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I'm learning complex analysis, specifically series applications, and need help with the following problem:

Let $f(z)$ be an entire function which is not a polynomial. Show that for every $c \in \mathbb{C}$ there exists a complex sequence $\{z_n\}$ with $z_n \to \infty$ such that $f(z_n) \to c$.

Since $f(z)$ is holomorphic in the whole complex plane, then $f(z)$ can be expanded in a Taylor series around $z_0 = 0$, i.e.

$$f(z) = \sum_{n = 0}^{\infty}a_nz^n.$$

I suspect the solution to this exercise has to do with the Casorati-Weierstrass theorem but I don't know how to apply it in this context. If $f(z)$ is not a polynomial, can we say something of interest about the Taylor coefficients $a_n$ of $f(z)$ and why?

In a previous exercise a hint was given to consider the series expansion of $g(z) = f(1/z)$. My guess is that, in order to apply the Casorati-Weierstrass theorem, $g(z)$ must have a essential singularity at $z_0 = 0$. I'd like to understand why this is so and possibly some help with the application of the Casorati-Weierstrass theorem.

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  • $\begingroup$ Yes, there can not be any $n$ for which all $m>n : a_m=0$ because if it were then it would be a polynomial of order $n$. Also I think you may need to include negative exponents to "capture" all entire functions. $\endgroup$ – mathreadler Jun 29 '16 at 12:34
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    $\begingroup$ @mathreadler Thank you for your explanation about the coefficients $a_n$ of $f(z)$, makes perfect sense now. I think we can't include the negative exponents since the series expansion will then be a Laurent series which will not be analytic in $\mathbb{C}$. The function $f(z)$ will only be analytic in some punctured disk around $z_0 = 0$ but the assumption says that $f(z)$ is entire. $\endgroup$ – Von Kar Jun 29 '16 at 12:57
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Some hints/steps :

  1. If an entire function has a non-essential singularity at $\infty$, then it is a polynomial. In order to prove this, consider $g(z)=f(\dfrac{1}{z})$, and compare power series expansions.

  2. From (1), conclude that $\infty$ is an essential singularity of $f$.

  3. Casorati-Weierstrass tells you that the image of a deleted neighbourhood of a essential singularity is dense in $\mathbb{C}$. Every set of the form $\{z:|z|>n\}$ is a deleted neighbourhood of $\infty$.

  4. Pick a point $c\in \mathbb{C}$. Pick neighbourhoods $B(c,\dfrac{1}{n})$, and observe that for each $n\in \mathbb{N}$, $f(\{z:|z|>n\}\cap B(c,\dfrac{1}{n}) \neq \emptyset$.

  5. Pick $z_n$ in $\{z:|z|>n\}$.

Alternatively, argue contrapositively. Suppose $\not \exists ~z_n\to \infty$ such that $f(z_n)\to c$. Consider $g(z)=\dfrac{1}{f(\frac{1}{z})-c}$. Show that $g$ is either meromorphic or holomorphic at $0$, and conclude.

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