I'm studying for a Complex Analysis midterm, and there's a problem I don't quite understand how to do. This is from Marsden's Basic Complex Analysis (#1.6.11):
Find the minimum of $|e^{z^{2}}|$ for those $z$ with $|z|\le 1$.
The book says the answer is $\frac{1}{e}$ at $z = \pm i$, but I don't really understand how to get there. Is it because the minimum of the function is at $\frac{1}{e}$, and that is for $z^2 = -1$? I think I'm missing something, or am I just overthinking it? Any guidance would be great. Thank you.
The key point here is that $|e^{a+ib}|=e^a$ (for $a,b\in\mathbb{R}$), so to minimize $|e^{z^2}|$, you just want to minimize the real part of $z^2$. If $z=x+iy$, the real part of $z^2$ is $x^2-y^2$. To minimize this, you want $|x|$ as small as possible and $|y|$ as large as possible, and so it is clear that $z=\pm i$ is the best choice given the restriction $|z|\leq 1$.