# Singular Integral

## singular integral

[′siŋ·gyə·lər ¦int·ə·grəl əv ə ‚dif·ə′ren·chəl i′kwā·zhən]
(mathematics)

## 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 ?
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.
Some of these problems lead to obtain the boundedness of certain singular integral operators and this drives one to the classical and modern Caldern-Zygmund theory, the paradigm of Harmonic Analysis.
Secondly, the singular integral is evaluated using the present method.
Satisfying boundary conditions (5) by the functions (13)-(14) on the faces of cracks with end zones, we get a system of singular integral equations with respect to the unknown functions [g.
Suppose that T is a Calderon-Zygmund singular integral operator.
2004) Numerical solution of singular integral equations using cubic spline interpolation, India journal of applied mathematics, Vol.
Prabhakar, "A singular integral equation with a generalized Mittag-Leffler func-tion in the Kernel," Yokohama Mathematical Journal, vol.
Singular Integral Equations in Diffraction Theory, Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Vol.
Then, we use the Toeplitz matrix method and product Nystrom method, as the best two methods, to obtain the solution of the singular integral equation numerically.
The solution of a system of the singular integral Equations (30) and (31) will be constructed by the method of edge sources [10]
Here we present the investigation of the photonic periodic waveguide structures using the Singular Integral Equations' Method.

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