prob01 - 2 i 1 i 1 6 8 Prove that if n is a positive...

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Problem Set 1 Due January 28 rd , 2007 1. Evaluate (8 + 2 i ) - (1 - i ) (2 + i ) 2 2 - 3 i 1 + 2 i - 8 + i 6 - i 2. Let z be a complex number such that Im z > 0, show that Im(1 /z ) < 0. 3. Show that if | z | = 1( z n = 1), then Re 1 / (1 - z ) = 1 / 2. 4. Find the principal value of the arguement for following complex numbers: - 1 2 , - 3 + 3 i, ( 3 - i ) 4 , - 1 + 3 i 2 + 2 i , (1 - i )( - 3 + i ) 5. Given two points P = (1 , 2) and Q = (2 , 0), ±nd the equation for the line trough P and Q in terms of z and ¯ z . 6. Solve the equation ( z + 1) 5 = z 5 . 7. Find all the values of ( - 16) 1 / 4 and
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Unformatted text preview: ( 2 i 1 + i ) 1 / 6 . 8. Prove that if n is a positive integer, then v v sin( nθ/ 2) sin( θ/ 2) v v ≤ n ( θ n = 0 , ± 2 π, ± 4 π, ··· ) Hint: if z = e iθ , then the LHS above equals | (1-z n ) / (1-z ) | . 9. Consider the regular decagon (= 10-gon) of radius 10. Prove that the product p of the distance from any one vertex to each of the others is given by P = 10 , 000 , 000 , 000...
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This note was uploaded on 02/23/2010 for the course MATH 3007 taught by Professor Hou during the Spring '08 term at Columbia.

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