Mathematik
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== Analysis == | == Analysis == | ||
| - | <math>a \cdot x^2 + b \cdot x + c = 0 \Rightarrow x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4\cdot a\cdot c}} {2\cdot a}</math> | + | <math>a \cdot x^2 + b \cdot x + c = 0 \quad \Rightarrow \quad x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4\cdot a\cdot c}} {2\cdot a}</math> |
| + | '''Binome, Trinome''' | ||
| + | |||
| + | {| | ||
| + | |<math>(a+b)^2=a^2+2ab+b^2</math> | ||
| + | |<math>(a+b)^3=a^3+3a^2b+3ab^2+b^3</math> | ||
| + | |- | ||
| + | |<math>(a-b)^2=a^2-2ab+b^2</math> | ||
| + | |<math>(a-b)^3=a^3-3a^2b+3ab^2-b^3</math> | ||
| + | |- | ||
| + | |<math>(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc</math> | ||
| + | |<math>a^3+b^3=(a+b)(a^2-ab+b^2)</math> | ||
| + | |- | ||
| + | |<math>a^2-b^2=(a+b)(a-b)</math> | ||
| + | |<math>a^3-b^3=(a-b)(a^2+ab+b^2)</math> | ||
| + | |} | ||
| + | <math>a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\cdots+ab^{n-2}+b^{n-1})</math> | ||
| + | |||
| + | <math>n!=1\cdot2\cdot3\cdot...\cdot(n-1)\cdot n,\ n\in N\setminus\{1\}</math> | ||
== Geometrie == | == Geometrie == | ||
| Line 9: | Line 27: | ||
| - | + | Satz des Pythagoras: | |
| - | + | <math>a^2+b^2=c^2</math> | |
| + | |||
| + | == komplexe Zahlen == | ||
| + | '''Normalform''' | ||
{| | {| | ||
|<math>z = x\, +\, \mathrm i y \quad</math> | |<math>z = x\, +\, \mathrm i y \quad</math> | ||
| Line 27: | Line 48: | ||
|} | |} | ||
| - | Polarform: | + | '''Polarform''': |
| - | + | ||
{| | {| | ||
|<math>z = |z| \cdot E(\varphi)</math> | |<math>z = |z| \cdot E(\varphi)</math> | ||
| Line 44: | Line 64: | ||
|Potenz: | |Potenz: | ||
|<math>{z}^n = |z|^n E(n\, \varphi) = |z|^n (\cos\, n \varphi + \mathrm i\,\sin\, n \varphi)</math> | |<math>{z}^n = |z|^n E(n\, \varphi) = |z|^n (\cos\, n \varphi + \mathrm i\,\sin\, n \varphi)</math> | ||
| + | |} | ||
| + | |||
| + | '''<math>E(\varphi)</math>-Rechenregeln:''' | ||
| + | {| | ||
| + | |<math>E(\varphi+k\cdot2\phi) = E(\varphi)</math> | ||
| + | |<math>|E(\varphi)| = 1</math> | ||
| + | |- | ||
| + | |<math>E(\varphi_1)\cdot E(\varphi_2) = E(\varphi_1 + \varphi_2)</math> | ||
| + | |<math>E(\varphi)\cdot E(-\varphi) = 1</math> | ||
| + | |- | ||
| + | |<math>[E(\varphi)]^n = E(N\, \varphi)</math> | ||
| + | |- | ||
| + | |<math>E(\varphi) = e^{\mathrm i\varphi}</math> | ||
| + | |<math>e^{2\pi\mathrm i} = 1</math> | ||
| + | |} | ||
| + | |||
| + | '''konjungierte komplexe Zahlen''': | ||
| + | |||
| + | {| | ||
| + | |<math>\bar z = \bar{x + \mathrm i y} = x - \mathrm i y</math> | ||
| + | |- | ||
| + | |<math>\bar{z_1+z_2} = \bar{z_1} + \bar{z_2}</math> | ||
| + | |<math>\bar{z_1-z_2} = \bar{z_1} - \bar{z_2}</math> | ||
| + | |- | ||
| + | |<math>\bar{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}</math> | ||
| + | |<math>\bar{\frac{z_1}{z_2}} = \frac{\bar{z_1}}{\bar{z_2}}</math> | ||
| + | |- | ||
| + | |<math>\bar{z^n} = (\bar z)^n</math> | ||
| + | |<math>z \cdot \bar z = |z|^2</math> | ||
| + | |<math>\bar{(\bar z)} = z</math> | ||
| + | |- | ||
| + | |<math>z + \bar z = 2 \cdot \operatorname Re(z)</math> | ||
| + | |<math>z - \bar z = 2\mathrm i \cdot \operatorname Im(z)</math> | ||
|} | |} | ||
Current revision
Analysis
Binome, Trinome
Geometrie
Satz des Pythagoras:
komplexe Zahlen
Normalform
| Summe | |
| Produkt |
Polarform:
| Produkt: | |
| Quotient: | |
| Potenz: |
-Rechenregeln:
konjungierte komplexe Zahlen:
