Mathematik

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Analysis

$MathImage$

Binome, Trinome

$MathImage$	$MathImage$
$MathImage$	$MathImage$
$MathImage$	$MathImage$
$MathImage$	$MathImage$

$MathImage$

Geometrie

$MathImage$

Satz des Pythagoras:

$MathImage$

komplexe Zahlen

Normalform

$MathImage$	$MathImage$ := $MathImage$
$MathImage$	$MathImage$
Summe	$MathImage$
Produkt	$MathImage$

Polarform:

$MathImage$	$MathImage$
$MathImage$	$MathImage$
Produkt:	$MathImage$
Quotient:	$MathImage$
Potenz:	$MathImage$

$MathImage$ -Rechenregeln:

$MathImage$	$MathImage$
$MathImage$	$MathImage$
$MathImage$
$MathImage$	$MathImage$

konjungierte komplexe Zahlen:

$MathImage$
$MathImage$	$MathImage$
$MathImage$	$MathImage$
$MathImage$	$MathImage$	$MathImage$
$MathImage$	$MathImage$

@@ Line 1: / Line 1: @@
-[[category:remember]]
 == Analysis ==
-<math>a \cdot x^2 + b \cdot x + c = 0</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>
-=>
-<math>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 ==
 <math>U = \pi \cdot D = 2 \cdot \pi \cdot r</math>
+Satz des Pythagoras:
+<math>a^2+b^2=c^2</math>
+== komplexe Zahlen ==
+'''Normalform'''
+{|
+|<math>z = x\, +\, \mathrm i y \quad</math>
+|<math>\mathrm i</math>:=<math>\sqrt{-1} \qquad i^2 = -1</math>
+|-
+|<math>x = \operatorname{Re}\, 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>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:remember]]

Mathematik

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