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Which of the following statements are true?

  1. There exists an entire function $f : \mathbb C\rightarrow \mathbb C$ which takes only real values and is such that $f(0) = 0$ and $f(1) = 1$.

  2. There exists an entire function $f : \mathbb C\rightarrow \mathbb C$ such that $f(n + {1\over n}) = 0$ for all positive integers $n$.

  3. There exists an entire function $f : \mathbb C\rightarrow \mathbb C$ which is onto and which is such that $f({1\over n}) = 0$ for all positive integers $n$.

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    What have you tried? What relevant facts do you know? (For example, are you familiar with Picard's theorems?)2012-08-02

2 Answers 2

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  1. No such function exists. By Little Picard's Theorem (see wikipedia), if $f$ is entire and non-constant, then $f$ can miss at most one value.

  2. Here I am assuming you want an entire function with zeros at $n + \frac{1}{n}$. Since $n + \frac{1}{n}$ does not have a limit point (goes to $\infty$), you can use the Hadamard product formula to produce an entire function with exactly those values as it zeros (and you can even control its order of growth).

  3. No such function exists. Since $(\frac{1}{n})_{n \in \omega}$ has a limit point in $\mathbb{C}$, by analytic continuation, this function must be the constant zero function.

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    Often called the "Identity Theorem". See e.g. [here](https://en.wikipedia.org/wiki/Identity_theorem).2017-10-15
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$1$ is false: Non-constant entire function takes every value in $\mathbb C$ with one possible exception.

$2$ is true: $f=0$.

$3$ is false: Here $f$ is a non-constant entire function & so $f$ takes every value in $\mathbb C$ with one possible exception. {$1\over n$} is a sequence of distinct points converging to $0\implies f=0$ $!$

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    Beginners always try to go for trivial examples :)2012-12-16