(03/09/2010, 01:31 PM)tommy1729 Wrote: [ -> ]i wrote sinh(x) but i meant 2*sinh(x) !!

tommy1729

I'm trying to understand Tommy's suggestions here, so I'm restating what you guys already know. It seems like a different version of sexp, but it might be interesting to graph it.

Tommy suggests calculating a superfunction of

The fixed point of 2sinh(z) is z=0. The behavior near the fixed point of z=0 is f(z)=2*z. Then the superfunction would be real valued at the real axis, which would lead to a straightforward real valued superfunction, based on iterating from the fixed point/function of

, although its not clear if Tommy was suggesting generating the limits from the fixed point. It seems to work in an excel spreadsheet, and the super function would be generated from the limit:

Once this super function starts growing, it grows super-exponentially. I'm assuming Tommy is suggested modifying this into a

by iterating natural logarithms. This is similar to what Bo is referring to when using the Abel function (inverse of the super function) to convert from the Abel function of exp(z)-1 to the Abel function of exp(z), which is also the "base change" equation. Here the Abel function is for 2sinh, so its a different sexp_e equation than all of the other sexp_e definitions we've seen so far.

Later edit: By the way, Bo, thanks for the info on Levy's Abel function equation, that you posted earlier in this thread! I now understand Abel function notation! So the "change of base" idea has been around for a long time ....

bo198214 Wrote:Actually Paul Lévy [1] showed how to obtain an iteration of e^x if we have an iteration of e^x-1.

Say is an Abel function of e^x-1, then

Continuing on, we take the natural logarithm "m" times of the superfunc(z+m+k). The constant k has to be calculated from the inverse superfunction (Abel function) to normalize the equation so that sexp_e(0)=1. Then we have something like:

Anyway, the limits appear to work, and this may be what Tommy or Bo have in mind. Because the e^-z term in 2sinh(z) decays rapidly to zero, this leaves us with an exponential with the same base as the iterated logarithms. So one could hope that this definition would not fall prey to the "base change" version of tetration, which gives a smooth function that is infinitely differentiable but seems to be nowhere analytic. On the other hand, it may fall into the same smooth but nowhere analytic class that the base change definition falls into, due to an infinite number of singularities generated from iterated logarithms in the complex plane. Either way, it could be difficult to make sense of the infinitely dense thicket of superexponential windings and logarithmic singularities.

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)? Does it grow superexponentially negative at imag(z)=i*pi/ln(2)? 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)?

(06/02/2010, 02:58 AM)bo198214 Wrote: [ -> ].... if you provide some values for your solution we can compare with Lévy's solution.

Back to converting this superfunction into sexp_e .... using the limit equation I just posted above, I was able to use Tommy's 2*sinh solution to generate the half iterate. I calculate:

TommySexp_e(-0.5)=0.49874336

The published value from Dimitrii's taylor series coefficients is

DimitriiSexp_e(-0.5)=0.49856329

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.

- Sheldon