Newton's Identities and the Laplace Transform

The relations now named Newton's identities or the Girard-Newton formulas were obtained by A. Girard in 1629 and, independently, by I. Newton in 1666 and published by Newton in his book Arithmetica Universalis (1707). They connect the sums of powers of the roots of a polynomial, counted with th...

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Bibliographic Details
Published inThe American mathematical monthly Vol. 117; no. 1; pp. 67 - 71
Main Author Cîrnu, Mircea I.
Format Journal Article
LanguageEnglish
Published Washington Taylor & Francis 01.01.2010
Mathematical Association of America
Taylor & Francis Ltd
Subjects
Algebra
Differential equations
Discrete mathematics
Laplace transformation
Linear algebra
Logical proofs
Mathematical functions
Mathematical theorems
Mathematics
Matrix identities
Polynomials
Proof theory
Theorems
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Summary:The relations now named Newton's identities or the Girard-Newton formulas were obtained by A. Girard in 1629 and, independently, by I. Newton in 1666 and published by Newton in his book Arithmetica Universalis (1707). They connect the sums of powers of the roots of a polynomial, counted with their multiplicities, with the polynomial's coefficients and also by Viete's formulas with the symmetric polynomials in the roots. There are many proofs for Newton's identities, using a variety of approaches. Here, Cirnu gives a new proof, based on the Laplace transform. This transform, introduced by L. Euler in 1737 and extensively used by P. S. Laplace in his book Theorie Analytique des Probabilites has many applications in mathematics, physics, optics, electrical and control engineering, signal processing, and probability theory. Moreover, Cirnu presents an explicit formula for the solution of the initial value problem for homogeneous linear differential equations with constant coefficients, obtained by the Laplace transform method.
ISSN:0002-9890
1930-0972
DOI:10.4169/000298910X474998