Cesàro means
(0.010 seconds)
11—20 of 66 matching pages
11: 7.20 Mathematical Applications
12: How to Cite
13: 4.1 Special Notation
14: Bille C. Carlson
15: 23.22 Methods of Computation
In the general case, given by , we compute the roots , , , say, of the cubic equation ; see §1.11(iii). These roots are necessarily distinct and represent , , in some order.
If and are real, and the discriminant is positive, that is , then , , can be identified via (23.5.1), and , obtained from (23.6.16).
If , or and are not both real, then we label , , so that the triangle with vertices , , is positively oriented and is its longest side (chosen arbitrarily if there is more than one). In particular, if , , are collinear, then we label them so that is on the line segment . In consequence, , satisfy (with strict inequality unless , , are collinear); also , .
Finally, on taking the principal square roots of and we obtain values for and that lie in the 1st and 4th quadrants, respectively, and , are given by
where denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs (, ), corresponding to the 2 possible choices of the square root.