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math185-hw1 - MATH 185 COMPLEX ANALYSIS FALL 2007/08...

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 1 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 . (b) cos 1 4 π + i cos 3 4 π + · · · + i n cos( 2 n +1 4 ) π + · · · + i 40 cos 81 4 π. (c) 1 + 2 i + 3 i 2 + · · · + ( m + 1) i m where m is divisible by 4. 2. Use the exponential form of cos θ and sin θ to show the following. (a) Show that 1 + n cos θ + · · · + n ! r !( n - r )! cos + · · · + cos = (2 cos 1 2 θ ) n cos 1 2 nθ. Prove that, as n → ∞ , the series converges to 0 if 2 3 π < θ < 4 3 π . (b) If sin θ = α sin( θ + β ), where α and β are real constants, prove that e 2 = 1 - αe - 1 - αe . Hence prove that θ = n =1 α n n sin nβ. State the range of values of α for which the series is valid. 3. Express the roots of the equation z 7 - 1 = 0 in the form cos θ + i sin θ . Hence show that the roots of the equation u 3 + u 2 - 2 u - 1 = 0 are 2 cos 2 π 7 , 2 cos 4 π 7 , 2 cos 6 π 7 ,
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