Could be tetration if this integral converges

[JmsNxn]

I totally agree. Although I added a minus sign to it, which I assume was a typo.

So lets think about \( \vartheta(-x) \).

Since the Taylor coefficients of \( \vartheta(-x) \) decay EXTREMELY FAST , I consider this as a function that is well approximated by a polynomial for a long time.
( many remainder theorems for Taylor series imply this )

This means the main behaviour of this \( \vartheta(-x) \) is like \( (-1)^n a_n x^n \) where n increases slowly with x.

This implies that \( \vartheta(-x) \) is not bounded by a polynomial and also that \( \vartheta(-x) \) = 0 infinitely often.

Therefore the integral diverges.

Even if we consider taking the limit of x going to +oo as the limit of the sequence x_i with \( \vartheta(-x_i) \) = 0.

regards

tommy1729

yes yes, I'm quite aware it diverges. That's only another trick we need to come up with to handle that. I have a few but I need to look deeper into the laplace transform.