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A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
With the n-th polynomial normalized to give P n (1) = 1, the i-th Gauss node, x i, is the i-th root of P n and the weights are given by the formula [3] = [′ ()]. Some low-order quadrature rules are tabulated below (over interval [−1, 1] , see the section below for other intervals).
Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. [ 1 ] : 13–15 Other integrals can be approximated by versions of the Gaussian integral.
In the previous two integrals, n!! is the double factorial: for even n it is equal to the product of all even numbers from 2 to n, and for odd n it is the product of all odd numbers from 1 to n; additionally it is assumed that 0!! = (−1)!! = 1.
The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration. It is a variant of Gaussian quadrature , in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information produced by the computation of a less accurate approximation.
Methods such as Gaussian quadrature and Clenshaw–Curtis quadrature with unequally spaced points (clustered at the endpoints of the integration interval) are stable and much more accurate, and are normally preferred to Newton–Cotes. If these methods cannot be used, because the integrand is only given at the fixed equidistributed grid, then ...
The formula = ((())) expresses the Fubini law. ... The following formulas for Gaussian integrals are used often in the path integral formulation of quantum field theory:
The integral / for some real constants a, b and c > 0 can be calculated by putting it into the form of a Gaussian integral.First, the constant a can simply be factored out of the integral.