to compute a taylor series , we need to know that it is justified.

concretely that means we need to prove that

, where D^m is the mth derivative with respect to x(*) , holds.

( with respect to z should give the same result , but seems harder at first sight )

analytic continuation preserves periodicity , which is 2pi i.

that is also the reason why we apparantly cant extent this method to bases between eta and e^(1/2). ( hints : fourier , entire , think about it )

a further remark and actually request is that i would love to see a plot of all the solutions , including branches of

exp(x) = t

exp(exp(x)) = t

...

exp^[n] = t

where t = 0 +/- 1.895494239i is one of the nonzero fixpoint of 2sinh.

it might give me new insight or ideas.

thanks in advance.

regards

tommy1729