic weakly singular integral equations of the second kind in case a = 0 has been undertaken in .
A periodic distribution of constantly loaded mode III slant cracks was analyzed by Li and Lee  using a collocation method and quadrature formula to solve numerically the governing singular integral equations.
where the corresponding singular integral at the right-hand side of the relation is a principle value integral and [z.sub.k](t), k = {0, 1}, are written in the form of truncated Fourier polynomials:
Prabhakar, "A singular integral equation with a generalized Mittag Leffler function in the kernel," Yokohama Mathematical Journal, vol.
Finally, as applications, we prove some singular integral operators are bounded from [H[??].sup.[alpha],p.sub.q,L]([R.sup.n]) to Herz spaces [[??].sup.[alpha],p.sub.q]([R.sup.n]) and also bounded on [H[??].sup.[alpha],p.sub.q,L]([R.sup.n]).
Meng, "Some new weakly singular integral inequalities and their applications to fractional differential equations," Journal of Inequalities and Applications, vol.
In this paper, we propose that an alternative computational scheme by combining the new traction BIE and displacement BIE, that is, the displacement BIE, is collocated on the no-crack boundary, and the traction BIE only with the singular integral kernel in order of 1/r is collocated along the entire crack surface (no need for the separate discretization of the upper and lower crack surfaces).
By assuming perfect adhesion between film and half plane together with membrane behaviour of the film, the compatibility condition between the coating and substrate leads to a singular integral equation with Cauchy kernel.
Jalilian, Existence of solutions for a class of nonlinear Volterra singular integral equations, Comput.
Mathematical modeling of controlling the coating process of the structural elements based on singular integral equations.

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