ON SOME HYPERSINGULAR INTEGRALS

Summary: The paper examines different types of hypersingular integrals with the Cauchy kernel on a segment and a unit circle and defines them using specific methods. It presents more general definitions for one-dimensional hypersingular integrals with the Cauchy kernel based on Hadamard's integral in the sense of a finite part. The paper also establishes the existence theorems of these hypersingular integrals and formulas, which demonstrates the accuracy of the resulting integrals that are applied in various applications and engineering problem-solving. The proposed formulas are straightforward to calculate, making the new approximate method reliable and easy to apply and the obtained numerical results demonstrate the stability and efficiency of the approach.