On numerical integration methods with T-distribution weight function

摘要

A class of weighted quadrature rules whose weight function corresponds to T distribution, i.e. K(1+x2)-(p-12),x∈(-∞,∞), is introduced and investigated. The integration formulas, given in this work, are generally in the following form:∫-∞∞(1+x2)-(p-12)f(x)dx=∑i=1nwif(xi)+Rn[f],where xi is the zeros of orthogonal polynomials with respect to the introduced weight function, wi is the related coefficient and Rn[f] is the error function. It is important to point out that the above mentioned formula is valid only for the finite values of n. In other words, p > {max n} + 1 must be satisfied in order that the above integration formula is applicable. Some analytical examples are finally given and compared.