I need to find the Image of the region $\{z\in\mathbb{C}:\Re(z)>\Im(z)>0\}$ under the map $z\to e^{z^2}$
$z=x+iy\Rightarrow z^2=x^2-y^2+2ixy \Rightarrow e^{z^2}=e^{x^2-y^2}[\cos(2xy)+i\sin(2xy)]$
What I know is $x>y\Rightarrow x^2-y^2>0\Rightarrow e^{x^2-y^2}>0$ and infact as $x^2=y^2$ so $e^{x^2-y^2}>1$ and the modulas of any complex number in the image plane is always $=e^{x^2-y^2}>1$ so can I conclude the image set as $\{w\in\mathbb{C}:|w|>1\}$?
Please tell If I am wrong in my logic anywhere. Thank you for discussion.