Conformal Mapping (unit circle => ellipse)

In summary, the speaker is seeking help with mapping the open unit circle to an open ellipse using the text "Complex Var. and Applications" by Ward and Churchill. They mention a table of mappings in the text and suggest composing different mappings to solve the problem. They also mention the possibility of mapping a closed rectangle or a half-annulus to a half-ellipse, but are unsure how to map the unit circle. The suggested solution is the Joukowsky transformation, which involves using the function f(z) = (1/2)(A+B)z + (1/2)(A-B)(1/z).
  • #1
I'd like to map the open unit circle to the open ellipse x/A^2 + y/B^2 = 1. How would I go about doing this? I really have no idea how to go about doing these mappings.

I'm working with the text Complex Var. and Applications by Ward and Churchill which has a table of mappings in the back. The best I can do is to compose different mappings (not sure if this is how these problems are generally attacked).

From what I can see (and I am probably way off base) I can either map a closed rectangle along and above the x-axis to a half-ellipse, or map a half-annulus (centered at the origin) to a half-ellipse. I'm not sure how I could map the unit circle into either of those two, the box or annulus.

Thanks in advance, any help is appreciated.

PS if this is in the wrong section please move it.
 
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  • #2
The mapping:
f(z) = (1/2)(A+B)z + (1/2)(A-B)(1/z)
will do the trick. This is known as the Joukowsky transformation.
http://en.wikipedia.org/wiki/Joukowski_transformation" [Broken]
 
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1. What is conformal mapping?

Conformal mapping is a mathematical transformation that preserves angles on a surface. In other words, it maintains the same shape and orientation of small elements on the surface.

2. How is the unit circle transformed into an ellipse through conformal mapping?

The unit circle can be transformed into an ellipse through a complex-valued function, known as a conformal mapping function. This function maps each point on the unit circle to a corresponding point on the ellipse, preserving angles and shapes.

3. What are the applications of conformal mapping?

Conformal mapping has various applications in mathematics, physics, and engineering. It is commonly used in the study of fluid dynamics, electrostatics, and heat flow. It is also used in cartography to represent the curved surface of the Earth on a flat map.

4. Can conformal mapping be applied to any shape?

No, conformal mapping is only applicable to surfaces that are conformally equivalent, meaning they can be transformed into each other without changing the angles. This includes shapes such as circles, ellipses, and rectangles, but not irregular shapes.

5. How does conformal mapping relate to complex analysis?

Conformal mapping is a fundamental concept in complex analysis, which is the study of functions of complex numbers. It is closely related to the concept of analytic functions, which are functions that can be represented by a power series. Conformal mapping is used to describe the behavior of analytic functions on curved surfaces.

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