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Question: Let f(z) = (az + b)/(cz + d) and ad – bc + 0. If this transformation maps zo → wo, Z1 → w, and z2 → w2 (in other words, if three points and their images under this bi-linear transformation is specified) show that f (z) can be found uniquely. (Hint: Write W; = (az¡ + b)/(cz; + d), i = 0,1,2 and convert this three equation to a linear equation system of three equations, whose unknowns b c d are a b c d and the right hand side includes terms having zo,Z1, Z2, wo,W1, W2. Then, solve ,-,- in a'a a'a'a terms of zo, Z1, Z2, Wo, W1, W2 and find f(z) = (z +)/Ez+).)

Question: Let f(z) = (az + b)/(cz + d) and ad – bc + 0. If this transformation maps zo → wo, Z1 → w, and z2 → w2 (in other words, if three points and their images under this bi-linear transformation is specified) show that f (z) can be found uniquely. (Hint: Write W; = (az¡ + b)/(cz; + d), i = 0,1,2 and convert this three equation to a linear equation system of three equations, whose unknowns b c d are a b c d and the right hand side includes terms having zo,Z1, Z2, wo,W1, W2. Then, solve ,-,- in a'a a'a'a terms of zo, Z1, Z2, Wo, W1, W2 and find f(z) = (z +)/Ez+).)

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: Let f(z) = (az + b)/(cz + d) and ad – bc ± 0. If this transformation maps zo → wo, → w2 (in other words, if three points and their images under this bi-linear Z1 → W1 and z2 transformation is specified) show that f (z) can be found uniquely. (Hint: Write W¡ = (az; + b)/(cz¡ + d), i = 0,1,2 and convert this three equation to a linear equation system of three equations, whose unknowns b c d are and the right hand side includes terms having z0,Z1, Z2, Wo, W1, W2. Then, solve b c d in а а' а а а' а terms of zo, Z1, Z2, Wo, W1, W2 and find f(z) = (z +)/z +).)
Transcribed Image Text:Question: Let f(z) = (az + b)/(cz + d) and ad – bc ± 0. If this transformation maps zo → wo, → w2 (in other words, if three points and their images under this bi-linear Z1 → W1 and z2 transformation is specified) show that f (z) can be found uniquely. (Hint: Write W¡ = (az; + b)/(cz¡ + d), i = 0,1,2 and convert this three equation to a linear equation system of three equations, whose unknowns b c d are and the right hand side includes terms having z0,Z1, Z2, Wo, W1, W2. Then, solve b c d in а а' а а а' а terms of zo, Z1, Z2, Wo, W1, W2 and find f(z) = (z +)/z +).)
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