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exsheet2

# exsheet2 - Sivakumar M407 Example Sheet 2 1(a Suppose that...

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Unformatted text preview: Sivakumar M407 Example Sheet 2 1. (a) Suppose that R is a fixed positive number. Show that the solutions of the equation z 2 = R are given by ± √ R . (This shows that there is no loss of consistency when passing to complex roots of positive real numbers.) (b) Solve each of the following equations for z : (i) (2 z- 1) 3 = 1 + i (ii) z 4 =- 8- 8 √ 3 i (iii) z 2 + √ 32 iz- 6 i = 0 ( Suggestion: Complete squares.) (iv) (1 + z ) 5 = (1- z ) 5 2. Suppose that z and w are complex numbers. Establish the parallelogram identity : | z + w | 2 + | z- w | 2 = 2( | z | 2 + | w | 2 ) . 3. Find all possible values of θ such that | 1- cos θ- i sin θ | = 2. Verify your solution geometrically. 4. Suppose that n is a positive integer and z is a complex number. Establish the following identity: 1 + z + ··· + z n- 1 = n, if z = 1; 1- z n 1- z , if z 6 = 1. ( Suggestion: For the second part, write s = 1 + z + ··· + z n- 1 and consider the difference s- zs .) 5. Suppose that n is a fixed positive integer and recall that...
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exsheet2 - Sivakumar M407 Example Sheet 2 1(a Suppose that...

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