Trigonometric Quadrature Research For Singular Integrals With Cauchy Kernel

By use of the method of the change of trigonometric variable and π (antiperiodic) periodic trigonometric interpolation, fundamental study on numerical computation for singular integrals with Cauchy kernel is discussed in the present thesis which consists of five chapters.At first, the results of π (antiperiodic) periodic trigonometric polynomials interpolation by Delvos are introduced, specially the expression of the Lagrange π periodic trigonometric interpolation with the nonequdistant nodes on [0, k) provides the necessary accessories for our research. Next, the idea and result by Kim and Choi are introduced, which a singular integral with Cauchy kernel is transformed to a singular integral with a new kernel using the method of the change of trigonometric variable, and it is described that the quadrature formula of the singular integral with the new kernel with 2n equidistant nodes on [0,π) is constructed. And then, the new interpolatory-type quadrature formulae for the Cauchy singular integral is constructed, which the density function of the new singular integrals is approximated by π (antiperiodic) periodic trigonometric interpolation polynomials with nonequdistant nodes by Delvos. The different explicit expressions of the quadrature formulae are established depending on the odd and even numbers of the nodes. The recurrence relations of the quadrature coefficients of the quadrature formulae are derived. It is pointed out that the quadrature formulae are exact in the inner product space spanned by some trigonometric functions. The restriction of the 2n number nodes and the equdistance of the nodes in Kim and Choi’s research is cancelled in our result, which provides convenience for the practical engineering application of the quadrature formulae. At end, numerical examples are realized with Matlab, the numerical results and images illustrate the asymptotic error of the quadrature formulae.