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Let $f(x, y) = (e^x \cos y, e^x \sin y)$. Suppose $O$ is an open set. Is it true that $f(O)$ is open?

Partial results I have so far:

  1. $f$ is locally invertible (for every $r_0 \in R^2$ there is an $\epsilon$ such that $f$ is invertible on $\{r | d(r, r_0) < \epsilon \}$ where $d$ is the euclidean distance function

  2. If $O$ is the product of two open intervals, then $f(O)$ is open

zodiac
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  • Do you know that $f$ is locally invertible or that it's locally a diffeomorphism? If the second, then you can deduce the result from that quite easily. – Mauro Dec 22 '15 at 04:06
  • This is a basic multivariable calculus course, so I haven't encountered diffeomorphisms yet, but I'll look into it – zodiac Dec 22 '15 at 04:07
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    If you have 2 above - aren't you done? Any open set $O$ can be covered by products of open intervals (i.e., you can fit a square in $O$ around each point that is in $O$), so every point in $f(O)$ has an open neighborhood inside of $f(O)$, so $f(O)$ is open. (This was @Mauro's 'second' comment, I think.) – peter a g Dec 22 '15 at 04:09
  • @peterag - yes, I think that works. I am kicking myself for not seeing this. If you write is as an answer I'll accept it – zodiac Dec 22 '15 at 04:15
  • Actually, also take a look at the statement of theorem,and the example (!), of https://en.wikipedia.org/wiki/Inverse_function_theorem. That would be the more standard way of doing it. (Ignore any words you don't yet know on the page.) don't worry about accepting - I have to sign off for now... – peter a g Dec 22 '15 at 04:19

2 Answers2

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Did you consider that $f(x+iy) = e^x(\cos y + i \sin y) = e^{x+iy}$ ??

Look at how the exponential map behaves in the complex plane. Take a normal-shaped rectangle to a wedge.

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cactus314
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  • thanks, I did see this after searching for the question, but this is a question from my real multivariable calculus course (no complex analysis) so I'm sure there's a way to do this without proving that analytic functions map open sets to open sets – zodiac Dec 22 '15 at 05:10
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$f(z)=e^z$ is an analytic function and analytic function is an open map.

K_user
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