math185f08-hw1 - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 1 Throughout the problem set, i = - 1; and whenever we write a + bi , it is implicit that a,b R . 1. Determine the values of the following (without the aid of any electronic devices). (a) (1 + i ) 200- (1- i ) 200 . (b) cos 1 4 + i cos 3 4 + + i n cos( 2 n +1 4 ) + + i 400 cos 801 4 . (c) 1 + 2 i + 3 i 2 + + ( m + 1) i m where m is divisible by 4. 2. Let z C be such that Im( z ) 6 = 0 and Im 1 + z + z 2 1- z + z 2 = 0 . Prove that | z | = 1. 3. Let z 1 ,z 2 C . (a) Prove that | z 1 | - | z 2 | | z 1 + z 2 | | z 1 | + | z 2 | and that | z 1 | - | z 2 | | z 1- z 2 | | z 1 | + | z 2 | . (b) Suppose | z 1 | = | z 2 | = 1. Prove that | z 1 + 1 | + | z 2 + 1 | + | z 1 z 2 + 1 | 2 . 4. (a) Let b,c C . Let , C be the roots of z 2 + bz + c = 0 and , C be the roots of z 2 + | b | z + | c | = 0 ....
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This note was uploaded on 09/21/2008 for the course MATH 185 taught by Professor Lim during the Spring '07 term at University of California, Berkeley.

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math185f08-hw1 - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

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