Gauss-Hermite quadrature

Gauss-Hermite quadrature is a form of Gaussian quadrature for approximating integrals over the whole real line in which the function is multiplied by a weight of e to the power minus x squared. The integral is replaced by a weighted sum of the function's values at n sample points. The sample points are the roots of the physicists' version of the Hermite polynomial of degree n, and each point carries a weight given by a closed formula involving factorials, the square root of pi and the Hermite polynomial of the next lower degree. A common use is the expectation of a function of a normally distributed variable: after a change of variable that scales and shifts it, the expectation takes the form the rule handles. In the simplest non-trivial case, with two points, the rule reduces to the average of the function's values one standard deviation below and above the mean. Published tables list the abscissae and weights up to order 32.