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135-a10-2011

# 135-a10-2011 - = 4 i 4 If z,w ∈ C prove that | z w | 2 |...

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Assignment 10 – Questions 1. Prove that (1 + i 2 n )(1 + i n ) is either 0 or 4 for n P . Describe the values of n that yield the various answers. 2. Suppose z = x + iy with x, y R . Determine all complex numbers z for which z 2 = 2 z . 3. Solve the following pairs of equations for z, w C . (a) z + i w = 2 i z + 2 w = 3 (b) i z + (1 + i ) w = - 3 + i (2 + i ) z + (3 - 2 i ) w
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Unformatted text preview: = 4 i 4. If z,w ∈ C , prove that | z + w | 2 + | z-w | 2 = 2 | z | 2 + 2 | w | 2 . 5. Prove that, for all z 1 ,z 2 ∈ C | z 1-z 2 | ≥ | z 1 | - | z 2 | . What is the geometric interpretation of this inequality, and when does this equality hold? 1...
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