math185f09-hw1sol

math185f09-hw1sol - MATH 185 COMPLEX ANALYSIS FALL 2009/10...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 1 SOLUTIONS Throughout the problem set, i = √- 1; and whenever we write α + βi , it is implicit that α,β ∈ R . 1. Determine the values of the following (without the aid of any electronic devices). (a) (1 + i ) 20- (1- i ) 20 . Solution. (1 + i ) 20- (1- i ) 20 = [(1 + i ) 2 ] 10- [(1- i ) 2 ] 10 = (2 i ) 10- (- 2 i ) 10 = 0 . (b) cos 1 4 π + i cos 3 4 π + ··· + i n cos( 2 n +1 4 ) π + ··· + i 40 cos 81 4 π. Solution. Write a n = i n cos( 2 n +1 4 ) π . Note that a n +2 =- i n cos ( 2 n +1 4 ) π + π = i n cos( 2 n +1 4 ) π = a n . So a = a 2 = ··· = a 40 , a 1 = a 3 = ··· = a 39 , and a + a 1 + ··· + a 40 = 21 a + 20 a 1 = √ 2 2 (21- 20 i ) . (c) 1 + 2 i + 3 i 2 + ··· + ( m + 1) i m where m is divisible by 4. Solution. Let S be the sum. Then S = 1 + 2 i + 3 i 2 + ··· + ( m + 1) i m , iS = i + 2 i 2 + ··· + mi m + ( m + 1) i m +1 . Subtracting the second equation from the first yields (1- i ) S = 1 + i + i 2 + ··· + i m- ( m + 1) i m +1 = 1- i m +1 1- i- ( m + 1) i m +1 = 1- ( m + 1) i since i m = 1 if m is divisible by 4. Hence S = 1- ( m + 1) i 1- i × 1 + i 1 + i = 1 2 ( m + 2- mi ) ....
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This note was uploaded on 02/17/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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math185f09-hw1sol - MATH 185 COMPLEX ANALYSIS FALL 2009/10...

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