Principal Value and Its Applications for One Type of Cauchy—Fantappiè Integral and the System of Equations

It is known that principal value plays vital roles in the study of singular integral. In this paper, one type of Cauchy—Fantappiè integral with multiple indexes is introduced. Since the structure of the kernel is complexity, the Jacobian determinant included in the kernel is expanded for making clearly the expression of the kernel. Moreover, one differential operator is utilized for setting up relations between integrals with higher and usual orders. The work also concerns the convergent properties of the integral. In order to study Hadamard principal value and composite formula of this integral, finite and divergent parts will be estimated and separated. As an application, solvability of the system of integral equations with higher order singularity kernel is discussed.