# [Complex Analysis] prove non-existence of conformal map

1. Jun 22, 2011

### nonequilibrium

1. The problem statement, all variables and given/known data
"Show that there is no conformal map from D(0,1) to \mathbb C"
and D(0,1) means the (open) unit disk

2. Relevant equations
Conformal maps preserve angles

3. The attempt at a solution
I don't have a clue. I thought the clou might be that D(0,1) has a boundary, and C doesn't, so I tried to draw some circles/lines on D(0,1) that couldn't be imaged onto C whilst preserving angles, but to no avail.
1. The problem statement, all variables and given/known data

2. Jun 22, 2011

### Dick

I'm guessing you mean a bijective function. Then the inverse map would map C to D(0,1). Think about what you might conclude from that.

Last edited: Jun 22, 2011
3. Jun 22, 2011

### nonequilibrium

Oh, does a conformal map have to be bijective?

(in that case: the inverse would be bounded ==(Liouville)==> constant, contradiction)
(Now I'm also assuming differentiability! So does a conformal map have to be bijective AND/OR analytical?)

4. Jun 22, 2011

### Dick

You ask good questions. Those are the questions I was asking myself after I posted. Looking things up, indeed a map in C is conformal iff it's analytic and it has a nonzero derivative. I'm still working on justifying my reckless post. Anything in your course to help?

5. Jun 22, 2011

### nonequilibrium

the confusing this: in my course itself (the free course by ash & novinger), it says a conformal map only has to be locally invertible (which is actually a consequence of the non-vanishing derivative). But in pen I had written down that a conformal map is a bijection (and the way it was written makes me think it was dictated by the professor)