Does there exist an example

  • Thread starter esisk
  • Start date
  • #1
esisk
44
0
Is it possible to map the unit disk onto the complex plane C holomorphically?
This is not a homework question. Thank you for your help
 

Answers and Replies

  • #2
Marin
193
0
Hi there!

This is an interesting question.

The Riemannian mapping thm. actually gives a partial answer to that question - it is possible to find such a mapping, if the domain is not all of C, so the statement is seriously doubted - otherwise Riemann would have stated his thm in a more general way

However, I'll try to sketch a counter proof:

Assume that f: C -> D is holomorphic. It is evident that f is then an entire function. What is more, for all z in C it is true that |f(z)|<=1, i.e. f is bounded, because D is bounded. Now the Liouville's thm implies that f must be constant, implying the statement is incorrect.
(the Liouville's thm is that very nice tool also used for proving the Fundamental Theorem of Algebra in less that 5 lines)


regards,
marin
 
  • #3
esisk
44
0
Hi Marin,
I thank you for the response, but I believe you have misread the question. I am looking for a function from D to C. Not the other way around; and, yes, otherwise Liousville's Theorem would make it impossible.

Again: Can you map the unit disk onto C?
Thanks
 
  • #4
quasar987
Science Advisor
Homework Helper
Gold Member
4,790
19
Holomorphic maps are continuous. The unit disk is compact. Continuous maps apply compact set to compact set. The complex plane is not compact. Hence, you can't map continuously (or holomorphically) D onto C.
 
  • #5
g_edgar
607
0
Holomorphic maps are continuous. The unit disk is compact. Continuous maps apply compact set to compact set. The complex plane is not compact. Hence, you can't map continuously (or holomorphically) D onto C.

Of course he means the open disk. The disk and the plane are homeomorphic. But not conformally equivalent (as noted, by Liouville's theorem). So what about mapping the open disk onto the complex plane in a many-to-one manner?
 
  • #6
quasar987
Science Advisor
Homework Helper
Gold Member
4,790
19
For some reason I didn't even consider that the OP might mean the open disk!
 
  • #7
esisk
44
0
Thank you both of you. I retrospect, I should have made it clearer by saying the open disk.

And, yes, following the suggestion from Edgar
I can use (z-i)^2 to map the upper falf plane onto C and

Cayley map to map the open disk onto the upper half plane. I believe this will do it. Thsnk you again.
 

Suggested for: Does there exist an example

Replies
10
Views
18K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
995
  • Last Post
Replies
7
Views
1K
Replies
4
Views
3K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
7
Views
4K
  • Last Post
Replies
2
Views
909
  • Last Post
Replies
3
Views
1K
Top