# American Institute of Mathematical Sciences

July  2015, 20(5): 1355-1375. doi: 10.3934/dcdsb.2015.20.1355

## Euler-Maclaurin expansions and approximations of hypersingular integrals

 1 College of Mathematics, Sichuan University, Chengdu,Sichuan, 610064, China, China 2 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65401, United States

Received  March 2013 Revised  January 2015 Published  May 2015

This article presents the Euler-Maclaurin expansions of the hypersingular integrals $\int_{a}^{b}\frac{g(x)}{|x-t|^{m+1}}dx$ and $\int_{a}^{b}% \frac{g(x)}{(x-t)^{m+1}}dx$ with arbitrary singular point $t$ and arbitrary non-negative integer $m$. These general expansions are applicable to a large range of hypersingular integrals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.
Citation: Chaolang Hu, Xiaoming He, Tao Lü. Euler-Maclaurin expansions and approximations of hypersingular integrals. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1355-1375. doi: 10.3934/dcdsb.2015.20.1355
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