Analytic Function vs Exponential Order function

Question

We say that a function $f$ is of exponential order $\alpha$ if there exist constants: $M$, $\alpha$, $T$ such that for $x>T$ $$f(x)<M\cdot e^{\alpha x}$$ Polynomials are of exponential order. Then, is it true that analytic functions in a neighborhood of the origin are of exponential order?

I want to know this to see if it is legal to calculate the Laplace transform of analytic functions.

Answer 1

In general no, take for example $e^{x^2}$.