03/22/2008, 11:59 AM
Ivars Wrote:@Henryk
The next questions to provoke further thinking is :
b[4](i*pi/2)=b[4]ln(i)?
b[4](i/2)=?
b[4](-i)=?
As I am not so interested in particular values perhaps you need to stretch your own brain. The formula I used was:
\( f^{\circ t}(x)=\lim_{n\to\infty} f^{\circ -n}(a(1-q^t) + q^t f^{\circ n}(x)) \), \( q=f'(a) \).
where \( a \) is the attracting fixed point and \( f \) is the function to be iterated.
Specialized to our case using the lower fixed point \( a \) we get the base by \( b=a^{1/a} \) and \( f'(a)=(b^x)'(a)=\ln(b)b^a=\ln(b)a=\ln(b^a)=\ln(a) \), so:
\( b[4]z=\exp_b^{\circ z}(1)=\lim_{n\to\infty} \log_b^{\circ n}(a(1-\ln(a)^z) + \ln(a)^z \exp_b^{\circ n}(1)) \)
