Help my buddy out…
"I have a test tomorrow and this is one of the questions. The test is open book and open notes and I am writing down all of the equations for 2-6 section Chebyshev transformers and I can’t figure out the polynomial equations for a 5 and 6 section polynomial. Any help would be GREATLY appreciated.
Thanks
Pope"
There are different forms, but I think these are most common:
T[sub]5[/sub] = 5x - 20x[sup]3[/sup] + 16x[sup]5[/sup]
T[sub]6[/sub] = -1 + 18 x[sup]2[/sup] - 48x[sup]4[/sup] + 32x[sup]6[/sup]
x[sup]5[/sup] = (10T[sub]1[/sub] + 5T[sub]3[/sub] + T[sub]5[/sub])/16
x[sup]6[/sup] = (10T[sub]0[/sub] + 15T[sub]2[/sub] + 6T[sub]4[/sub] + T[sub]6[/sub])/32
The answer to your question is about three-fifths of the way down this MathWorld page. It just confirms what JonF said, but I wanted an excuse to plug the site.
Those are Chebyshev Polynomials of the First Kind. For Chebyshev Polynomials of the Second Kind, look three-fifths of the way down this page.
Eric Weisstein’s World of Mathematics site has been back on line for just over a week. It is a fabulous resource for any mathematical question, issue, or concept.
Your question regarding Chebyshev polynomials is answered here.
Sorry, simulpost.
Yeah, but I got 'em up through order 20 here {grin}. Figured 'em out about 25 years ago and I think I’ve referred to 'em about three times, including this time.
well, sure. doesn’t everyone, but I’m not gonna do your work for you
Neener neener neener
1 - 200x[sup]2[/sup] + 6600x[sup]4[/sup] - 84480x[sup]6[/sup] + 549120x[sup]8[/sup] - 2050048x[sup]10[/sup] + 4659200x[sup]12[/sup] - 6553600x[sup]14[/sup] + 5570560x[sup]16[/sup] - 2621440x[sup]18[/sup] + 524288x[sup]20[/sup], right?
I have the coefficient of x[sup]4[/sup] as 6606, otherwise I agree.
If we only disagree on one coefficient, it should be easy enough to check, right? Since T[sub]n/sub = 1. Of course, I’d never even heard of these things until a few days ago, so I don’t want to say with any certainty.