Gauss of the formula

1) quadrature g. f. - formula of the form

in which knots xi, and coefficients A_k do not depend on function f(x) and are selected so that the formula is precise (i.e. R_n = 0) for the arbitrary polynomial of the degree of 2n-1. In contrast to the quadrature formulas of Newton- Cotes, the knots in quadrature g. f., generally speaking, are not equidistant. If

that for any natural p is located unity, quadrature g. f. these formulas have high practical value, since in a number of cases they give considerably high accuracy, the better the formulas with the same number of equidistant it is main. Itself Gauss investigated (1816) case of p(x)=l.

2) g. f., which expresses total curvature to the surface through the coefficients of its linear element; in the coordinates, for which g. f. takes the form

This formula was published. it shows in 1827 that the total curvature does not change with the bending of surface. It composes content of one of osn of the proposals of the internal geometry of surface created by Gauss.

3) g. f. for the sums of Gauss:

This formula was used by Gauss (1801) in one of the proofs of the principle of reversibility of the quadratic residue

where r and q - odd prime numbers, and - Legendre symbol. It was the first example of the application of a method of trigonometric sums in the theory of the numbers. This method was developed further in the works G. Weyl and especially i. M. Vinogradov is one of the most powerful methods of the analytical theory of the numbers.

4) g. f. for the sum of hyper-geometric series. If Re(c - b - A) > 0, then

where G(dg) - gamma-function. It is published Into 1812. S. B. stechkin.