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Solutions for assignment 3

# Solutions for assignment 3 - Solutions for assignment 3 1...

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Section 7.4 Question 1: You can apply the standard precalculus techniques for inverting a function to do this problem. For example, you can solve the equation w = f_1(z) for z to obtain the inverse mapping z = f_1^{-1}(w). In this case, though, there is a substantial shortcut available, because of the identity f_1(z) = f_2(z+2). This means that f_1^{-1}(f_2(z)) = f_1^{-1}(f_1(z-2)) = z-2. Question 14: We first consider the simpler problem of whether the transformation (12) maps the unit circle |z|=1 to the unit circle |w|=1. In other words, we want to show that |w| = 1 whenever z is of the form z = exp(i beta). (Actually, because (12) is a Mobius transform, one only needs to check this claim for three values of z, such as z = 1, -1, and i, but for this question this trick won't save too much effort). If z is of the above form, then w = exp(i theta) (exp(i beta) - alpha) / (alpha* exp(i beta) - 1), where alpha* is the conjugate of alpha, and it is a routine matter of algebra to check that ww* = 1, so that |w| = 1, as desired. To finish the question we have to show that the unit disk |z| < 1 maps to the unit disk |w| < 1. Since we knw the unit circle maps to itself, the inside of the unit circle must map to either the inside of the unit circle or the outside of the unit circle. So it suffices to check what happens to just one point of the unit disk, say the origin.

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Solutions for assignment 3 - Solutions for assignment 3 1...

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