Point $(e,\pi)$ on the graph satisfies the famous unifying Euler relation $e^{i\pi}+1=0 $ and we know that these irrational numbers can be individually expressed in power series of $(e^x, \tan^{-1}x)$ or from among many other series given by Euler in Analysis.
I would like to see the common exp/log unification basis of such complex numbers played out in a generalization to the pair $(u,v)$.
$$ u^{iv(u)}+1=0 \rightarrow v(u)=\frac{\pi}{ln\,u}$$
Barring integer values on the $(u,v)$ graph can any number on abscissa or ordinate be expressed by means of a convergent series? If so by means of which series can they be expressed as another case.. and if they cannot be, why not?
Do identities/relations exist in complex analysis or elsewhere satisfied by such irrational point pairs?
For narrowing the discussion.. in particular we take an integer from the geometric series sum:
$$ 2 = 1+ 1/2 + 1/2^2+ 1/2^3 + 1/2^4 +.. $$
and in particular when $v(2)= n\approx 4.53236 $
Can we set up a convergent series for this $n\approx 4.53236?$
Thanks in advance for all related comments.