(This is my first post on stackexchange. Please tell me, if I made any formatting errors and such.)
This question is about how the absolute value function works with the complex exponential.
We have to determine, what $|\exp (z^2)|$ is. Since we know that $z=x+i y$ and $|\exp(z)| = \exp(Re(z))$, after some calculation, it arises that $|\exp(z^2)| = \exp(x^2 - y^2)$.
Does the absolute value of the left side of the equation influence the right side? How I take it, since $|\exp(z)| = \exp(Re(z))$ i.e. it doesn't. But as I said, we're unsure.
P.S: Yes, I know that the right side still needs to be calculated further, but right now, I'd primarily like to know, how the abs. value works in this situation.
Not many calculations, actually. You know that, for every $w\in\mathbb{C}$, $$ \lvert\exp(w)\rvert=\exp(\operatorname{Re}(w)) $$ If $z=x+yi$ and $w=z^2=(x^2-y^2)+2xyi$, you immediately get $$ \lvert\exp(z^2)\rvert=\exp(x^2-y^2) $$ You might write this as $$ \lvert\exp(z^2)\rvert=\exp(\operatorname{Re}(z)^2-\operatorname{Im}(z)^2) $$ in order to express the result only in terms of $z$.