04/21/2008, 01:33 AM
jaydfox Wrote:...
Such a solution \( f \) (even if analytic and strictly increasing) is generally not unique because for example the solution \( g(x):=f(x+\frac{1}{2\pi}\sin(2\pi x)) \) is also an analytic strictly increasing solution, by
\( g(x+1)=f(x+1+\frac{1}{2\pi}\sin(2\pi + 2\pi x))=F(f(x+\frac{1}{2\pi}\sin(2\pi x))=F(g(x)) \) and
\( g'(x)=f'(x+\frac{1}{2\pi}\sin(2\pi x))(1+\frac{1}{2\pi}\cos(2\pi x)2\pi)=\underbrace{f'(x+\frac{1}{2\pi}\sin(2\pi x))}_{>0}\underbrace{(1+\cos(2\pi x))}_{\ge 0}>0 \)
If we're only concerned about a real function, then this issue of a cyclic shift of the input to the sexp function is an important one.
However, what about for a complex function? The sin and cos functions grow exponentially (in absolute value) as we move away from the real line (growth is dictated by sinh and cosh, in fact). More importantly, the magnitude of the difference between the "crest" and "trough" is increasing exponentially.
...
In the case of complex domain, the uniqueness of tetration \( F \) seems to be provided by the axiom about the asymptotic behavior of \( F(z) \) at \( \Im(z) \rightarrow +\infty \):
\( F( z )= L + {\mathcal O}\Big( \exp( L z ) \Big) \)
where \( L \) is eigenvalue of logarithm. See details (and pics) at
http://www.ils.uec.ac.jp/~dima/PAPERS/2008analuxp.pdf
However, there is continuum of other tetrations that grow in the direction of imaginary axix.
P.S. Henryk Trappmann had invited me here. I hope, his message
was not a trap.