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COMPLEX ANALYSIS I Exercise 2, spring 2011

1. Show that |z1 − z2| = |1− z¯1z2|, for all z1, z2 ∈ C with |z1| = 1 or

|z2| = 1.

2. Let a1, a2,· · · , an ∈ R ; a0 > a1 > a2 > · · · > an > 0 and p(z) = a0 + a1z + · · · + anzn, z ∈ C. Suppose that p(z0) = 0. Show that

|z0| > 1.

3. Find the polarcoordinates of z ∈ C when a) z =−3i, b) z =

√

3−i, c) z = 2−i

√ 12.

4. Calculate (1−i

√

3)15 and (1 +i)11 and (1 +i)5 (1−i

√ 3)7.

5. Let z ∈ C,|z| = 1, z 6= −1. Show that z can be given in the form z = 1 +it

1−it with t∈ R.

6. Find the solutions

a) z4 = −1, b) z6 = 1, c) z3 = −i.

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