Is integrating $e^{iz^{2}}$, along the real axis in the complex plane the same as integrating the riemann integral of $e^{x^2}$?

Question

In the title, $z\in \mathbb{C}$ and $x\in\mathbb{R}$. More specific to my problem, I am hoping that $\int_{0}^{R}e^{iz^{2}}dz=\int_{0}^{R}e^{x^{2}}dx$. Maybe this is obvious but I want to make sure.

Answer 1

No they are not the same. The $i$ before $z^2$ (that is not present before $x^2$) changes things.

For one, the modulus of $e^{i\theta}$ is always one whenever $\theta \in \mathbb{R}$, and since your integral is along the real line, $z$ (and thus $z^2$) assumes only real values.