Conformal mapping in Complex Analysis

• rhia
In summary, conformal mapping is an analytic map from a domain in the complex plane to the complex plane that preserves angles and is also orientation preserving. This concept is closely related to complex analysis as it involves investigating transforms for which the mapping function is analytic. The purpose of using complex analysis is to prove that all conformal isomorphisms of the complex plane extend to complex automorphisms of the extended complex plane. Examples of conformal mappings often use complex analysis in proofs, but not necessarily in the actual examples.

rhia

I would appreciate if someone could explain Conformal Mapping using Complex Analysis using an example. I get the rough idea but have no clue how complex analysis comes into the picture.

Thank You!

Can you give an example to show how it works?
I am finding it difficult to visualize.

No, I cannot since I cannot understand what you need to do in order to understand it. A map is conformal at a point if its derivative doesn't vanish. It is a definition. Look up how angles transform (in terms of derivatives) to see why. I've no idea what you mean ny "how it works", sorry.

The map z goes to az+b (a=/=0) is a conformal mapping (ie map conformal at all points of its domain) of C to C. it's a rotation, scaling and translation, obviously it preserves angles.

using complex analysis one can prove that all conformal isomorphisms of the "complex plane", extend to complex automorphisms of the extended complex plane (the compelx projective "line"), hence have form (az+b)/(cz+d).

For example, the map sending z to w = (z-i)/(z+i) is an isomorphism of the extended complex plane, which sends the points z which are closer to i than to -i, to those points w of norm less than one.

I.e. this is an isomorphism from the upper half plane, onto the open unit disc.

Examples of conformal mappings seldom use complex analysis, but proofs that they have a certain form do so.

(I am using the word conformal here in the sense of not just angle preserving, which is the correct meaning, but also orientation preserving, hence complex holomorphic.)

From the complex point of view, the porpuse is to investigate in more general terms the character of transforms for which the mapping function $$w = u(x,y) + i\nu(x,y)$$ is analytic.

Since w = f(z) is analytic, substituting into the jacobian determinant, we get

$$J \left ( \frac{u,\nu}{x,y} \right ) = \begin{vmatrix} \frac{\partial u}{\partial x} & - \frac{\partial \nu}{\partial x} \\ \frac{\partial \nu}{\partial x} & \frac{\partial u}{\partial x} \end{vmatrix} = |f'(z)|^2$$

From here there are 4 theorems which can be deduced...

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1. What is conformal mapping?

Conformal mapping is a technique used in complex analysis to map one complex plane onto another in a way that preserves angles and shapes. This means that the mapping does not distort the shape of the original figure, and the angles between any two intersecting curves are preserved after the mapping.

2. What are some applications of conformal mapping in complex analysis?

Conformal mapping has various applications in engineering, physics, and mathematics. It is used in the study of fluid dynamics, electromagnetism, and quantum mechanics. It also has applications in image processing, computer graphics, and cartography.

3. How is conformal mapping different from other types of transformations?

Conformal mapping is different from other types of transformations, such as affine or projective transformations, because it preserves angles and shapes. Other transformations may change the size and shape of the original figure, but conformal mapping maintains these properties.

4. What are some commonly used conformal mappings?

Some commonly used conformal mappings include the exponential mapping, the logarithmic mapping, the Mobius transformation, and the bilinear transformation. These mappings are often used in solving complex analysis problems and have different properties and applications.

5. What are some limitations of conformal mapping?

While conformal mapping is a powerful tool in complex analysis, it has some limitations. It can only be applied to regions that are simply connected, meaning there are no holes or gaps in the region. Additionally, conformal mapping cannot preserve all properties of a figure, such as area or perimeter.