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Wed 8 Dec, 2004 10:18 pm
So if I were to take a function like Sin that intercepts the x-axis regularly every pi radians, would I be able to approximate the function by graphing the equation as (x-n*pi)...(x-2pi)(x-pi)x(x+pi)(x+2pi)...(x+n*pi) which could be simplified a bit to x(x-pi^2)(x-4*pi^2)...(x-(n^2)(pi^2)) for large n, or is that just going to approximate it at the x intercepts. I would try it myself, but my calculator doesn't want to be nice to me and I figure someone else might have a bit better of a technique they could use.
The amplitude needs to be kept uniform and bounded.
Taylor expansion
Why can't you use the standard polynomial expansion for sin x?
sin x = x - x^3/3! + x^5/5! - x^7/7! + ...
Because I hadn't heard about it before. It uses odd numbers though, correct? All the ones you have up there are prime.
Yes, Odd numbers
"It uses odd numbers though, correct? All the ones you have up there are prime."
Yes, it just continues the series with the sign alternating. Even numbers are used in the cosine expansion.
cos x = 1 - x^2/2! + x^4/4! - x^6/6! ....
The polynomial approximation for sin is given by power series:
sin x = x - (x^3)/3! + ... + [(-1)^n x^(2n+1)]/(2n+1)!
Power series can be found in any calculus book. There are also series for other trig functions, pi, and even the number e.
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