03/12/2016, 01:12 PM

I changed my opinion almost 100 %.

To do the m-test we need to think in terms of a starting point near - but not on - the real line. ( and not An infinitesimal imag part ! ).

Now consider the iterations as relocations on 2sinh^[r].

( r > 0 , noninteger ).

Then by the chaotic nature of the iterated exp we get near fixpoints of 2sinh , exp , cyclic points of exp and 2sinh , ... And also the singularities or places where the equation fails ( double composition not exp ).

I used to consider that to be solved by continuation , but it seems that continuation merely solves ln(exp) = id and the alike ( removing false periodicity and other invariants , resolving the bound of ln * having im max 2 pi - i think that is all it does ! - ) , at least when starting off the real line.

These relocations are app not destroyed by continuation but rather imposed.

But then the logs of the relocations bring us in trouble.

So seeing my method like this Will make the m-test fail i think.

AND SEEING IT CORRECTLY Will make the m-test impossible ; the m-test requires the starting point(s) ( (s) from radius or such ) to be nonreal !

So the only way is to Keep thinking on the real line.

Because the chaotic behaviour of exp has no effect on ln^[oo] 2sinh ( exp^[oo](z) = exp(z) !!

Similarly - as illustration - ln^[oo] ( 3^^oo (z) ) = z + const.

BUT fails the m-test.

Yet z + const continues to analytic , even entire.

So i do not believe in the m-test anymore.

And im not thinking 2sinh^[r](z) is analytic anymore ... But how to prove it ?!?

So it seems my trip to complex analysis failed and im back at real-analysis / calculus.

This is a confusing function !

I orig switched from real to complex analysis because i felt stuck ...

But after trying m-test , cauchy's contour , Mittag-leffler and a few others i feel i might need something completely different from real or complex analysis .... ?!?!

What could that be ??

I also tried matrix methods but they also failed.

I believe it is non-analytic now , and thus the nth derivative must blow up on the real line.

But i do not know how to show it.

My apologies to sheldon.

Regards

Tommy1729

To do the m-test we need to think in terms of a starting point near - but not on - the real line. ( and not An infinitesimal imag part ! ).

Now consider the iterations as relocations on 2sinh^[r].

( r > 0 , noninteger ).

Then by the chaotic nature of the iterated exp we get near fixpoints of 2sinh , exp , cyclic points of exp and 2sinh , ... And also the singularities or places where the equation fails ( double composition not exp ).

I used to consider that to be solved by continuation , but it seems that continuation merely solves ln(exp) = id and the alike ( removing false periodicity and other invariants , resolving the bound of ln * having im max 2 pi - i think that is all it does ! - ) , at least when starting off the real line.

These relocations are app not destroyed by continuation but rather imposed.

But then the logs of the relocations bring us in trouble.

So seeing my method like this Will make the m-test fail i think.

AND SEEING IT CORRECTLY Will make the m-test impossible ; the m-test requires the starting point(s) ( (s) from radius or such ) to be nonreal !

So the only way is to Keep thinking on the real line.

Because the chaotic behaviour of exp has no effect on ln^[oo] 2sinh ( exp^[oo](z) = exp(z) !!

Similarly - as illustration - ln^[oo] ( 3^^oo (z) ) = z + const.

BUT fails the m-test.

Yet z + const continues to analytic , even entire.

So i do not believe in the m-test anymore.

And im not thinking 2sinh^[r](z) is analytic anymore ... But how to prove it ?!?

So it seems my trip to complex analysis failed and im back at real-analysis / calculus.

This is a confusing function !

I orig switched from real to complex analysis because i felt stuck ...

But after trying m-test , cauchy's contour , Mittag-leffler and a few others i feel i might need something completely different from real or complex analysis .... ?!?!

What could that be ??

I also tried matrix methods but they also failed.

I believe it is non-analytic now , and thus the nth derivative must blow up on the real line.

But i do not know how to show it.

My apologies to sheldon.

Regards

Tommy1729