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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.
If the interval [a, b] is subdivided, the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at the midpoint for odd numbers of evaluation points), and thus the integrand must be evaluated at every point. Gauss–Kronrod formulas are extensions of the Gauss quadrature formulas generated by ...
The integration problem can be expressed in a slightly more general way by introducing a positive weight function ω into the integrand, and allowing an interval other than [−1, 1]. That is, the problem is to calculate ∫ a b ω ( x ) f ( x ) d x {\displaystyle \int _{a}^{b}\omega (x)\,f(x)\,dx} for some choices of a , b , and ω .
Gauss–Jacobi quadrature can be used to approximate integrals of the form () (+) where ƒ is a smooth function on [−1, 1] and α, β > −1. The interval [−1, 1] can be replaced by any other interval by a linear transformation. Thus, Gauss–Jacobi quadrature can be used to approximate integrals with singularities at the end points.
"Table of zeros and Gaussian Weights of certain Associated Laguerre Polynomials and the related Hermite Polynomials". Mathematics of Computation. 18 (88): 598– 616. doi: 10.1090/S0025-5718-1964-0166397-1. JSTOR 2002946. MR 0166397. Ehrich, S. (2002). "On stratified extensions of Gauss-Laguerre and Gauss-Hermite quadrature formulas".
Gauss–Legendre quadrature is optimal in a very narrow sense for computing integrals of a function f over [−1, 1], since no other quadrature rule integrates all degree 2n − 1 polynomials exactly when using n sample points. However, this measure of accuracy is not generally a very useful one---polynomials are very simple to integrate and ...
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. Fourier integrals are also considered.
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.