# Prove, that $f: S_{k} \rightarrow \mathbb{C}-\{0\}$ is a surjection.

For $k>0$ define

$S_{k} := \{z=x+iy\in\mathbb{C}\mid |z|<k,\ \ \ \ k\cdot y>|x|\}\subset\mathbb{C}$

Let $f(z)=\exp(1/z)\ \ \ \text{for}\ \ \ z\neq 0$ Prove, that $f: S_{k} \rightarrow \mathbb{C}-\{0\}$ is a surjection.

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The equation $w=\exp(1/z)$ has the solutions $$z=\frac{1}{\log w} = \frac{1}{\ln|w| +i\arg w}=\frac{\ln|w|-i\arg w}{(\ln|w|)^2+(\arg w)^2},$$ where $\arg w$ is any argument of $w$. By choosing $\arg w$ large enough negative, $|z| \approx 0$ and the real part of $z$ will be small compared to the imaginary part. Hence $z$ will be in $S_k$.