# Great Picard's Theorem and infinite number of poles

Let $$f(z)$$ be a meromorphic function on the complex plane $$\mathbb{C}$$.

(1). If $$f(z)$$ has a finite number of poles and $$\infty$$ is not an essential singularity, then $$f(z)$$ is a rational function.

(2). If $$f(z)$$ has at most a finite number of poles and has an essential singularity at $$z=\infty$$(for example, $$f(z)=\frac{e^z}{1+z^2}$$), Great Picard's Theorem says that $$f(z)$$ takes on all possible complex values on $$\{z\in\mathbb{C}\mid |z|>R>0\}$$, with at most a single exception.

(3). If $$f(z)$$ has infinte number of poles， say $$\{z_n\}$$. Then $$z_n\to\infty$$ and $$\infty$$ is not an essential singularity. For example, $$f(z)=1/\sin z$$, $$\tan z$$.

My Question: In the case (3), Does $$f(z)$$ also takes on all possible complex values on $$\{z\in\mathbb{C}\mid |z|>R>0\}$$, with at most two exceptions($$f(z)$$ takes $$\infty$$ at poles). Here $$\{z\in\mathbb{C}\mid |z|>R>0\}$$ is a neighbourhood of $$\infty$$.

Can anyone recommend some references?