Show that all the zeros of $$f(z)=\int_0^z \exp(-w^2/2)\, \mathrm{d}w$$ lie in the domain $\{z:\mathrm{Re}(z^2)<0\}$ except $0$.
Also with some confusing hints: Consider the function $f$ maps the diagonals $y=x$ and $y=-x$ to the Cornu's spiral. Verify this and consider the boundary curve of the domain $z:\mathrm{Re}(z^2)>0$ and $0<r<|z|<R$ for some small $r$.