Although the concept of a hypersingular integral was introduced by Hadamard at the beginning of the 20th century, it began to be put into practical use only in the second half of the century. The theory of hypersingular integral equations has been widely developed in recent decades and this is due to the fact that they describe the governing equations of many applied problems in various fields: elasticity theory, fracture mechanics, wave diffraction theory, electrodynamics, nuclear physics, geophysics, theory vibrator antennas, aerodynamics, etc. It is analytically possible to calculate the hypersingular integral only for a very narrow class of functions; therefore, approximate methods for calculating such an integral are always in the field of view of researchers and are a rapidly developing area of computational mathematics. There are a very large number of papers devoted to this subject, in which various approaches are proposed both to approximate calculation of the hypersingular integral and to the solution of hypersingular integral equations, mainly taking into account the specifics of the behavior of the densi-ty of the hypersingular integral. In this paper, quadrature formulas are obtained for a hypersingular integral whose density is the product of the Hölder continuous function on the closed interval [–1, 1], and weight function of the Jacobi polynomials . It is assumed that the exponents α and β can be arbitrary complex numbers that satisfy the condition of non-negativity of the real part. The numerical examples show the convergence of the quadrature formula to the true value of the hypersingular integral. The possibility of applying the mechanical quadrature method to the solution of various, including hypersingular, integral equations is indicated.
APPROXIMATE METHODS OF SOLVING HYPERSINGULAR INTEGRAL EQUATIONS OF FIRST KIND WITH SECOND-ORDER PECULIARITIES ON CLASSES OF FUNCTIONS WITH WEIGHTS (1 t2 )1/2