Skip to content

Complex Numbers#

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.

\[\cx z := a + b\i = r \cdot \exp(\i \varphi)\]

with Imaginary Unit \(\i = \sqrt{-1}\).

Complex Plane

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

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

with \(\varphi \in [0, 2 \pi]\)


\(\i^{2n} = -1^n\)    \(\i^{2n+1} = -\i^n\)    \(\i^{-1} = -\i\)

Conjugate: \(\cxc z = a - b\i\)     \(\exp(\overline{i\varphi}) = \exp({-i\varphi})\) \(\cx z \ol{\cx z} = \abs{\cx z}^2 = a^2+b^2\)

Inverse: \(\displaystyle \cx z^{-1} = \frac{\cxc z}{\cxc z} \cx z=\frac{a - b\i}{a^2+b^2}\)

Multiplikation: \(\cx z_1\cdot \cx z_2=r_1r_2 ( \cos ( \varphi_1 + \varphi_2) + \mathbf{i} \sin (\varphi_1 + \varphi_2))\)

Division: \(\frac{\cx z_1}{\cx z_2}=\frac{r_1}{r_2} ( \cos ( \varphi_1 - \varphi_2) + \mathbf{i} \sin (\varphi_1 - \varphi_2))\)

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

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