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math185f08-hw1

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

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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 π + i cos 3 4 π + · · · + i n cos( 2 n +1 4 ) π + · · · + i 400 cos 801 4 π. (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 α, β C be the roots of z 2 + bz + c = 0 and γ, δ C be the roots of z 2 + | b | z + | c | = 0 . Show that if | α | = | β | = 1, then | γ | = | δ | = 1. (b) Let c 0 , . . . , c 4 R . Suppose the polynomial equation c 4 z 4 + ic 3 z 3 + c 2 z 2 + ic 1 z + c 0 = 0 has a root given by a + ib . Show that - a + ib is also a root but the complex conjugate a - ib may not be a root. Why doesn’t this contradict what you’ve learnt about roots occurring in conjugate pairs? 5. Show that if 2 3 π < θ < 4 3 π , then the limit lim n →∞ n X r =1 n r cos exists. Note that for a given θ ( 2 3 π, 4 3 π ) and r N , we do

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

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