Gaussian Formulas definition of Gaussian Formulas in the Free Online Encyclopedia.

Gaussian Formulas

formulas relating to various branches of mathematics and bearing the name of K. Gauss.

(1) Gaussian quadrature formulas —formulas of the type

in which the nodes X_k and the coefficients A_k do not depend on the function f(x) and are chosen in such a way that the formula is accurate (that is, R_n = 0) for an arbitrary polynomial of degree 2n — 1. In contrast to the Newton-Cotes quadrature formulas, the nodes in Gaussian quadrature formulas generally are not equally spaced. If p(x) ≥ 0 and , then for any natural n there is only one Gaussian quadrature formula. These formulas are of great practical importance, since in many cases they result in much greater accuracy than do quadrature formulas having the same number of equidistant nodes. In 1816, Gauss himself investigated the case p(x) ≡ 1 .

(2) The Gaussian formula that expresses the total curvature K of a surface in terms of the coefficients of its linear element; in coordinates for which ds² = λ(du² + dv²), this Gaussian formula has the form

This formula, published in 1827, shows that the total curvature does not change as the surface is bent. It constitutes the content of one of the main propositions of the intrinsic geometry of a surface devised by Gauss.

(3) The Gaussian formula for Gaussian sums:

This formula was used by Gauss in 1801 in one of the proofs of the reciprocity principle of quadratic residues

where p and q are odd primes and (p/q) is the Legendre symbol. This formula was the first example of the use of the method of trigonometric sums in number theory. This method was further developed in the works of H. Weyl and especially of I. M. Vinogradov and represents one of the most powerful methods of analytic number theory.

(4) The Gaussian formula for the sum of a hypergeometric series. If Re(c — b — a) > 0, then

where Γ(x) is the gamma function. This formula was published in 1812.

S. B. STECHKIN

Mentioned in	?		References in periodicals archive	?		Encyclopedia browser	?		Full browser	?
No references found	It is known that Gaussian formulas ensure a higher order of accuracy and have better stability properties than Newton-Cotes formulas [2]. Gaussian direct quadrature methods for double delay Volterra integral ... by Cardone, Angelamaria; Del Prete, Ida; Nitsch, And Claudia / Electronic Transactions on Numerical Analysis The closest parallelism with classical Gaussian formulas and orthogonal polynomials will be specially emphasized. Orthogonal Laurent polynomials and quadratures on the unit circle and ... by Cruz-Barroso, Ruyman; Gonzalez-Vera, Pablo / Electronic Transactions on Numerical Analysis	Gauss test Gauss Theorem Gauss' error curve Gauss' hypergeometric equation Gauss' law of flux Gauss' law of the arithmetic mean Gauss' mean value theorem Gauss' principle of least constraint Gauss' theorem Gauss, Carl Friedrich Gauss, Karl Friedrich Gauss-Bonnet theorem Gauss-Codazzi equations Gauss-Jordan elimination Gauss-Legendre rule Gauss-Seidel method Gaussian beam Gaussian blur Gaussian complex integers Gaussian Constant Gaussian curvature Gaussian curve Gaussian distribution Gaussian elimination Gaussian error Gaussian Formulas Gaussian integer Gaussian noise Gaussian noise generator Gaussian optics Gaussian pulse Gaussian reduction Gaussian representation Gaussian system Gaussian System of Units Gaussian units Gaussian weighing method Gaussian year gaussmeter Gautama Gauteng Gauthiot, Robert Gautier Gautier, Émile Félix Gautier, Theophile Gautier, Théophile Gauze gauze tones gavage Gavarni Gavarni, Paul	Gaussian Beam Model Gaussian blur Gaussian Broadcast Channel Gaussian Charge Polarizable Model Gaussian Co-Channel Interferer Gaussian Collision Channel Gaussian complex integers Gaussian constant Gaussian Correction Functions Gaussian curvature Gaussian curve Gaussian curve Gaussian density Gaussian density Gaussian density Gaussian density Gaussian Differential Binary Phase Shift Keying Gaussian distribution Gaussian distribution Gaussian distribution Gaussian distribution Gaussian distribution Gaussian Distributions Gaussian Distributions Gaussian Distributions Gaussian Distributions Gaussian elimination Gaussian Emulation Machine for Sensitivity Analysis Gaussian error Gaussian filter Gaussian Formulas Gaussian Frequency Shift Keying Gaussian function Gaussian function Gaussian Impulse Antenna Gaussian integer Gaussian integers Gaussian Interference Channel Gaussian Interference Channel with Common Information Gaussian interpolation Gaussian Least Square Differential Correction Gaussian Low-Pass Filter Gaussian Markov Random Field Gaussian Maximum Likelihood Classifier Gaussian Method Gaussian Minimum Shift Keying Gaussian Minimum Shift Keying - Frequency Modulation Gaussian Mixture Bi-Gram Model Gaussian mixture model Gaussian Mixture Noise Gaussian Multiple-Access Channel Gaussian Network Model Gaussian noise Gaussian noise generator Gaussian normal distribution Gaussian normal distribution Gaussian normal distribution Gaussian normal distribution Gaussian optics

Mentioned in

References in periodicals archive

Encyclopedia browser

Full browser

No references found

It is known that Gaussian formulas ensure a higher order of accuracy and have better stability properties than Newton-Cotes formulas [2].

Gaussian direct quadrature methods for double delay Volterra integral ... by Cardone, Angelamaria; Del Prete, Ida; Nitsch, And Claudia / Electronic Transactions on Numerical Analysis

The closest parallelism with classical Gaussian formulas and orthogonal polynomials will be specially emphasized.

Orthogonal Laurent polynomials and quadratures on the unit circle and ... by Cruz-Barroso, Ruyman; Gonzalez-Vera, Pablo / Electronic Transactions on Numerical Analysis

Gauss test

Gauss Theorem

Gauss' error curve