A new interpretation of Cauchy type singular integrals with an application to singular integral equations

Cauchy type integrals were given the interpretation of the principal value for points inside the integration interval. Here this interpretation is modified and generalized in a very simple manner. The new interpretation in general is not equivalent to the classical one. The relationship between the new interpretation and the classical one is investigated and various applications of the new interpretation (to the Plemelj formulas, the Riemann-Hilbert boundary value problem, singular integral equations, the inversion formula, quadrature rules and interface crack problems) are presented.