Searching for an asymptotic to exp[0.5]

[tommy1729]

Im probably crazy but

Is there an entire asymptotic to exp , G(z) , that has a pseudoperiod of pi i instead of period 2 pi i ?

The inverse would then by an asymptotic to ln.

I was thinking about sqrt( (e^2)^x ) but it either reduces to exp(x) or it has sqrt branches.

Let g(x) be asymptotic to 1/2 for real x < 0 and asymptotic to 1 for x > 1.

Then g(x) ln(x) seems to be the inverse of G(z).

I ask this for better insight to the connections of complex analysis and fake function theory.
This could help to formalize things.

regards

tommy1729