ACIT4330-Page/content/Definitions/Complex Analysis/Complex Conjugation.md

Definition

Let z = x + iy. Then its complex conjugate is

\bar{z} := x-iy.

Properties

The following hold:

1. \mid z \mid^2 = z \bar{z}, (=r^2)
2. z + \bar{z} = 2 \mathrm{Re} z,
3. z - \bar{z} = 2i \mathrm{Im} z,
4. \overline{r e^{i \phi}} = r e^{-i \phi}.

[!note]+ Note that (4) implies that

\mid \bar{z} \mid = \mid z \mid.

Also write that z = r e^{i \phi} and z' = r' e^{i \phi'}.

Then zz' = rr'e^{i(\phi + \phi')}. Then (1) implies that

\mid z z' \mid = rr' = \mid z \mid \mid z' \mid.

(Nice interplay between complex multiplication with absolute values).