Conformal mapping problem(bilinear type)

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SUMMARY

The discussion focuses on a conformal mapping problem involving the transformation of a shaded area in the right half-plane to an annulus between two concentric circles. The initial suggestion of using the transformation \( w = \frac{1}{z+a} \) was proposed, but the correct transformation identified is \( w = \frac{1}{z + \sqrt{a^2 - b^2}} \). This transformation effectively maps the area while accounting for a gap centered at \( x = a \) with radius \( b \). The discussion highlights the importance of precise transformations in complex analysis.

PREREQUISITES
  • Understanding of complex analysis concepts, specifically conformal mappings.
  • Familiarity with bilinear transformations and their properties.
  • Knowledge of the geometric interpretation of transformations in the complex plane.
  • Ability to manipulate complex functions and apply them to specific regions.
NEXT STEPS
  • Study the properties of bilinear transformations in complex analysis.
  • Learn about the geometric interpretations of conformal mappings.
  • Explore examples of transformations that map regions in the complex plane.
  • Investigate the implications of gaps in regions when applying transformations.
USEFUL FOR

Students and professionals in mathematics, particularly those studying complex analysis, as well as educators seeking to understand the nuances of conformal mappings and their applications in transforming geometric shapes.

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Homework Statement


Hi all. We are asked to transform the shaded area in below figure to between two concentric circles, an annulus. Where these circles' center will be is not important, just transform the area to between any two concentric circles. As you see in figure, shaded area is whole right half plane except a gap having center at x=a and radius b.

Figure is here.
http://img189.imageshack.us/img189/6331/conform.gif
You may suggest [tex]\frac{z+\alpha}{z+\beta}[/tex] type transformations which may map to origin centered circles.
My suggestion is w=[tex]\frac{1}{z+a}[/tex], no b, looks simple? I think this formula maps the shaded area to between circles centered at [tex]\frac{1}{2a}[/tex].
My explanation is: Shift right whole plane "a" much. Take [tex]\frac{1}{z}[/tex]. If the gap were not there, taking [tex]\frac{1}{z}[/tex] yields a circle having center at[tex]\frac{1}{2a}[/tex] and being tangential to origin. The gap is there and it causes a little gap inside the former circle having center at [tex]\frac{1}{2a}[/tex]. So, [tex]\frac{1}{z+a}[/tex] ends up with two concentric circles with transformed are between them. Am I wrong? My instructor denies me without making a reasonable explanation.

Supporting or refuting explanations will be appreciated. Thanks in advance.
 
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I have noticed my mistake. Idea is correct but there is a little mistake. Correct transform is: 1/(z+sqrt(a^2-b^2)). Anybody, any comment?
 

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