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# Formelsammlung Mathematik: Komplexe Zahlen

## Darstellung

Kartesische Form
${\displaystyle z=a+b\mathrm {i} }$
Polarform (trigonometrische Darstellung)
${\displaystyle z=r(\cos \varphi +\mathrm {i} \sin \varphi )}$
Polarform (Exponentialdarstellung)
${\displaystyle z=r\mathrm {e} ^{\mathrm {i} \varphi }}$

## Elementare Operationen

Name Operation Polarform kartesische Form
Identität ${\displaystyle z}$ ${\displaystyle =r\mathrm {e} ^{\mathrm {i} \varphi }}$ ${\displaystyle =a+b\mathrm {i} }$
Identität ${\displaystyle z_{1}}$ ${\displaystyle =r_{1}\mathrm {e} ^{\mathrm {i} \varphi _{1}}}$ ${\displaystyle =a_{1}+b_{1}\mathrm {i} }$
Identität ${\displaystyle z_{2}}$ ${\displaystyle =r_{2}\mathrm {e} ^{\mathrm {i} \varphi _{2}}}$ ${\displaystyle =a_{2}+b_{2}\mathrm {i} }$
Addition ${\displaystyle z_{1}+z_{2}}$ ${\displaystyle =(a_{1}+a_{2})+(b_{1}+b_{2})\mathrm {i} }$
Subtraktion ${\displaystyle z_{1}-z_{2}}$ ${\displaystyle =(a_{1}-a_{2})+(b_{1}-b_{2})\mathrm {i} }$
Multiplikation ${\displaystyle z_{1}z_{2}}$ ${\displaystyle =r_{1}r_{2}\mathrm {e} ^{\mathrm {i} (\varphi _{1}+\varphi _{2})}}$ ${\displaystyle =(a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+a_{2}b_{1})\mathrm {i} }$
Division ${\displaystyle {\frac {z_{1}}{z_{2}}}}$ ${\displaystyle ={\frac {r_{1}}{r_{2}}}\mathrm {e} ^{\mathrm {i} (\varphi _{1}-\varphi _{2})}}$ ${\displaystyle ={\frac {a_{1}a_{2}+b_{1}b_{2}}{a_{2}^{2}+b_{2}^{2}}}+{\frac {a_{2}b_{1}-a_{1}b_{2}}{a_{2}^{2}+b_{2}^{2}}}\mathrm {i} }$
Kehrwert ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle ={\frac {1}{r}}\mathrm {e} ^{-\mathrm {i} \varphi }}$ ${\displaystyle ={\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}\mathrm {i} }$
Potenzierung ${\displaystyle z^{n}}$ ${\displaystyle =r^{n}\mathrm {e} ^{n\mathrm {i} \varphi }}$
Konjugation ${\displaystyle {\overline {z}}}$ ${\displaystyle =r\mathrm {e} ^{-\mathrm {i} \varphi }}$ ${\displaystyle =a-b\mathrm {i} }$
Realteil ${\displaystyle \mathrm {Re} (z)}$ ${\displaystyle =r\cos \varphi }$ ${\displaystyle =a}$
Imaginärteil ${\displaystyle \mathrm {Im} (z)}$ ${\displaystyle =r\sin \varphi }$ ${\displaystyle =b}$
Betrag ${\displaystyle |z|}$ ${\displaystyle =r}$ ${\displaystyle ={\sqrt {a^{2}+b^{2}}}}$
Argument ${\displaystyle \mathrm {arg} (z)}$ ${\displaystyle =\varphi }$ ${\displaystyle =s(b)\arccos {\Big (}{\frac {a}{r}}{\Big )}}$

${\displaystyle s(b):={\begin{cases}+1&{\text{wenn}}\;b\geq 0,\\-1&{\text{wenn}}\;b<0\end{cases}}}$

Rechenweg zur Division:

${\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {z_{1}{\overline {z_{2}}}}{z_{2}{\overline {z_{2}}}}}={\frac {z_{1}{\overline {z}}_{2}}{|z_{2}|^{2}}}}$
${\displaystyle {\frac {1}{z}}={\frac {\overline {z}}{z\,{\overline {z}}}}={\frac {\overline {z}}{|z|^{2}}}}$

## Konjugation

Für alle ${\displaystyle z,z_{1},z_{2}\in \mathbb {C} }$ gilt:

 ${\displaystyle {\overline {z_{1}+z_{2}}}={\bar {z}}_{1}+{\bar {z}}_{2}}$ ${\displaystyle {\overline {z_{1}-z_{2}}}={\bar {z}}_{1}-{\bar {z}}_{2}}$ ${\displaystyle {\overline {z_{1}z_{2}}}={\bar {z}}_{1}\,{\bar {z}}_{2}}$ ${\displaystyle z_{2}\neq 0\implies {\overline {{\Big (}{\frac {z_{1}}{z_{2}}}{\Big )}}}={\frac {{\bar {z}}_{1}}{{\bar {z}}_{2}}}}$ ${\displaystyle |{\bar {z}}|=|z|}$ ${\displaystyle {\bar {\bar {z}}}=z}$ ${\displaystyle z{\bar {z}}=|z|^{2}}$ ${\displaystyle \operatorname {Re} (z)={\frac {z+{\bar {z}}}{2}}}$ ${\displaystyle \operatorname {Im} (z)={\frac {z-{\bar {z}}}{2i}}}$ ${\displaystyle {\overline {\mathrm {e} ^{z}}}=\mathrm {e} ^{\bar {z}}}$ ${\displaystyle {\overline {\sin(z)}}=\sin({\bar {z}})}$ ${\displaystyle {\overline {\cos(z)}}=\cos({\bar {z}})}$

Für alle ${\displaystyle z\in \mathbb {C} \setminus \{x\in \mathbb {R} \mid x\leq 0\}}$ und ${\displaystyle x\in \mathbb {C} }$ gilt:

${\displaystyle {\overline {z^{x}}}={\bar {z}}^{\bar {x}}}$
${\displaystyle {\overline {\ln(z)}}=\ln({\bar {z}})}$
${\displaystyle \arg({\bar {z}})=-\arg(z)}$

## Argument

Für alle ${\displaystyle r>0}$, ${\displaystyle z,z_{1},z_{2}\in \mathbb {C} \setminus \{0\}}$ und ${\displaystyle x\in \mathbb {C} }$ gilt:

${\displaystyle \arg(rz)=\arg(z)}$
${\displaystyle \arg(z_{1}z_{2})\equiv \arg(z_{1})+\arg(z_{2}){\pmod {2\pi }}}$
${\displaystyle \arg {\Big (}{\frac {z_{1}}{z_{2}}}{\Big )}\equiv \arg(z_{1})-\arg(z_{2}){\pmod {2\pi }}}$
${\displaystyle \arg {\Big (}{\frac {1}{z}}{\Big )}\equiv -\arg(z){\pmod {2\pi }}}$
${\displaystyle \arg({\bar {z}})\equiv -\arg(z){\pmod {2\pi }}}$
${\displaystyle \arg(z^{x})\equiv \arg(z)\operatorname {Re} (x)+\ln(|z|)\operatorname {Im} (x){\pmod {2\pi }}}$

Für alle ${\displaystyle z\in \mathbb {C} \setminus \{x\in \mathbb {R} \mid x\leq 0\}}$ gilt:

${\displaystyle \arg({\bar {z}})=-\arg(z)}$
${\displaystyle \arg {\Big (}{\frac {1}{z}}{\Big )}=-\arg(z)}$

## Potenzen

Definitionen:

${\displaystyle \mathrm {e} ^{z}:=\mathrm {e} ^{a}\cos(b)+\mathrm {i} \,\mathrm {e} ^{a}\sin(b)\qquad (z=a+b\mathrm {i} )}$
${\displaystyle \ln(z):=\ln(|z|)+\operatorname {arg} (z)\,\mathrm {i} \qquad (z\neq 0)}$
${\displaystyle z^{w}:=\mathrm {e} ^{w\ln(z)}\qquad (z\neq 0)}$

Für alle ${\displaystyle z,z_{1},z_{2}\in \mathbb {C} }$ gilt:

${\displaystyle \mathrm {e} ^{z_{1}+z_{2}}=\mathrm {e} ^{z_{1}}\mathrm {e} ^{z_{2}}}$
${\displaystyle \mathrm {e} ^{-z}={\frac {1}{\mathrm {e^{z}} }}}$
${\displaystyle \mathrm {e} ^{z}\neq 0}$
${\displaystyle \mathrm {e} ^{\mathrm {i} z}=\cos(z)+\mathrm {i} \sin(z)}$

${\displaystyle \forall k\in \mathbb {Z} \colon \;\mathrm {e} ^{2k\pi \mathrm {i} }=1}$

${\displaystyle \forall k\in \mathbb {Z} \colon \;\mathrm {e} ^{(2k+1)\pi \mathrm {i} }+1=0}$

Für alle ${\displaystyle z\in \mathbb {C} \setminus \{0\}}$ und ${\displaystyle x,y\in \mathbb {C} }$ gilt:

${\displaystyle z^{x+y}=z^{x}z^{y}}$
${\displaystyle z^{x-y}={\frac {z^{x}}{z^{y}}}}$
${\displaystyle z^{-x}={\frac {1}{z^{x}}}}$
${\displaystyle z^{0}=1}$

Für alle ${\displaystyle r>0}$, ${\displaystyle z\in \mathbb {C} \setminus \{0\}}$ und ${\displaystyle x\in \mathbb {C} }$ gilt:

${\displaystyle (rz)^{x}=r^{x}z^{x}}$
${\displaystyle {\Big (}{\frac {z}{r}}{\Big )}^{x}={\frac {z^{x}}{r^{x}}}}$
${\displaystyle {\Big (}{\frac {1}{r}}{\Big )}^{x}={\frac {1}{r^{x}}}=r^{-x}}$

Für alle ${\displaystyle r>0}$, ${\displaystyle z\in \mathbb {C} \setminus \{x\in \mathbb {R} \mid x\leq 0\}}$ und ${\displaystyle x\in \mathbb {C} }$ gilt:

${\displaystyle {\Big (}{\frac {r}{z}}{\Big )}^{x}={\frac {r^{x}}{z^{x}}}}$

## Wurzeln

Sei ${\displaystyle \varphi :=\arg(z)}$. Für alle ${\displaystyle n\in \mathbb {N} }$ gilt:

${\displaystyle z=w^{n}\iff w={\sqrt[{n}]{|z|}}\exp {\Big (}{\frac {\mathrm {i} \varphi +2k\pi \mathrm {i} }{n}}{\Big )},\;k\in \{0,1,\ldots ,n-1\}.}$

Hauptwert:

${\displaystyle {\sqrt[{n}]{z}}={\sqrt[{n}]{|z|}}\exp {\Big (}{\frac {\mathrm {i} \varphi }{n}}{\Big )}.}$

Hauptwert, allgemein für ${\displaystyle z,x\in \mathbb {C} \setminus \{0\}}$:

${\displaystyle {\sqrt[{x}]{z}}:=\exp {\bigg (}{\frac {\ln(z)}{x}}{\bigg )}.}$

## Logarithmen

Definitionen:

${\displaystyle \ln(z):=\ln(|z|)+\operatorname {arg} (z)\,\mathrm {i} \qquad (z\neq 0)}$
${\displaystyle \log _{b}(a):={\frac {\ln(a)}{\ln(b)}}\qquad (a,b\in \mathbb {C} \setminus \{0\})}$

Logarithmus als Urbild der Exponentialfunktion:

${\displaystyle \operatorname {Ln} (z):=\{w\mid \exp(z)=w\}}$
${\displaystyle \operatorname {Ln} (z)=\{w\mid w=\ln(z)+2k\pi \mathrm {i} ,\;k\in \mathbb {Z} \}}$

Für alle ${\displaystyle r>0}$ und ${\displaystyle z\in \mathbb {C} \setminus \{0\}}$ gilt:

${\displaystyle \ln(rz)=\ln(r)+\ln(z)}$

Für alle ${\displaystyle z\in \mathbb {C} \setminus \{x\in \mathbb {R} \mid x\leq 0\}}$ gilt:

${\displaystyle \ln {\Big (}{\frac {1}{z}}{\Big )}=-\ln(z)}$

Für alle ${\displaystyle x,y\in \mathbb {C} \setminus \{0\}}$ gilt:

${\displaystyle \ln(xy)\equiv \ln(x)+\ln(y)\quad {\pmod {2\pi \mathrm {i} }}}$

Für alle ${\displaystyle z\in \mathbb {C} \setminus \{0\}}$ und ${\displaystyle x\in \mathbb {C} }$ gilt:

${\displaystyle \ln(z^{x})\equiv x\ln(z)\quad {\pmod {2\pi \mathrm {i} }}}$

## Aufgaben

### Aufgabe 1

Ist ${\displaystyle \alpha \,}$ eine fest vorgegebene komplexe Zahl und ist ${\displaystyle z\,}$ eine komplexe Variable, so gilt ${\displaystyle \left|z^{\alpha }\right|\in \Theta \left(|z|^{{\text{Re}}\,\alpha }\right)}$ für ${\displaystyle |z|\to \infty \,}$. (${\displaystyle \Theta }$: Landau-Symbol)

### Aufgabe 2

Sind ${\displaystyle z_{1},z_{2}\,}$ komplexe Zahlen mit positivem Realteil und ist ${\displaystyle \alpha \,}$ irgendeine komplexe Zahl, so ist ${\displaystyle (z_{1}\cdot z_{2})^{\alpha }=z_{1}^{\alpha }\cdot z_{2}^{\alpha }}$ und ${\displaystyle \left({\frac {z_{1}}{z_{2}}}\right)^{\alpha }={\frac {z_{1}^{\alpha }}{z_{2}^{\alpha }}}}$.

### Aufgabe 3

Ist ${\displaystyle z\,}$ eine komplexe Zahl, so ist ${\displaystyle 0^{z}=\left\{{\begin{matrix}0&{\text{Re}}(z)>0\\1&z=0\\{\text{nicht definiert}}&{\text{sonst}}\end{matrix}}\right.}$.

### Aufgabe 4

${\displaystyle (a^{2}+b^{2})(c^{2}+d^{2})=(ac-bd)^{2}+(ad+bc)^{2}\,}$

### Aufgabe 5

${\displaystyle {\sqrt {a+ib}}={\sqrt {\frac {{\sqrt {a^{2}+b^{2}}}+a}{2}}}+i\,\Theta (b)\,{\sqrt {\frac {{\sqrt {a^{2}+b^{2}}}-a}{2}}}}$   , mit ${\displaystyle \Theta (b)=\left\{{\begin{matrix}1&,&b\geq 0\\\\-1&,&b<0\end{matrix}}\right.}$

Vergleich verschiedener Darstellungen zum Thema bei Wikibooks

Die komplexen Zahlen werden in folgenden Büchern von Wikibooks behandelt:

Einzelne Kapitel anderer Bücher richten sich an bestimmte Zielgruppen: