Chapter A.7, Problem 30E

(a)

To determine

Show that $z+\overline{z}=2\mathrm{Re}\left(z\right)$.

(b)

To determine

Show that $z-\overline{z}=2i\mathrm{Im}\left(z\right)$.

(c)

To determine

Show that $\left|\mathrm{Re}\left(z\right)\right|\le \left|z\right|$.

(d)

To determine

Show that ${\left|z_1+z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2+2\text{Re}\left(z_1{\overline{z}}_2\right)$.

(e)

To determine

Show that $\left|z_1+z_2\right|\le \left|z_1\right|+\left|z_2\right|$.