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Did we study this before ?
Im not sure if we studied this before or not.

dexp(x) = exp(x) - 1.

exp^[1/2](x) - dexp^[1/2](x) = a_1 x^-1 + a_2 x^-2 + ...

That last equation is conjectured.

A much weaker conjecture is that it holds asymptotically for large x.
( and yes I could involve fake function theory now , but I prefer not , its not the main issue here )

In fact , it is not difficult to show that IF exp^[1/2](x) - dexp^[1/2](x) is a laurent series then the x^0 , x^1 , ... terms are all 0.

This all looks pretty familiar and Im not sure if we considered this before or not.

Since dexp has a parabolic fixpoint , I assume the equation only holds asymptotically for large x.

The reason for that is that if exp^[0.5] is laurent and dexp^[0.5] is not then their difference cannot be laurent.

One could also study exp^[0.5] - 2 sinh^[0.5] but that difference grows to 0 very fast.

exp^[1.5] fast ??

Anyway those differences between half-iterates of similar functions fascinates me.

But in this specific dexp case its quite likely that my fascination comes from stuff I have temporarily forgotten and/or from not understanding dexp^[0.5] well.

So back to dexp^[0.5].

the fixpoint is parabolic.
Now recently I mentioned using fake function theory to get a Taylor anyway.
But again the focus here is not on fake function theory.

What I was intrested in was something like this :

dexp^[0.5] = f ( g(x) ).

where f is a Taylor series. Preferably entire.

And g is an analytic function that is not entire.

Or similar.

For instance

dexp^[0.5](x) = f ( sqrt(x) )
dexp^[0.5](x) = f ( ln(x^3 + 1) )

I think that the number of pettals gives A such that

parabolicfixpointfunction^[0.5](x) = f ( x^(1/A) ) or something.

Maybe I need to reconsider parabolic fixpoints again.

Notice the once again returning idea of fake function theory :

parabolic fixpointfunction^[0.5](x) = f ( fake x^(1/A) ).

thereby giving a systematic way of computing fake half-iterates at parabolic fixpoints.

Im probably typing more than thinking and should start thinking and perhaps reading now.

But even if I bump my head in 5 min from now , saying it was trivial , I still want to share this.


5 min have passed.
Still no aha moment so I must be either good or bad today.



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