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 8: | Line 26: | ||
<math>U = \pi \cdot D = 2 \cdot \pi \cdot r</math> | <math>U = \pi \cdot D = 2 \cdot \pi \cdot r</math> | ||
| + | |||
| + | Satz des Pythagoras: | ||
| + | |||
| + | <math>a^2+b^2=c^2</math> | ||
== komplexe Zahlen == | == komplexe Zahlen == | ||
| + | '''Normalform''' | ||
{| | {| | ||
|<math>z = x\, +\, \mathrm i y \quad</math> | |<math>z = x\, +\, \mathrm i y \quad</math> | ||
| - | |<math>\mathrm i</math>:=<math>\sqrt{-1} | + | |<math>\mathrm i</math>:=<math>\sqrt{-1} \qquad i^2 = -1</math> |
| - | + | ||
|- | |- | ||
|<math>x = \operatorname{Re}\, z</math> | |<math>x = \operatorname{Re}\, z</math> | ||
|<math>y = \operatorname{Im}\, z</math> | |<math>y = \operatorname{Im}\, z</math> | ||
| + | |- | ||
| + | |Summe | ||
| + | |<math>z_1 + z_2 = (x_1 + x_2) + \mathrm i(y_1 + y_2)</math> | ||
| + | |- | ||
| + | |Produkt | ||
| + | |<math>z_1 \cdot z_2 = (x_1x_2-y_1y_2) + \mathrm i(x_1y_2 + x_2_y_1)</math> | ||
|} | |} | ||
| - | + | '''Polarform''': | |
| + | {| | ||
| + | |<math>z = |z| \cdot E(\varphi)</math> | ||
| + | |<math>E(\varphi) = |z| \cdot(\cos\varphi + \mathrm i\,\sin\varphi)</math> | ||
| + | |- | ||
| + | |<math>\varphi = arc(z)</math> | ||
| + | |<math>\tan\varphi = \frac y x</math> | ||
| + | |- | ||
| + | |Produkt: | ||
| + | |<math>z_1 \cdot z_2 = |z_1||z_2| \cdot E(\varphi_1+\varphi_2) = |z_1||z_2| (\cos(\varphi_1+\varphi2) + \mathrm i\, \sin(\varphi_1+\varphi_2))</math> | ||
| + | |- | ||
| + | |Quotient: | ||
| + | |<math>\frac{z_1}{z_2} = \frac{|z_1|}{|z_2|} E(\varphi_1-\varphi_2) = \frac{|z_1|}{|z_2|} (\cos(\varphi_1-\varphi2) + \mathrm i\, \sin(\varphi_1-\varphi_2))</math> | ||
| + | |- | ||
| + | |Potenz: | ||
| + | |<math>{z}^n = |z|^n E(n\, \varphi) = |z|^n (\cos\, n \varphi + \mathrm i\,\sin\, n \varphi)</math> | ||
| + | |} | ||
| - | <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> | ||
| + | |} | ||
[[category:Mathematik]] | [[category:Mathematik]] | ||
[[category:remember]] | [[category:remember]] | ||
Current revision
Analysis
Binome, Trinome
Geometrie
Satz des Pythagoras:
komplexe Zahlen
Normalform
| Summe | |
| Produkt |
Polarform:
| Produkt: | |
| Quotient: | |
| Potenz: |
-Rechenregeln:
konjungierte komplexe Zahlen:
