A regularization of Fredholm type singular integral equations (Q5945780)

The paper is devoted to singular integral equations with Cauchy kernel NEWLINE\[NEWLINE\int^{b}_{a}\frac{K(x,\xi)}{x-\xi}y(\xi) dx=f(x),\quad x\in (a,b),\tag{1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEy(x)=\int^{b}_{a}\frac{K(x,\xi)}{x-\xi}y(\xi) d\xi+f(x),\quad x\in (a,b),\tag{2}NEWLINE\]NEWLINE on a finite interval \((a,b)\) of the real line. Using the Poincaré-Bertrand formula of interchanging the order of integration in double singular integrals with Cauchy kernels, equations (1) and (2) are reduced to other singular integral equations.NEWLINENEWLINENEWLINENote. 1) Equations (1) and (2) are well known as singular integral equations with Cauchy kernel. For example, see the book by \textit{F. D. Gakhov} [Boundary value problems (1966; Zbl 0141.08001) and (1977; Zbl 0449.30030), and (1990; Zbl 0830.30026)]. 2) By the relation \(K(x,\xi)=K(x,x)+[K(x,\xi)-K(x,x)]\), equations (1) and (2) can be represented as complete singular integral equations with Cauchy kernels, for which the regularization is well known; see the above.