# Conformal mapping between two half space

In summary, there is a conformal mapping between the bumped half-space and the flat upper half-space by Riemann's theorem. To find an explicit one, the region can be divided into smaller pieces and worked on individually. However, since the region is unbounded, this may not be possible. The boundary shape is known but could be a piecewise function with respect to x or y.
Hi all,

Suppose there is a bump at the origin, is there a conformal mapping between the bumped half-space (y>|b-x|, |x|<b && y>0, |x|>b) and the flat upper half space (y>0)? Anyone has a hint? Thanks in advance.

Regards,
Tony

of course the answer is yes by riemann's theorem. to find an explicit one i suppose you could chop your region up into pieces and work on each piece. you have two quarter planes and a strip with a triangle removed, it looks like.

mathwonk said:
of course the answer is yes by riemann's theorem. to find an explicit one i suppose you could chop your region up into pieces and work on each piece. you have two quarter planes and a strip with a triangle removed, it looks like.

Thanks. I don't think I can chop the region up into pieces since that it is unbounded. I do know there is a conformal mapping cause we can imagine the coordinate grid lines in the bumped half space. However, I cannot find it although I know the boundary (line) shape (could be a piecewise function with respect to x or y).

## 1. What is a conformal mapping between two half spaces?

A conformal mapping is a mathematical technique used to transform or map a region in one space onto another space while preserving angles. In the case of two half spaces, it is a mapping between two regions that are mirror images of each other and are divided by a boundary, such as a line or plane.

## 2. How is a conformal mapping between two half spaces used in science?

Conformal mapping between two half spaces has various applications in physics and engineering. It can be used to solve problems involving potential flow, electrostatics, and heat transfer in half space geometries. It is also useful in the study of boundary value problems in mathematical physics.

## 3. What is the difference between a conformal mapping and a conformal transformation?

A conformal mapping is a specific type of transformation that preserves angles, while a conformal transformation is a generalization of this concept and can include transformations that do not necessarily preserve angles. In the context of two half spaces, a conformal mapping is a specific type of conformal transformation.

## 4. Can conformal mapping between two half spaces be extended to mapping between more complex shapes?

Yes, conformal mapping techniques can be extended to more complex shapes, such as mapping between different types of curved surfaces or mapping between three-dimensional spaces. However, the mathematics involved become more complex and the resulting conformal mapping may not always be possible to find analytically.

## 5. What are some limitations of conformal mapping between two half spaces?

One limitation of conformal mapping between two half spaces is that it can only be applied to regions that are mirror images of each other and are divided by a simple boundary. It also assumes that the two half spaces have the same physical properties, such as conductivity or temperature, which may not always be the case in real-world applications.

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