06/03/2010, 08:00 AM

Sheldon, you are really a blessing! Nice to have you on the forum.

I couldnt have it better explained.

Exactly! Seems Paul Lévy being an old Fox (in the double meaning haha).

Ya every regular superfunction at a hyperbolic fixed point has the period , . This is because it can be expressed as , is the inverse Schröder function. You can find details in

[1] Kouznetsov, D., & Trappmann, H. (2010). Portrait of the four regular super-exponentials to base sqrt(2). Math. Comp., 79, 1727–1756.

provides the superexponential growth on the real axis, so falls superexponentially. Again details in [1], at every fixed point we have these *two* regular superfunctions.

Ya and we can construct lots of others by choosing instead of any function that has similar growth like for !

I couldnt have it better explained.

(06/02/2010, 04:47 PM)sheldonison Wrote: So the "change of base" idea has been around for a long time ....

Exactly! Seems Paul Lévy being an old Fox (in the double meaning haha).

Quote:But it would be interesting to graph the superfunction of 2sinh in the complex plane, which is well defined. I believe it would be periodic in the imaginary direction. Is the period i*2pi/ln(2)?

Ya every regular superfunction at a hyperbolic fixed point has the period , . This is because it can be expressed as , is the inverse Schröder function. You can find details in

[1] Kouznetsov, D., & Trappmann, H. (2010). Portrait of the four regular super-exponentials to base sqrt(2). Math. Comp., 79, 1727–1756.

Quote: Does it grow superexponentially negative at imag(z)=i*pi/ln(2)?Yes, by the above representation of the superfunction:

provides the superexponential growth on the real axis, so falls superexponentially. Again details in [1], at every fixed point we have these *two* regular superfunctions.

Quote: Could one extend this 2sinh superfunction definition to other real bases? Which ones? My quick initial guess is that it would be limited to bases greater than e^(0.5)?You mean base in which then would approach for ?

Quote:As might be expected, Tommy's sexp gives different values. The function also has different values than the base change solution, discussed earlier in this forum, which Bo points out is equivalent to Lévy's solution. So, it seems we have yet another super exponential, different from all the others.

Ya and we can construct lots of others by choosing instead of any function that has similar growth like for !