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Unformatted text preview: Exam 1 Review Problems 1. Let z =- 2 1- 3 i . (a) Calculate the rectangular forms of z , z , and z- 1 . (b) Calculate | z | , Arg( z ) and the polar form of z . (c) Calculate the rectangular form of z 42 . ( Hint: Use your answer from part (b).) 2. Solve the equation | e i- 1 | = 2 for real. 3. Prove that 1 e i = e- i , where is a real number. 4. Find all the cube roots of w = 3 3 + 3 i . 5. Prove that 1 z 2- 1 1 3 for all z on the circle of radius 2 centered at the origin. What can you say about 1 z 2- 1 on the circle of radius R centered at the origin? 6. Solve each of the following equations or explain why no solution exists. (a) z 3 =- 27 (b) e z = 2- 2 i (c) Log z = 2 e i/ 6 (d) z 1 / 3 = i 7. Write the function f ( z ) = z 3 + z + 1 in the form f ( z ) = u ( x,y ) + iv ( x,y ). 8. Suppose that f ( z ) = ( x 2- y 2- 2 y ) + i (2 x- 2 xy ), where z = x + iy . Use the fact that x = z + z 2 and y = z- z 2 i to express f ( z ) in terms of z , and simplify the result. 9. A fixed point of a function f is a complex number, z in the domain of f such that f ( z ) = z . Find any and all of the fixed points of the function....
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This note was uploaded on 03/21/2010 for the course MATH 375 taught by Professor Marchesi during the Spring '10 term at Binghamton University.
- Spring '10