09/08/2007, 07:03 PM
jaydfox Wrote:I wonder, however, if there are "mixed" fixed points. You know, alternate between two branches for every other logarithm, or three branches for every third, etc. What type of values do we settle on?
I dont think so. We have in each rectangle of values \( x+iy \) with \( x>0 \), \( 2\pi k < y < 2\pi k + \frac{1}{2}\pi \), \( k\ge 0 \) exactly one fixed point and these together with its conjugates are all fixed points of \( e^x \).
I think the fixed point with imaginary part in \( 2\pi k ... 2\pi k + \frac{1}{2}\pi \) is the limit of the iterated \( \ln(x)+2\pi i k \) for a starting value with positive imaginary part.
