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On Kutt’s Gaussian quadrature rule for finite-part integrals. (English) Zbl 0688.41030

Let us consider the divergent integrals of the forms \(I_ 1=\oint^{1}_{0}x^{\lambda}f(x)dx\), \(I_ 2=\oint^{1}_{0}| x-y|^{\lambda}f(x)dx\), \(y\in (0,1)\), \(\lambda\leq -1\), and defined in the sense of finite-part integrals. For the numerical evaluation of these integrals quadrature rules are available. For instance, it is well- known the Kutt’s Gaussian quadrature rule [H. R. Kutt, “Quadrature formulae for finite-part integrals”, CSIR Special Rept. WISK 178. Pretoria National Research Institute for Mathematical Sciences (1975; Zbl 0327.65027)]. The author of the paper under review establishes the relationship of this rule to the classical Gauss-Jacobi quadrature formula in the case when \(\lambda\) is not a negative integer. In fact it is shown that Kutt’s Gaussian quadrature rule for \(I_ 1\) coincides with the Gauss-Jacobi formula with shifted Jacobi polynomials. Finally, it is mentioned that the integral \(I_ 2\) can be directly evaluated by means of two integrals of the form \(I_ 1\).

MSC:

41A55 Approximate quadratures
41A35 Approximation by operators (in particular, by integral operators)

Citations:

Zbl 0327.65027
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References:

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