Image of annulus around essential singularity

Question

I found an exercise in functions of one complex variable of Conway where asks you to find $f(\{z:0<|z|<\delta\})$ for arbitrary small values of $\delta>0$ for the functions $$f(z)=(1-e^z)^{-1},\,g(z)=z^n\sin(1/z).$$ But I don't know how to approach it. I know by Casorati-Weierstrass that it must be dense in $\mathbb{C}$. Any suggestions?

Answer 1

I should warn that this is not a complete solution.

There's actually a stronger theorem called (Great) Picard Theorem where if you have an essential singularity, on any punctured neighborhood, it misses at most two points of $\mathbb{C}\cup\{\infty\}$. This is a big theorem in complex analysis.

Actually, for the first function $f$, it's not even an essential singularity. It's actually a simple pole with residue $1$. (Check $\lim_{z\to 0} \frac{1}{1-e^z}-\frac{1}{z}=0$). Perhaps there was a typo in the function.

For the second function $g$, this is indeed essential at $z=0$.You can write the Laurent series for $\sin(1/z)$ to see this. Actually you can see that it misses $\infty$ if you notice that $\sin(w)$ is holomorphic on $\mathbb{C}$. Firstly you can see that $\infty$ is missed because the function has no pole on the punctured disk.

Does it miss one more value of the extended complex plane $\mathbb{C}\cup\{\infty\}$?　You can perhaps try to use the argument principle of $h(z)=\sin(w)/w^n-\lambda$ on a large contour but I think this is going to be rather messy. For $n=0$, it is pretty clear that $\sin(w)$ attains all complex values as you can solve an appropriate quadratic on $e^w$ to solve $e^w-e^{-w}=2i\lambda$.