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Question 5. Suppose that f is a Möbius transformation which fixes the real and imag- inary axes, i.e. f({ze C: Im(z)=0}) = {ze C: Im(z)=0} and f({ze C: Re(z)=0}) = { z EC: Re(z)=0}. Prove that f maps circles centred at the origin to circles centred at the origin (not necessarily of the same radius), i.e. for all r₁ > 0, there exists r2 > 0 such that f({ze C:|2|=r₁}) = {z EC: |z| = r₂}. A geometric argument is sufficient, as long as you clearly state any properties of Möbius transformations that you use. You may assume, should you wish, that a circle is centred on a line if and only if it intersects the line at right angles.

Question 5. Suppose that f is a Möbius transformation which fixes the real and imag- inary axes, i.e. f({ze C: Im(z)=0}) = {ze C: Im(z)=0} and f({ze C: Re(z)=0}) = { z EC: Re(z)=0}. Prove that f maps circles centred at the origin to circles centred at the origin (not necessarily of the same radius), i.e. for all r₁ > 0, there exists r2 > 0 such that f({ze C:|2|=r₁}) = {z EC: |z| = r₂}. A geometric argument is sufficient, as long as you clearly state any properties of Möbius transformations that you use. You may assume, should you wish, that a circle is centred on a line if and only if it intersects the line at right angles.

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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Question 5. Suppose that f is a Möbius transformation which fixes the real and imag- inary axes, i.e. f({ z = C: Im(z) = 0 }) = { z = C: Im(z)=0} and ƒ({ z € C: Re(z) = 0 }) = { z = C: Re(z) = 0}. Prove that f maps circles centred at the origin to circles centred at the origin (not necessarily of the same radius), i.e. for all r₁ > 0, there exists r2 > 0 such that ƒ({ z ≤ C : |z| = r₁ }) = { z € C : |Z| = r₂}. A geometric argument is sufficient, as long as you clearly state any properties of Möbius transformations that you use. You may assume, should you wish, that a circle is centred on a line if and only if it intersects the line at right angles.
Transcribed Image Text:Question 5. Suppose that f is a Möbius transformation which fixes the real and imag- inary axes, i.e. f({ z = C: Im(z) = 0 }) = { z = C: Im(z)=0} and ƒ({ z € C: Re(z) = 0 }) = { z = C: Re(z) = 0}. Prove that f maps circles centred at the origin to circles centred at the origin (not necessarily of the same radius), i.e. for all r₁ > 0, there exists r2 > 0 such that ƒ({ z ≤ C : |z| = r₁ }) = { z € C : |Z| = r₂}. A geometric argument is sufficient, as long as you clearly state any properties of Möbius transformations that you use. You may assume, should you wish, that a circle is centred on a line if and only if it intersects the line at right angles.
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