This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full Document
 Fall '08
 Staff
 Real Numbers, Complex number, positive integer

Click to edit the document details