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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).
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It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric " bell curve " shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation , sometimes called the Gaussian RMS width) controls the width of the "bell".
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
The gauss is the unit of magnetic flux density B in the system of Gaussian units and is equal to Mx/cm 2 or g/Bi/s 2, while the oersted is the unit of H-field. One tesla (T) corresponds to 10 4 gauss, and one ampere (A) per metre corresponds to 4π × 10 −3 oersted.
The standard Gaussian measure on . is a Borel measure (in fact, as remarked above, it is defined on the completion of the Borel sigma algebra, which is a finer structure);; is equivalent to Lebesgue measure: , where stands for absolute continuity of measures;
In mathematics, Gauss congruence is a property held by certain sequences of integers, including the Lucas numbers and the divisor sum sequence. Sequences satisfying this property are also known as Dold sequences, Fermat sequences, Newton sequences, and realizable sequences. [ 1 ]
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.