Complex Analysis Handbook/Basic Arithmetic

Complex Conjugate

Let: $\bar{z}$ be the complex conjugate of $z$ then

$\overline{(z_{1} \pm z_{2})} = \bar{z_{1}} \pm \bar{z_{2}}$

$\overline{(z_{1} z_{2})} = \bar{z_{1}} \bar{z_{2}}$

$p = x + i y$

$r : = \sqrt{x^{2} + y^{2}}$ , $θ : = \arctan (y / x)$

$p$	$= r (\cos θ + i \sin θ) = r c i s θ$
	$= r e^{i θ}$

We call $r$ absolute value (or modulus) and $θ$ argument of $z$

$z_{1} z_{2} = r_{1} r_{2} c i s (θ_{1} + θ_{2}) = r_{1} r_{2} (\cos (θ_{1} + θ_{2}) + i \sin (θ_{1} + θ_{2}))$

$\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}} c i s (θ_{1} - θ_{2})$

$\frac{1}{z} = \frac{1}{r} c i s (- θ)$

So

$z^{n} = r^{n} c i s (n θ) = r^{n} (\cos n θ + i \sin n θ)$

For any complex variable $z = x + i y$ it is possible to express $x$ and $y$ in terms of $z$ and $\bar{z}$ , as follows:

$x = \frac{z + \bar{z}}{2}$ and $y = \frac{z - \bar{z}}{2 i}$

$| z_{1} + z_{2} | \leq | z_{1} | + | z_{2} |$

$| z_{1} - z_{2} | \geq | | z_{1} | - | z_{2} | | \geq 0$