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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 ${\text{i}}= \sqrt{-1}$.

Complex Plane
Complex Plane

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!)

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