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

• nonequilibrium
In summary, there is no conformal map from D(0,1) to \mathbb C because conformal maps preserve angles and there is no way to preserve angles when mapping the open unit disk, which has a boundary, onto the complex plane, which does not have a boundary. Additionally, a conformal map must be bijective and have a non-zero derivative, which is not possible in this case.
nonequilibrium

## Homework Statement

"Show that there is no conformal map from D(0,1) to \mathbb C"
and D(0,1) means the (open) unit disk

## Homework Equations

Conformal maps preserve angles

## 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.

## Homework Statement

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:
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?)

mr. vodka said:
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?)

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?

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)

## 1. What is a conformal map?

A conformal map is a function that preserves angles between intersecting curves at each point. In other words, it preserves the shape of curves, but not necessarily their size or orientation.

## 2. Why is proving non-existence of a conformal map important?

Proving the non-existence of a conformal map is important because it helps us understand the limitations of complex analysis. By identifying cases where a conformal map does not exist, we can gain a deeper understanding of the properties and behaviors of complex functions.

## 3. How is the non-existence of a conformal map proven?

The non-existence of a conformal map can be proven using various techniques, including the Cauchy-Riemann equations, the Schwarz lemma, and the maximum modulus principle. These methods involve analyzing the behavior of complex functions and their derivatives to determine if they satisfy the conditions for conformality.

## 4. Can a function be conformal in some regions but not others?

Yes, a function can be conformal in some regions but not others. In order for a function to be conformal, it must satisfy the Cauchy-Riemann equations at every point in the region of interest. If there are points where these equations are not satisfied, the function will not be conformal in those regions.

## 5. What are some real-world applications of conformal maps?

Conformal maps have many practical applications in fields such as engineering, physics, and computer graphics. They are used to model and analyze complex systems, such as fluid flow, electromagnetic fields, and 3D objects. They are also used in map-making and navigation, as they preserve the shape of land masses and coastlines.

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