Prove the items in Lemma 1.4 that were not proved in the lecture notes.

Prove Proposition 1.14.

Draw the following set of points

1. \(\{z \in \mathbb{C}\, |\, |z-1|=|z+1|\},\)

2. \(\{z \in \mathbb{C}\, |\, 1<|z-\mathrm{i}|<2\},\)

3. \(\{z \in \mathbb{C}\, |\, |z|\geqslant 1, |\mathrm{Re}(z)|\leqslant \frac{1}{2}, \mathrm{Im}(z)>0\}.\)

Decompose the following complex numbers into real and imaginary parts:

1. \[\frac{1}{1+\mathrm{i}}+\frac{1}{2+\mathrm{i}}+\frac{1}{3+\mathrm{i}},\]

2. \[\frac{2-3\mathrm{i}}{2+\mathrm{i}}+\frac{1-\mathrm{i}}{1+3\mathrm{i}},\]

3. \[\left(\frac{1+\mathrm{i}}{1-\mathrm{i}}\right)^k,\,\,k\in\mathbb{Z}.\]