232sheet4

# 232sheet4 - w-α w-β = k z-α z-β where k = a-cα a-cβ 5...

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B U Department of Mathematics Math 232 Introduction To Complex Analysis Spring 2008 Exercise Sheet 4 1 1. Find the M¨ obius transformation f satsifying f (0) = 1, f (1 - i ) = i and f (2) = . 2. Find the M¨ obius transformation mapping 2, i , 0 to 1, - 1, 5 i , respectively. Under this mapping, what is the image of the line given by Re z = Im z ? 3. Find the M¨ obius transformation mapping 3, 0, to i , , 2, respectively. 4. Let f : z 7→ w = az + b cz + d , ( ad - bc 6 = 0) be a M¨ obius transformation other than the identity map. Prove that f has either one or two ﬁxed points. Furthermore, suppose f has two distinct ﬁxed points, say, α and β . Prove that
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Unformatted text preview: w-α w-β = k z-α z-β where k = a-cα a-cβ . 5. Find the ﬁxed points of the M¨ obius transformation z 7→ w when w is given by (i) z-1 z +1 , (ii) 3 z-4 z-1 , (iii) iz , (iv) 2 z-1 z . Find the image under each of these mappings of a) the circle | z | = 1, b) the real axis, c) the imaginary axis. 6. Suppose the four points z 1 , z 2 , z 3 and z 4 lie on a circle, show that ( z 1-z 3 )( z 2-z 4 ) ( z 1-z 4 )( z 2-z 3 ) is real. Notes 1 Please visit: http://www.math.boun.edu.tr/deptcourses/math232...
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