math185f09-hw1

# math185f09-hw1 - i i Express your solutions in the form α...

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MATH 185: COMPLEX ANALYSIS FALL 2009/10 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 = X 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 , and ﬁnd the roots of 8 w 3 + 4 w 2 - 4 w - 1 = 0 . 4. (a) Find all possible values of

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Unformatted text preview: i i . Express your solutions in the form α + βi . (b) Find all values of θ ∈ [0 , 2 π ) for which the following limit exists lim r →∞ e re iθ . 5. Let a ,...,a 4 ∈ R . Suppose the polynomial equation a 4 z 4 + ia 3 z 3 + a 2 z 2 + ia 1 z + a = 0 has a root given by z = α + βi . Find another root of the equation. Your answer should only depend on α,β . Date : September 5, 2009 (Version 1.0); due: September 11, 2009. 1 6. Let z n ,w n ∈ C for every n ∈ N . Show that (a) If ∑ ∞ n =1 z n and ∑ ∞ n =1 w n are both convergent, then so is ∞ X n =1 λz n + μw n for any λ,μ ∈ C . (b) If ∑ ∞ n =1 z n is convergent, then lim n →∞ z n = 0 . (c) If ∑ ∞ n =1 | z n | is convergent, then so is ∑ ∞ n =1 z n . 2...
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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-hw1 - i i Express your solutions in the form α...

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