Conformal Mapping: Sketch Regions & Find Mapping

In summary, conformal mapping is a mathematical technique used in complex analysis to map one complex plane onto another while preserving angles and infinitesimal shapes. It is important because it simplifies the analysis of complex functions and has applications in various fields such as fluid dynamics, electromagnetics, and cartography. To sketch regions using conformal mapping, one must identify the boundary of the region and use known functions to transform it. The mapping function for a given region can be found by setting up a system of equations or applying transformations to known functions. Some real-life applications of conformal mapping include creating accurate maps, analyzing fluid flow, and predicting the behavior of electrical circuits.
  • #1
diorific
19
0
Hi, I need to sketche ach of the following regions: R = {z :|z| < √2, 7π/16 < Argz<9π/16}, R1 = {z :|z| < 16, Rez>0} and write down a one-one conformal mapping f1 from R onto R1.

Here is my sketch https://onedrive.live.com/redir?resid=4cdf33ffa97631ef%2110238

But I'm finding hard to find the mapping. So if someone can help me out that'd be appreciated.
 
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  • #2
As a start, I'd consider the mappings ##z \to z^\alpha## for various powers ##\alpha## and see what effect they have on ##R##.
 
  • #3
Ok thank you I think it's z→z^8
 

1. What is conformal mapping?

Conformal mapping is a mathematical technique used in complex analysis to map one complex plane onto another while preserving the angles between intersecting curves. It is a type of transformation that maintains local angles and preserves infinitesimal shapes.

2. Why is conformal mapping important?

Conformal mapping is important because it allows us to analyze complex functions and their behaviors in a more simplified and intuitive way. It is also widely used in many fields of science and engineering, such as fluid dynamics, electromagnetics, and cartography.

3. How do you sketch regions using conformal mapping?

To sketch regions using conformal mapping, you first need to identify the boundary of the region on the complex plane. Then, you can use known conformal mapping functions, such as the exponential, logarithmic, or trigonometric functions, to transform the original region onto a simpler one. Finally, you can use your knowledge of the transformed region to sketch its shape and features.

4. How do you find the mapping function for a given region?

To find the mapping function for a given region, you can use the properties of conformal mapping, such as preserving angles and infinitesimal shapes. This allows you to set up a system of equations and solve for the unknown mapping function. You can also use known conformal mapping functions and apply transformations to them to find the desired mapping function.

5. What are some real-life applications of conformal mapping?

Conformal mapping has various real-life applications, such as in cartography to create accurate maps, in fluid dynamics to analyze the flow of fluids, in electromagnetics to study the behavior of electric and magnetic fields, and in computer graphics to create realistic 3D images. It is also used in designing aircraft wings, analyzing weather patterns, and predicting the behavior of electrical circuits.

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