For every $w\ne0$, on every small disc $B=B(0,\epsilon)$, there is some $z$ in $B$ with $e^{1/z}=w$

Question

Let $w$ be a nonzero number in the set of complex numbers. Show that for $f(z)= e^{\frac{1}{z}}$, on every small disc $B= B(0,\epsilon)$, there is a $z$ in $B$ with $f(z)=w$.

I'm a bit stuck on how to approach this problem. Should I perhaps try to find the Laurent series of $f(z)$ and make some assumptions from there?

Answer 1

Choose a branch of the logarithm function, then choose an integer $n$ such that $|\log w+2\pi in|>\epsilon^{-1}$. Let $z=(\log w+2\pi in)^{-1}$. Then $|z|<\epsilon$, and $e^{1/z}=w$, as requested.