Weight functions for Chebyshev quadrature

Abstract

Abstract.In this paper, we investigate if the weight function (1 -x2)~1/2R(x), where R(x) is a rational function of order (1,1), admits Chebyshev quadratures. Many positive examples are provided. In particular, we have proved that the answer is affirmative if R(x) -1 + bx, \b\ < 0.27846....

Fast and accurate evaluation of three‐center, two‐electron Coulomb, hybrid, and three‐center nuclear attraction integrals over Slater‐type orbitals using the SD transformation

“…The Gaunt coefficients are defined as 50–54 These Gaunt coefficients linearize the product of two spherical harmonics: where the symbol ∑ indicates that the summation proceeds in steps of two. The summation limits, which follow from the selection rules satisfied by the Gaunt coefficients, are 51 The reduced Bessel function $\hat{k}$ n −(1/2) ( z ) is defined by 8, 14, 55 Reduced Bessel functions satisfy the recurrence relation: The product of two reduced Bessel functions is given by 55 Another useful property of reduced Bessel functions is 55 The spherical Bessel function j n ( z ) is defined by 48 These functions satisfy the following recurrence relation: The Γ function for integer and half‐integer arguments is defined by 48 The Pochammer symbol, (α) n , is given by 48 The Rayleigh expansion of plane wave functions is defined by 56 The Fourier integral representation of the Coulomb operator is given by Multicenter integral expansions with the Fourier transform method of B functions are based over the following relation <...>…”

Chebyshev-type quadrature and partial sums of the exponential series

“…The question of Chebyshev quadrature for wa(t) has been discussed by Xu [16]. He proved that wa(t) admits Chebyshev quadrature if \a\ < y = 0.2784645... , where y is the unique positive solution of xex+x = 1.…”