= Solution
The exponent in the PDF is $2z^2$. A <complex exponential> equals one precisely when its exponent is an integer multiple of $2\pi i$: its real part must be zero, and its imaginary part must be a full rotation. Therefore $z^2=\pi i k$ for $k\in\mathbb Z$. Taking both square roots yields \b[all solutions]
$$
\boxed{\{0\}\ \cup\ \left\{\pm\sqrt{\frac{\pi k}2}(1+i):k=1,2,\ldots\right\}
\ \cup\ \left\{\pm\sqrt{\frac{\pi k}2}(1-i):k=1,2,\ldots\right\}.}
$$
The first nonzero family corresponds to positive exponent index, and the second to negative index. Both signs are required.
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