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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.
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.
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.
The integral of this Gaussian function over the whole -dimensional space is given as =. It can be easily calculated by diagonalizing the matrix C {\displaystyle C} and changing the integration variables to the eigenvectors of C {\displaystyle C} .
List of integrals of Gaussian functions; Gradshteyn, Ryzhik, Geronimus, ... An even larger, multivolume table is the Integrals and Series by Prudnikov, ...
The problem in numerical integration is to approximate definite integrals of the form ∫ a b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\,dx.} Such integrals can be approximated, for example, by n -point Gaussian quadrature
Weights versus x i for four choices of n. In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind:
In numerical analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind: + (). In this case