Error expansion of classical mid-point rectangle rule for computing Cauchy principal value integrals on an interval

References

• S. Amari, Evaluation of Cauchy principal-value integrals using modified Simpson rules, Appl. Math. Lett. 7 (1994), pp. 19–23. doi: 10.1016/0893-9659(94)90106-6
   Web of Science ®Google Scholar
• L.C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed., pnMcGraw-Hill, Inc., New York, 1992.
   Google Scholar
• G. Behforooz, Approximation of Cauchy principal value integrals by piecewise Hermite quartic polynomials by spline, Appl. Math. Lett. 5 (1992), pp. 75–78. doi: 10.1016/0893-9659(92)90141-U
   Web of Science ®Google Scholar
• G. Criscuolo, Gaussian formulas for computing Cauchy principal value integrals, and their convergence, Calcolo 22 (1985), pp. 391–411. doi: 10.1007/BF02600384
   Google Scholar
• G. Criscuolo and G. Mastroianni, On the uniform convergence of gaussian quadrature rules for Cauchy principal value integrals, Numer. Math. 54 (1989), pp. 445–461. doi: 10.1007/BF01396323
   Web of Science ®Google Scholar
• C. Dagnino and E. Santi, On the evaluation of one-dimensional Cauchy principal value integrals by rules based on cubic spline interpolation, Computing 43 (1990), pp. 267–276. doi: 10.1007/BF02242921
   Web of Science ®Google Scholar
• K. Diethelm, Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: Basic properties and error estimates, J. Comput. Appl. Math. 65 (1995), pp. 97–114. doi: 10.1016/0377-0427(95)00103-4
   Web of Science ®Google Scholar
• K. Diethelm, Asymptotically sharp error bounds for a quadrature rule for Cauchy principal value integrals based on piecewise linear interpolation, J. Approx. Theory Appl. 11 (1995), pp. 78–89.
   Google Scholar
• K. Diethelm, Modified compound quadrature rules for strongly singular integrals, Computing 52 (1994), pp. 337–354. doi: 10.1007/BF02276881
   Web of Science ®Google Scholar
• T. Hasegawa, Uniform approximations to finite Hilbert transform and its derivative, J. Comput. Appl. Math. 163 (2004), pp. 127–138. doi: 10.1016/j.cam.2003.08.059
   Web of Science ®Google Scholar
• N.I. Ioakimidis, On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives, Math. Comp. 44 (1985), pp. 191–198. doi: 10.1090/S0025-5718-1985-0771040-8
   Web of Science ®Google Scholar
• P. Junghanns and B. Silbermann, Numerical analysis for one-dimensional Cauchy singular integral equations, J. Comput. Appl. Math. 125 (2000), pp. 395–421. doi: 10.1016/S0377-0427(00)00482-9
   Web of Science ®Google Scholar
• P. Koler, On the error of quadrature formulae for Cauchy principal value integrals based on piecewise interpolation, J. Approx. Theory Appl. 13 (1997), pp. 58–69.
   Google Scholar
• P. Kim and U.J. Choi, A quadrature rule of interpolatory type for Cauchy integrals, J. Comput. Appl. Math. 126 (2000), pp. 207–220. doi: 10.1016/S0377-0427(99)00354-4
   Web of Science ®Google Scholar
• P. Kim and B.I. Yun, On the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals, J. Comput. Appl. Math. 149 (2002), pp. 381–395. doi: 10.1016/S0377-0427(02)00481-8
   Web of Science ®Google Scholar
• R. Kress, I.H. Sloan, and F. Stenger, A sinc quadrature method for the double-layer integral equation in planar domains with corners, J. Integr. Eqn Appl. 10 (1998), pp. 291–317. doi: 10.1216/jiea/1181074232
   Google Scholar
• J. Li and D.H. Yu, The superconvergence of certain two-dimensional cauchy principal value integrals, CMES 71(4) (2011), pp. 331–346.
   Web of Science ®Google Scholar
• J. Li and D.H. Yu, The superconvergence of certain two-dimensional Hilbert singular integrals, CMES 82(4) (2011), pp. 233–252.
   Web of Science ®Google Scholar
• I. Lifanov and L. Poltavskii, On the numerical solution of hypersingular and singular integral equations on the circle, Differ. Equ. 39 (2003), pp. 1175–1197. doi: 10.1023/B:DIEQ.0000011292.27341.04
   Web of Science ®Google Scholar
• P. Linz, On the approximate computation of certain strongly singular integrals, Computing 35 (1985), pp. 345–353. doi: 10.1007/BF02240199
   Web of Science ®Google Scholar
• D.J. Liu, X.P. Zhang, and J.M. Wu, The superconvergence of the Newton–Cotes rule for Cauchy principal value integrals, J. Comput. Appl. Math. 235(3) (2010), pp. 696–707. doi: 10.1016/j.cam.2010.06.023
   Web of Science ®Google Scholar
• G. Monegato, Convergence of product formulas for the numerical evaluation of certain two-dimensional cauchy principal value integrals, Numer. Math. 43 (1984), pp. 161–173. doi: 10.1007/BF01390121
   Web of Science ®Google Scholar
• A. Natarajan and N. Mohankumar, A comparison of some quadrature methods for approximating Cauchy principal value integrals, J. Comput. Phys. 116 (1995), pp. 365–368. doi: 10.1006/jcph.1995.1034
   Web of Science ®Google Scholar
• A.P. Orsi, Spline approximation for Cauchy principal value integrals, J. Comput. Appl. Math. 30 (1990), pp. 191–201. doi: 10.1016/0377-0427(90)90027-W
   Web of Science ®Google Scholar
• F. Stenger, Sinc approximation of Cauchy-type integrals over arcs, ANZIAM J. 42(1) (2000), pp. 87–97. doi: 10.1017/S1446181100011627
   Google Scholar
• J.M. Wu and W.W. Sun, The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval, Numer. Math. 109 (2008), pp. 143–165. doi: 10.1007/s00211-007-0125-7
   Web of Science ®Google Scholar
• C.F. Xu and Y.F. Yao, Numerical calculation of Cauchy principal value integrals using wavelet functions, Numer. Math. J. Chinese Univ. 20 (1998), pp. 97–105.
   Google Scholar
• D.H. Yu, Natural Boundary Integrals Method and its Applications, Kluwer Academic Publishers, Dordrecht, 2002.
   Google Scholar