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Homework 2 (MATH 1199 Fall 2019)
1. In the following problems, write in polar form and multiply z1 and z2, and then express that product in the form a+bi, where a,b∈R.
(a) z1=2+2i and z2=3i
(b) z1=−√3−i and z2=5
(c) z1=1−i and z2=5+5i
(d) z1=3i and z2=−4i
2. Find Arg(2i1+i) in any way you like.
3. Compute by hand and express the final answer in the form a+bi for a,b∈R.
(a) (−1+√3i)14
(b) (−2−2√3i)11
(c) (4−4i)−11
4. Find all fifth roots of 1.
5. Find all sixth roots of 1.
6. Find all cube roots of i and write them in the form a+bi for a,b∈R.
7. Find all fourth roots of 1+i.
8. Find all cube roots of −√3−i.
9. Find all fourth roots of −1 and write them in the form a+bi for a,b∈R.
10. Write the described function in the form u+iv where u,v:R×R→R.
(a) {f:C→Cf(z)=z2−2¯z
(b) {g:C∖{−1}→Cg(z)=zz+1
(c) {h:C∖{1+i}→Ch(z)=z2z−1−i
(d) {ℓ:C∖{0}→Cℓ(z)=¯z|z|