using sinh(x) ?

[tommy1729]

to go towards the taylor series you simply take the derivates you need of \( \ln^{[n]} (\2sinh^{[z]}(exp^{[n]}(x))) \)
and create the taylor series with real coefficients.

then just plug in complex Z instead of x and your done.

however the question is , what is the radius of that taylor series expanded at x = 0 or even elsewhere.

if that radius is nonzero , we could probably extend further by using analytic continuation , or mittag-leffler expansion.

and that is the way we go to the complex plane.

perhaps an intresting note is that if the ROC is large , we could check if that taylor series has the same period as exp(x) ( 2pi i ).

if it is indeed large enough and

1) the period is indeed 2pi i then the limit formula might hold for the complex plane and be equal to the taylor of the limit formula for the reals.

2) the period is not 2pi i then the limit formula will NOT hold for the complex plane and NOT be equal to the taylor of the limit formula for the reals.

also , if the radius is 0 everywhere , despite unlikely , then if the limit converges for all complex , then the function must have some local or global fractal or semi-fractal properties.

regards

tommy1729

perhaps constructing a fourier series is 'better' , assuming we have the period property of course.

'better ' in the sense of potentially easier to compute numerically , easier to compute 'symbolicly' ( four coeff ) and easier to prove related statements conjectures and properties.

this makes me doubt if taylor = four , if the period really exists and if the result only converges for x > 0 ... for reasons not yet explained ...

just saying that fourier expansion might be intresting imho.

if valid...

slightly off topic but i often think there should be a new type of series expansion designed for tetration ...

regards

tommy1729