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Newton's identities
Newton's identities are two different ways of describing the root of a polynomial. In particular, they relate sums of powers to elementary symmetric polynomials.
Explanation
They are computed on the roots of a polynomial P in one variable and allow the sums of the k-th powers of all the roots of P to be expressed in terms of the coefficients of P, without finding those roots themselves.
Example 1
In the calculation of Fourier's formula for π, the zeros of the sinc function are used in the form
Using Newton's identities, we calculate the coefficients of x2 as
HistoryThese identities were described around 1666 by the English mathematician Isaac Newton, apparently in ignorance of earlier work from 1629 by the French engineer Albert Girard. |