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(c) What is the image of the unit circle D = {|z| <1|z€ C} under the Mobius transforma- tion T(z) = -! (i.e., T(z) = )? Please show on your sketch what are the images of the segments {0

(c) What is the image of the unit circle D = {|z| <1|z€ C} under the Mobius transforma- tion T(z) = -! (i.e., T(z) = )? Please show on your sketch what are the images of the segments {0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help me with Question 3c (3c only). 

3. Assume that a, b, c, d e R such that ad - bc = 1. The transformation T: C\{-d/c} → C defined by az + b T(z) := cz + d is called Mobius transformation. (a) Show that the transformation S: C\{a/c} C defined by dz - b S(z) = %3D -cz + a is the inverse of T, i.e., we have S(T(z)) = T(S(z)) = z, for any z ¢ {a/c,-d/c}. %3D (b) Find a Mobius transformation T which satisfies: T (0) = 1, T (1) = 2, T (2) = 5. %3D (c) What is the image of the unit circle D = {|z| < 1|z € C} under the Mobius transforma- tion T(z) = - (i.e., T(z) = )? Please show on your sketch what are the images of the segments {0 <x< 1}, {iy|0 < y < 1} and {ren/4 |r < 1}. 1z+0
Transcribed Image Text:3. Assume that a, b, c, d e R such that ad - bc = 1. The transformation T: C\{-d/c} → C defined by az + b T(z) := cz + d is called Mobius transformation. (a) Show that the transformation S: C\{a/c} C defined by dz - b S(z) = %3D -cz + a is the inverse of T, i.e., we have S(T(z)) = T(S(z)) = z, for any z ¢ {a/c,-d/c}. %3D (b) Find a Mobius transformation T which satisfies: T (0) = 1, T (1) = 2, T (2) = 5. %3D (c) What is the image of the unit circle D = {|z| < 1|z € C} under the Mobius transforma- tion T(z) = - (i.e., T(z) = )? Please show on your sketch what are the images of the segments {0 <x< 1}, {iy|0 < y < 1} and {ren/4 |r < 1}. 1z+0
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