Image of punctured disk around essential singularity

Question

I am studying Picard's great theorem in complex analysis and I saw two different forms of theorem.For instance in convoy the theorem is stated as:

Great Picard's Theorem: If an analytic function $f$ has an essential singularity at a point $w$, then on any punctured neighborhood of $w, f(z)$ takes on all possible complex values, with at most a single exception, infinitely often.

While my text book foundations of complex analysis by S Ponnusamy states the same theorem as follows:

Suppose that in $f$ is analytic in $\Delta(z_0;r)$\{$z_0$}(punctured disk around $z_0$ with radius r) and $z=z_0$ is an essential singularity of $f$.Then $\mathbb{C}$\ $f$ ($\Delta(z_0;r)$ \{$z_0$}) is singleton.

Clearly according to first statement,the image of punctured disk around essential singularity may be either whole complex plane or a complex plane minus one point but according to second statement of the same theorem the the image of punctured disk around essential singularity is complex plane with one exception.If I suppose that statement of first book is more precise(convoy) then there is a possibility of a analytic function such that image of punctured disk around essential singularity is whole complex plane,if such a example exists then please provide it.

Thanks in advance!

Answer 1

One example is given by $$f(z)=\sin\frac1z$$ We must show that given $r>0$ and $w\in\mathbb{C}$ there exists a $\zeta\in\mathbb{C}$ with $|\zeta|<r$ and $\sin\frac1\zeta=w$ or what is the same thing, that there is a $z\in\mathbb{C}$ with $|z|>\frac1r$ and $\sin z =w$.

Solving for $z$, $$\begin{align} &\sin z=w\\ &e^{iz}-e^{-iz}=2iw\\ &e^{2iz}-2iwe^{iz}-1=0\\ &e^{iz}=\frac{2iw\pm\sqrt{-4w^2+4}}2 \end{align}$$ The right-hand side cannot be $0$, so there is a $z$ that satisfies the equation. Then by periodicity of $e^z$, there is a $z$ with arbitrarily large modulus that satisfies the equation.