[MSE] f(f(z)) = z , f(exp(z)) = exp(f(z))

[tommy1729]

The gaussian method idea variant method :

see : https://math.eretrandre.org/tetrationfor...p?tid=1339

R(s) = exp( tr(s) * R(s) )

and tr(s) strictly rises from 0 at s = -oo to 1 at s = + oo in a fast way.

The gaussian method had t(s) = ( 1 + erf(s) )/2.


R(s) uses tr(s) = ( 1 + erf(2sinh(c s)) )/2

For some real c that makes it analytic.

Now 2 sinh(c s) is periodic with period V.

So this V is isomorphic to a function fV(z) such that

fV(exp(x)) = exp(fV(x))

and fV(fV(x)) = x.


***

The interesting part is that this might relate to caleb ideas.

Maybe assuming the periodicity is continuation beyond a boundary, where boundary implies natural boundary ( not analytic ) , not converging or not satisfying the basis equations.

In that case one might get a " fake " solution.
( not to confuse with fake function theory )


regards

tommy1729