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Let $\displaystyle f$ be an entire function such that $$\lim_{|z|\rightarrow \infty} |f(z)| = \infty .$$ Then,

  1. $f(\frac {1}{z})$ has an essential singularity at 0.

  2. $f$ cannot be a polynomial.

  3. $f$ has finitely many zeros.

  4. $f(\frac {1}{z})$ has a pole at 0.

Please suggest which of the options seem correct.

I am thinking that $f$ can be a polynomial and so option (2) does not hold.

Further, if $f(z) = \sin z $ then it has infinitely many zeros... which rules out (3) while for $f(z) = z$ indicates that it has a simple pole at $0$ and option (4) seems correct.

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    Your reasoning looks fine to me as long as you can prove every claim made (for example, that $\,|\sin z|\to\infty \,\,if\,\,|z|\to\infty\,$...)2012-06-10

1 Answers 1