Calculating Supersingular Integrals Using Rithe Function And Complex Integrals

Since the birth of the natural boundary element method, supersingular integralscalculation becomes an important core issue. In this paper, we use orthogonal functionsand complex integrals to calculate the approximate and exact value of the supersingularintegral. Complex integrals method is first proposed to calculate singular integrals, and itis a very effective method. The main methods in this paper are as follows: we transformsupersingular integrals into the sum of the supersingular integrals whose exact value areknown by using the orthogonal functions to approximate density function, then obtainsupersingular integrals approximate value; through the generating function definition ofBernoulli numbers, the kernel of supersingular integrals on circle can be expanded intoLaurent series, and then apply the previous method to calculate its approximate value; bymaking full use of the Cauchy integral formula, we transfer the path of supersingularintegrals from the real axis to the complex plane, which makes the integrals path no longercontain singular points. So we can use the method of calculating normal complex integralsto calculate supersingular integrals, and then we move the supersingular integrals path tothe real axis from the complex plane, thereby we obtain the exact formulas ofsupersingular integrals.The paper is organized as follows:Firstly, supersingular integrals formulas are given when the density function is apolynomial or trigonometric function. And polynomial wavelet function constructionmethod is proposed to approximate density function, thus we can get supersingularintegrals approximate value. Meanwhile we make the further analysis to the error estimateand the convergence rate.Secondly, supersingular integrals on the circle is approximately calculated by thesupersingular integrals.Thirdly, we take advantage of the Cauchy integral formula and the Taylor formula toconvert the supersingular integrals whose density function is analytical into the complexintegrals without singularity point in the integral path. So that supersingular integrals is nodifference from normal integrals. According to the conclusion, we subsequently deduce the exact value formulas of supersingular integrals in real analysis, meanwhile the exactvalue formulas of supersingular integrals are presented when the density function isinverse tangent functions, inverse cotangent functions, logarithmic and exponentialfunctions.Finally, using the complex integrals method, we successfully solve the calculationproblem of the supersingular integrals on the circle when the density function is analytical,and further study the higher order supersingular integrals on the circle.