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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.
A more general formulation of a Gaussian function with a flat-top and Gaussian fall-off can be taken by raising the content of the exponent to a power : = ((())). This function is known as a super-Gaussian function and is often used for Gaussian beam formulation. [ 5 ]
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, [1] is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
For integrating f over [,] with Gauss–Legendre quadrature, the associated orthogonal polynomials are Legendre polynomials, denoted by P n (x). With the n-th polynomial normalized so that 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 [5]
Owen [1] has an extensive list of Gaussian-type integrals; only a subset is given below. Indefinite integrals
The term "numerical integration" first appears in 1915 in the publication A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb. [2] "Quadrature" is a historical mathematical term that means calculating area. Quadrature problems have served as one of the main sources of mathematical analysis.
It is similar to Gaussian quadrature with the following differences: The integration points include the end points of the integration interval. It is accurate for polynomials up to degree 2n – 3, where n is the number of integration points. [8]
The angular integration of an exponential in cylindrical coordinates can be written in terms of Bessel functions of the first kind [4] [5]: 113 ( ()) = and ( ()) = (). For applications of these integrals see Magnetic interaction between current loops in a simple plasma or electron gas .