I had an idea about cyclic functions, as they relate to a function f such that f(x+a)-f(x) = a for some a. For the purposes of this discussion, I'll assume that a=1.
I'm going to quote a post from a couple months ago:
http://math.eretrandre.org/tetrationforu...php?tid=19
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.
And as far as I can remember, any cyclic real function can be decomposed into a Fourier series of sin and cos functions. Therefore, no matter how small the shift in each cycle, if we get far enough off the real line, then the shift will be quite large over a full cycle.
Okay, this much I think is correct, but please let me know if I'm wrong.
Pushing forward (assuming I was correct), if we had a general idea of how the function should act for inputs with large imaginary part, then it would be easy to weed out solutions that had even very small cyclic shifts.
For the slog base e, I have just such a general idea. In the vicinity of the primary fixed point, it should behave like a logarithm with a complex base. The upper fixed point is approached as the imaginary part of the slog goes to positive infinity. The lower fixed point is approach as the imaginary part of the slog goes to negative infinity.
Luckily, this general idea is so simple that I think it is sufficient to give us the uniqueness criterion we've been looking for! For example, I think that my solution (using my change of base from base eta) will give us a very erratic slog in the vicinity of the fixed points.
I have yet to test this hypothesis with my solution, but I hope to show this in the next couple weeks. It'll be tricky, because I'll need to calculate a lot of points with very high precision to be able to get a useful power series expansion. I need a power series expansion because my change of base formula only works for real tetrational "exponents".
Moreover, I think this uniqueness criterion will set apart Andrew's slog as "the" correct solution, at least for base e.
I'm going to quote a post from a couple months ago:
http://math.eretrandre.org/tetrationforu...php?tid=19
bo198214 Wrote:Let \( F(x)=b^x \) for some base \( b \).
Then we demand that any tetration \( f(x)={}^x b \) is a solution of the Abel equation
\( f(x+1)=F(f(x)) \) and \( f(1)=b \).
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.
And as far as I can remember, any cyclic real function can be decomposed into a Fourier series of sin and cos functions. Therefore, no matter how small the shift in each cycle, if we get far enough off the real line, then the shift will be quite large over a full cycle.
Okay, this much I think is correct, but please let me know if I'm wrong.
Pushing forward (assuming I was correct), if we had a general idea of how the function should act for inputs with large imaginary part, then it would be easy to weed out solutions that had even very small cyclic shifts.
For the slog base e, I have just such a general idea. In the vicinity of the primary fixed point, it should behave like a logarithm with a complex base. The upper fixed point is approached as the imaginary part of the slog goes to positive infinity. The lower fixed point is approach as the imaginary part of the slog goes to negative infinity.
Luckily, this general idea is so simple that I think it is sufficient to give us the uniqueness criterion we've been looking for! For example, I think that my solution (using my change of base from base eta) will give us a very erratic slog in the vicinity of the fixed points.
I have yet to test this hypothesis with my solution, but I hope to show this in the next couple weeks. It'll be tricky, because I'll need to calculate a lot of points with very high precision to be able to get a useful power series expansion. I need a power series expansion because my change of base formula only works for real tetrational "exponents".
Moreover, I think this uniqueness criterion will set apart Andrew's slog as "the" correct solution, at least for base e.
~ Jay Daniel Fox