A First Course in Complex Analysis · Problem 1.5.22

Complex Numbers — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · properties-of-complex-conjugate

Problem

Prove Proposition 1.3.3: For any

z, z_1, z_2 \in \mathbb{C}

, ((a))

\overline{z_1 \pm z_2} = \overline{z_1} \pm \overline{z_2}

((b))

\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}

((c))

\overline{\left(\dfrac{z_1}{z_2}\right)} = \dfrac{\overline{z_1}}{\overline{z_2}}

((d))

\overline{\overline{z}} = z

((e))

|\overline{z}| = |z|

((f))

|z|^2 = z\overline{z}

((g))

\operatorname{Re}(z) = \dfrac{1}{2}(z + \overline{z})

((h))

\operatorname{Im}(z) = \dfrac{1}{2i}(z - \overline{z})

((i))

\overline{e^{i\phi}} = e^{-i\phi}