An Elementary Treatise On the Theory of Equations
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267.
SUMS OF THE POWERS OF THE ROOTS. 191 For ' \S'^ = a"' + r + c'"+... , ni + 2 Therefore SJ„^^, -S'^^, = a-b-(a-hy + a-G''{a-cy + hV{b-cy+. „ We will denote this by u^j so that .. =m. -. )-{-f(s)-*j(a"--}- Hence by proceeding as in Arts. 267 and 268 we may obtain the following results.
(1) If all the roots are real -*"±-^ can be brought as near as we please to the product of the two numerically greatest roots by increasing m sufliciently. , (2) If there are real roots numerically greater tha...n the modulus of any imaginary root, there is a limiting value of ^, namely, the product of the two greatest of these roots.
(3) If there be one or more moduli greater than the numori- cally greatest real root there is a limiting value of -'"—, namely, the square of the greatest of these moduli, that is, the product of the corresponding imaginary roots.
(4) Thus the only case in which "''^^ can fail to have a limit u n is when there is one real root, and only one, numerically greater than the greatest modulus of the imaginary roots.
What to read after An Elementary Treatise On the Theory of Equations?
You can find similar books in the "Read Also" column, or choose other free books by Todhunter, I. (Isaac), 1820-1884 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
267.
SUMS OF THE POWERS OF THE ROOTS. 191 For ' \S'^ = a"' + r + c'"+... , ni + 2 Therefore SJ„^^, -S'^^, = a-b-(a-hy + a-G''{a-cy + hV{b-cy+. „ We will denote this by u^j so that .. =m. -. )-{-f(s)-*j(a"--}- Hence by proceeding as in Arts. 267 and 268 we may obtain the following results.
(1) If all the roots are real -*"±-^ can be brought as near as we please to the product of the two numerically greatest roots by increasing m sufliciently. , (2) If there are real roots numerically greater tha...n the modulus of any imaginary root, there is a limiting value of ^, namely, the product of the two greatest of these roots.
(3) If there be one or more moduli greater than the numori- cally greatest real root there is a limiting value of -'"—, namely, the square of the greatest of these moduli, that is, the product of the corresponding imaginary roots.
(4) Thus the only case in which "''^^ can fail to have a limit u n is when there is one real root, and only one, numerically greater than the greatest modulus of the imaginary roots.
What to read after An Elementary Treatise On the Theory of Equations?
You can find similar books in the "Read Also" column, or choose other free books by Todhunter, I. (Isaac), 1820-1884 to read onlineMoreLess
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