In this paper, we present high accuracy quadrature formulas for hyper-singular integrals ∫abg(x)qα(x,t)dx\documentclass[12pt]{minimal}
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\begin{document}$\int_{a}^{b}g(x)q^{\alpha}(x,t)\, dx$\end{document}, where q(x,t)=|x−t|\documentclass[12pt]{minimal}
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\begin{document}$q(x,t)=|x-t|$\end{document} (or x−t\documentclass[12pt]{minimal}
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\begin{document}$x-t$\end{document}), t∈(a,b)\documentclass[12pt]{minimal}
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\begin{document}$t\in(a,b)$\end{document}, and α≤−1\documentclass[12pt]{minimal}
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\begin{document}$\alpha\leq-1$\end{document} (or α<−1\documentclass[12pt]{minimal}
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\begin{document}$\alpha<-1$\end{document}). If g(x)\documentclass[12pt]{minimal}
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\begin{document}$g(x)$\end{document} is 2m+1\documentclass[12pt]{minimal}
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\begin{document}$2m+1$\end{document} times differentiable on [a,b]\documentclass[12pt]{minimal}
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\begin{document}$[a,b]$\end{document}, the asymptotic expansions of the error show that the convergence order is O(h2μ+1+α)\documentclass[12pt]{minimal}
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\begin{document}$O(h^{2\mu+1+\alpha})$\end{document} with q(x,t)=|x−t|\documentclass[12pt]{minimal}
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\begin{document}$q(x,t)=|x-t|$\end{document} (or x−t\documentclass[12pt]{minimal}
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\begin{document}$x-t$\end{document}) for α≤−1\documentclass[12pt]{minimal}
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\begin{document}$\alpha\leq-1$\end{document} (or α<−1\documentclass[12pt]{minimal}
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\begin{document}$\alpha<-1$\end{document} and α being non-integer), and the error power is O(hη)\documentclass[12pt]{minimal}
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\begin{document}$O(h^{\eta})$\end{document} with q(x,t)=x−t\documentclass[12pt]{minimal}
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\begin{document}$q(x,t)=x-t$\end{document} for α being integers less than −1, where η=min(2μ,2μ+2+α)\documentclass[12pt]{minimal}
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\begin{document}$\eta =\min(2\mu,2\mu+2+\alpha)$\end{document} and μ=1,…,m\documentclass[12pt]{minimal}
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\begin{document}$\mu=1,\ldots,m$\end{document}. Since the derivatives of the density function g(x)\documentclass[12pt]{minimal}
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\begin{document}$g(x)$\end{document} in the quadrature formulas can be eliminated by means of the extrapolation method, the formulas can easily be applied to solving corresponding hyper-singular boundary integral equations. The reliability and efficiency of the proposed formulas in this paper are demonstrated by some numerical examples.
