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I came across the following problem that says:

Consider the functions $f,g\colon \mathbb C \rightarrow \mathbb C$ defined by $f(z)=e^z, g(z)=e^{iz}$. Let $S=\{z \in \mathbb C:Re(z)\in [-\pi,\pi]\}.$ Then which of the following options is correct?

$1$. $f$ is an onto entire function

$2$. $g$ is a bounded function on $\mathbb C$

$3$. $f$ is bounded on $S$

$4$. $g$ is bounded on $S$

Can someone point me in the right direction? Thanks in advance for your time.

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What does $[-z, z]$ mean? – Michael Albanese Feb 3 '13 at 3:12
I'm sure that you meant something else than $[-z,z]$, since $S$ makes no sense as it is. – Cameron Buie Feb 3 '13 at 3:36
sorry for the mistake. I have edited it. – user52976 Feb 3 '13 at 3:41
Thanks for remarking that point +1 – Babak S. Feb 4 '13 at 17:25
up vote 2 down vote accepted

For the first statement, can you find $z \in \mathbb{C}$ such that $f(z) = 0$? For the rest, use the fact that $$|e^z| = |e^{a+bi}| = |e^ae^{bi}| = |e^a||e^{bi}| = |e^a| = e^a = e^{\Re(z)}.$$

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Thanks a lot sir.I have got it. – user52976 Feb 3 '13 at 4:18
$f$ is bounded only – La Belle Noiseuse May 3 '13 at 6:43
Why $e^{iz}$ is not bounded in $\mathbb{C}$ @LaBelleNoiseuse – Kushal Bhuyan Oct 24 '15 at 11:12

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