Copyright © 2009 Kamal Aghigh and M. Masjed-Jamei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a finite class of weighted quadrature rules with the weight function |x|−2aexp⁡(−1/x2) on (−∞,∞) as ∫−∞∞|x|−2aexp⁡(−1/x2)f(x)dx=∑i=1nwif(xi)+Rn[f], where xi are the zeros of polynomials orthogonal with respect to the introduced weight function, wi are the corresponding coefficients, and Rn[f] is the error value. We show that the above formula is valid only for the finite values of n. In other words, the condition a≥{max⁡n}+1/2 must always be satisfied in order that one can apply the above quadrature rule. In this sense, some numerical and analytic examples are also given and compared.