/ math / algebra /

# Complex Numbers 

Imaginary Unit

##### Definition

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.

$\boldsymbol{z} := a + b\text{i}= r \cdot \exp(\text{i}\varphi)$

with Imaginary Unit $\text{i}= \sqrt{-1}$.

Polarkoordinaten: $\boldsymbol{z} = r \cdot (\cos(\varphi)+\mathbf{i}\sin(\varphi))=r\cdot e^{\varphi \mathbf{i}}$

$r=|\boldsymbol{z}|=\sqrt{a^2+b^2}\varphi=\arg(\boldsymbol{z})=\begin{cases}+\arccos \left( \frac{a}{r}\right), & b \ge 0 \\ -\arccos \left( \frac{a}{r}\right), & b<0 \end{cases}$

with $\varphi \in [0, 2 \pi]$

## Rules

$\text{i}^{2n} = -1^n$    $\text{i}^{2n+1} = -\text{i}^n$    $\text{i}^{-1} = -\text{i}$

Conjugate: $\boldsymbol{z}^{*} = a - b\text{i}$     $\exp(\overline{i\varphi}) = \exp({-i\varphi})$ $\boldsymbol{z} \overline{\boldsymbol{z}} = \left\vert{\boldsymbol{z}}\right\vert^2 = a^2+b^2$

Inverse: $\displaystyle \boldsymbol{z}^{-1} = \frac{\boldsymbol{z}^{*}}{\boldsymbol{z}^{*}} \boldsymbol{z}=\frac{a - b\text{i}}{a^2+b^2}$

Multiplikation: $\boldsymbol{z}_1\cdot \boldsymbol{z}_2=r_1r_2 ( \cos ( \varphi_1 + \varphi_2) + \mathbf{i} \sin (\varphi_1 + \varphi_2))$

Division: $\frac{\boldsymbol{z}_1}{\boldsymbol{z}_2}=\frac{r_1}{r_2} ( \cos ( \varphi_1 - \varphi_2) + \mathbf{i} \sin (\varphi_1 - \varphi_2))$

n-te Potenz: $\boldsymbol{z}^n=r^n\cdot e^{n\varphi \mathbf{i}}= r^n (\cos (n \varphi) + \mathbf{i} \sin (n \varphi))$

Logarithmus: $\ln(\boldsymbol{z})=\ln(r) + \mathbf{i}(\varphi + 2k\pi)$ (Nicht eindeutig!)