12/17/2009, 02:40 AM
Sorry for continuing to post more limit formulas but I found another that I do not think has been mentioned before.
\( f^n(x) = \lim_{k\to \infty} f^{-k}(\frac {f'(f^k(x))^n(f^k(x)-f(f^k(x)))+f(f^k(x))-f^k(x)f'(f^k(x))} {1-f'(f^k(x))}) \)
which is the same as
\( f^n(x) = \lim_{k\to \infty}f^{-k}(\frac {f'(u)^n(u-f(u))+f(u)-uf'(u)} {1-f'(u)})\\where\\u=f^k(x) \)
This works whenever a function has a regular attracting or repelling fixed point that it increases through. For it to work near a repelling fixed point you simply let k approach negative infinity.
Also note that \( \frac {f'(x)^n(x-f(x))+f(x)-xf'(x)} {1-f'(x)} \) gives a fairly decent aproximation of \( f^n(x) \) near a fixed point
Some pictures.
Red is \( f(x) \) blue is \( \frac {f'(x)^{1/2}(x-f(x))+f(x)-xf'(x)} {1-f'(x)} \) and green is \( f^{1/2}(x) \)
sin[x]
e^x-1, x<0
thanks.
\( f^n(x) = \lim_{k\to \infty} f^{-k}(\frac {f'(f^k(x))^n(f^k(x)-f(f^k(x)))+f(f^k(x))-f^k(x)f'(f^k(x))} {1-f'(f^k(x))}) \)
which is the same as
\( f^n(x) = \lim_{k\to \infty}f^{-k}(\frac {f'(u)^n(u-f(u))+f(u)-uf'(u)} {1-f'(u)})\\where\\u=f^k(x) \)
This works whenever a function has a regular attracting or repelling fixed point that it increases through. For it to work near a repelling fixed point you simply let k approach negative infinity.
Also note that \( \frac {f'(x)^n(x-f(x))+f(x)-xf'(x)} {1-f'(x)} \) gives a fairly decent aproximation of \( f^n(x) \) near a fixed point
Some pictures.
Red is \( f(x) \) blue is \( \frac {f'(x)^{1/2}(x-f(x))+f(x)-xf'(x)} {1-f'(x)} \) and green is \( f^{1/2}(x) \)
sin[x]
e^x-1, x<0
thanks.
