Complex Analysis

Complex Numbers

$z = a + b i$ where $a, b \in R$ and $i$ satisfies $i^{2} = - 1$ .

$\overset{―}{z} = a - b i$ .

Addition. $z_{1} + z_{2} = (a_{1} + a_{2}) + (b_{1} + b_{2}) i$
Multiplication. $z_{1} z_{2} = (a_{1} a_{2} - b_{1} b_{2}) + (a_{1} b_{2} + a_{2} b_{1}) i$
Division. $\frac{z_{1}}{z_{2}} = \frac{z_{1} {\overset{―}{z}}_{2}}{z_{2} {\overset{―}{z}}_{2}} = \frac{z_{1} {\overset{―}{z}}_{2}}{a_{2}^{2} + b_{2}^{2}} = \frac{(a_{1} a_{2} + b_{1} b_{2}) + (a_{2} b_{1} - a_{1} b_{2}) i}{a_{2}^{2} + b_{2}^{2}}$

$z = | z | \exp (i θ)$ , $θ = \arctan (b / a)$ , $| z | = \sqrt{z \overset{―}{z}}$

$\overset{―}{z} = | z | \exp (- i θ)$

`Complex Plane’

\textbf{Re}al and Complex (\textbf{Im}) axes.

Thus, complex numbers can be represented as a vector in the complex plane.

$+ (z_{1}, z_{2}) = (a_{1} + b_{1}) + (a_{2} + b_{2}) i$ - vector tail-tip addition

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

$- (z_{1}, z_{2}) = + (z_{1}, - z_{2}) = (a_{1} - b_{1}) + (a_{2} - b_{2}) i$

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

$z_{1} z_{2} = r_{1} r_{2} \exp (i (θ_{1} + θ_{2}))$