# Application of Casorati–Weierstrass theorem

Let $$f$$ be an entire function which is not a polynomial. If $$B$$ is any bounded subset of $$\mathbb{C}$$, then prove that the set $$f(\mathbb{C} \backslash B)$$ is dense in $$\mathbb C$$.

This problem came in isi phd entrance exam once,

(Casorati–Weierstrass) Suppose $$f$$ is holomorphic in the punctured disc $$D_{r}\left(z_{0}\right)-\left\{z_{0}\right\}$$ and has an essential singularity at $$z_{0}$$. Then, the image of $$D_{r}\left(z_{0}\right)-\left\{z_{0}\right\}$$ under $$f$$ is dense in the complex plane.

In my problem given that $$B$$ is any bounded subset of $$\mathbb C$$, then how can I use the casorati weierstrass theorem in my problem. (Also please tell me that, can this problem be solve without using this theorem)

Since $$f$$ is entire, there is a power series centered at $$0$$ such that $$f(z)=\sum_{n=0}^\infty a_nz^n$$ for each $$z\in\Bbb C$$. And, since $$f$$ is not polynomial, $$a_n\ne0$$ for infinitely many $$n$$'s. So, if $$g(z)=f\left(\frac1z\right)$$, $$0$$ is an essential singularity of $$g$$, and therefore $$g\left(\left\{\frac1z\,\middle|\,z\in\Bbb C\setminus B\right\}\right)$$ is dense in $$\Bbb C$$, by the Casorati-Weierstrass theorem. And I doubt that we can avoid using this theorem in order to prove what you want to prove.