Gauss quadrature rules for the evaluation of 2𝜋^{-1/2}∫₀^{∞}exp(-𝑥²)𝑓(𝑥)𝑑𝑥

Abstract

Gauss quadrature rules for evaluating integrals of the form \[ 2 π − 1 / 2 − ∫ 0 ∞ exp ⁡ ( − x 2 ) f ( x ) d x 2{\pi ^{ - 1/2}} - \smallint _0^\infty \exp ( - {x^2})f(x)dx \] have been calculated to 20S {\text {20S}} for one to twenty nodes. The coefficients for the threeterm recurrence relation of the first twenty orthogonal polynomials associated with the weight function exp ( − x 2 ) ( - {x^2}) on the interval [ 0 , ∞ ) [0, \infty ) are also tabulated to 20 S 20{\text {S}} .