3. Assume that a, b, c, d e R such that ad - bc = 1. The transformation T :C\{-d/c} → Cdefined byaz + bT(z) :=P + 25is called Mobius transformation.(a) Show that the transformation S: C\{a/c} C defined by9 - zp-cz + a= (2)Sis the inverse of T, i.e., we haveS(T(z)) = T(S(z))= z, for any z {a/c,-d/c}.(b) Find a Mobius transformation T which satisfies:T(0) = 1,T(1) = 2, T (2) = 5.(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 ofthe segments {0

Given that T(z)=az+bcz+dLet w=az+bcz+d⇒wcz+wd=az+b⇒wcz-az=b-wd⇒z(wc-a)=b-wd)⇒z=b-wdwc-a⇒z=-dw+b-a+cw…

3. Assume that a, b, c, d e R such that ad - bc 1. The transformation T : C\{-d/c} C defined by T(z) := 9 + 20 cz + d is called Mobius transformation. (a) Show that the transformation S: C\{a/c} • → C defined by dz - b = (2)S -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. (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

3. Assume that a, b, c, d e R such that ad - bc 1. The transformation T : C\{-d/c} C defined by T(z) := 9 + 20 cz + d is called Mobius transformation. (a) Show that the transformation S: C\{a/c} • → C defined by dz - b = (2)S -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. (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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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) :=
P + 25
is called Mobius transformation.
(a) Show that the transformation S: C\{a/c} C defined by
9 - zp
-cz + a
= (2)S
is the inverse of T, i.e., we have
S(T(z)) = T(S(z))
= z, for any z {a/c,-d/c}.
(b) Find a Mobius transformation T which satisfies:
T(0) = 1,T(1) = 2, T (2) = 5.
(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 {re'n/4 |r < 1}.
%3D
1z+0

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