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.