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MATH135 - Assignment9 Solutions

# MATH135 - Assignment9 Solutions - MATH 135 Fall 2007...

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Unformatted text preview: MATH 135 Fall 2007 Assignment #9 Due: Wednesday 28 November 2007, 8:20 a.m. N.B. This is the final regularly scheduled Assignment. There will be an Assignment 10 created and solutions posted. While you should not hand Assignment 10 in, working through the problems will be helpful in learning the material from Chapter 9. Hand-In Problems 1. (a) Convert- 3 √ 7- √ 21 i to polar form. (b) Convert (12 ,- 31 π/ 6) to standard form. 2. Determine the modulus and argument of (1- i ) 42 . 3. Express ( √ 2- √ 6 i ) 32 in standard form. 4. Determine all z ∈ C such that z 8 = 81 i . Plot your solutions in the complex plane. 5. Determine all z ∈ C such that iz 3 + 1 + √ 3 i = 0. Plot your solutions in the complex plane. 6. (a) Prove directly that if z = r cis( θ ) and w = s cis( φ ), then z w = r s cis( θ- φ ). (b) Using this result, evaluate 1- i √ 3 + i . Express your answer in polar form. 7. Determine all z ∈ C such that z 9 +8 iz 6 + z 3 +8 i = 0. Plot your solutions in the complex plane. 8. If a, b, c, d ∈ R with c + id = ( a + ib ) n , show that c 2 + d 2 = ( a 2 + b 2 ) n . 9. Suppose n ∈ P , with n ≥ 3. Suppose that z ∈ C has z = z n- 1 and z 6 = 0. (a) Determine | z | . (b) Determine all possible values for z (in terms of n ). 10. (a) Write 4 √ 3- 4 i in the form e u + iv where u, v ∈ R . (b) If z ∈ C and 1 2 ( e iz + e- iz ) =- 2 i , determine the two possible values of e iz . (c) If z = x + iy with x, y ∈ R , show that e iz = e- y cis( x ). (d) If 1 2 ( e iz + e- iz ) =- 2 i and z = x + iy with x, y ∈ R , determine all possible values of z . If z ∈ C , we define cos z = 1 2 ( e iz + e- iz ), so this part solves the equation cos z =- 2 i . In this part, you should explicitly find x and y . Do not try to take the “ln” of a complex number. (If you had e iz = 4 i (which you won’t), you should not write something like iz = ln(4 i ).) Recommended Problems 1. Text, page 219, #44 2. Text, page 219, #51 3. Text, page 219, #65 4. Text, page 219, #67 5. Text, page 220, #74 6. Text, page 222, #129 7. Text, page 222, #136 8. If z ∈ C and r, s ∈ P , we define z r/s to be an s th root of z r ....
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MATH135 - Assignment9 Solutions - MATH 135 Fall 2007...

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