Differentialrechnung

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		+	== Ableitungen/Integrale ==
		+
	{|border=1		{|border=1
	|'''Funktion <math>f(x)</math>'''		|'''Funktion <math>f(x)</math>'''
-	|'''Ableitung <math>f'(x)=\frac{d f(x)}{d x}d x</math>'''	+	|'''Ableitung <math>f'(x)=\frac{d f(x)}{d x}</math>'''
-	|'''Integral <math>F(x)=\int f(x)</math>'''	+	|'''Integral <math>F(x)=\int f(x) d x</math>'''
		+	|-
		+	|<math>b</math>
		+	|<math>0</math>
		+	|<math>0+C</math>
		+	|-
		+	|<math>a \cdot x + b</math>
		+	|<math>a</math>
		+	|<math>\frac 1 2 \cdot x^2</math>
		+	|-
		+	|<math>x^n</math>
		+	|<math>n \cdot x^{n-1}</math>
		+	|<math>\frac 1{n+1} \cdot x^{n+1}+C,\ n\ne-1</math>
		+	|-
		+	|<math>\frac 1 x</math>
		+	|<math>-\frac 1 {x^2}</math>
		+	|<math>\ln\, \left|x\right|+C</math>
		+	|-
		+	|<math>\frac 1{ax+b}</math>
		+	|
		+	|<math>\frac 1 a \ln|ax+b|+C</math>
		+	|-
		+	|<math>e^x</math>
		+	|<math>e^x</math>
		+	|<math>e^x+C</math>
		+	|-
		+	|<math>a^x</math>
		+	|<math>(\ln\,a) \cdot a^x</math>
		+	|<math>\frac 1{\ln a} \cdot a^x+C</math>
		+	|-
		+	|<math>\ln x</math>
		+	|<math>\frac 1 x</math>
		+	|<math>-x + x\,\ln\,x+C</math>
		+	|-
		+	|<math>\log_a x</math>
		+	|<math>\frac 1 {\ln a \cdot x} = log_a e \cdot \frac 1 x</math>
	|-		|-
	|<math>\sin x</math>		|<math>\sin x</math>
Line 14:		Line 51:
	|<math>\tan x</math>		|<math>\tan x</math>
	|<math>\frac 1 {\cos^2 x} = 1+\tan^2 x</math>		|<math>\frac 1 {\cos^2 x} = 1+\tan^2 x</math>
		+	|<math>-\ln|\cos\,x|+C</math>
	|-		|-
	|<math>\cot x</math>		|<math>\cot x</math>
	|<math>-\frac 1 {\sin^2 x} = -(1+\cot^2 x)</math>		|<math>-\frac 1 {\sin^2 x} = -(1+\cot^2 x)</math>
		+	|<math>\ln|\sin\,x|+C</math>
	|-		|-
	|<math>\arcsin x</math>		|<math>\arcsin x</math>
-	|<math>\frac 1 {1-x^2}</math>	+	|<math>\frac 1 {\sqrt{1-x^2}}</math>
-	|<math></math>	+
	|-		|-
	|<math>\arccos x</math>		|<math>\arccos x</math>
-	|<math>-\frac 1 {1-x^2}</math>	+	|<math>-\frac 1 {\sqrt{1-x^2}}</math>
-	|<math></math>	+
	|-		|-
	|<math>\arctan x</math>		|<math>\arctan x</math>
	|<math>\frac 1 {1+x^2}</math>		|<math>\frac 1 {1+x^2}</math>
-	|<math></math>
	|-		|-
	|<math>\operatorname{arccot} x</math>		|<math>\operatorname{arccot} x</math>
	|<math>-\frac 1 {1+x^2}</math>		|<math>-\frac 1 {1+x^2}</math>
-	|<math></math>
	|-		|-
-	|<math>e^x</math>	+	|<math>\sinh x</math>
-	|<math>e^x</math>	+	|<math>\cosh x</math>
-	|<math>e^x+C</math>	+	|<math>\cosh x+C</math>
	|-		|-
-	|<math>a^x</math>	+	|<math>\cosh x</math>
-	|<math>(\ln\,a) \cdot a^x</math>	+	|<math>\sinh x</math>
-	|<math></math>	+	|<math>\sinh x+C</math>
	|-		|-
-	|<math></math>	+	|<math>\tanh x</math>
-	|<math></math>	+	|<math>\frac 1 {\cosh^2 x} = 1-\tanh^2 x</math>
-	|<math></math>	+	|<math>\ln\,\cosh x+C</math>
	|-		|-
-	|<math></math>	+	|<math>\coth x</math>
-	|<math></math>	+	|<math>-\frac 1 {sinh^2 x} = 1-coth^2 x</math>
-	|<math></math>	+	|<math>\ln\,\left|\sinh x\right|+C</math>
	|-		|-
-	|<math></math>	+	|<math>\operatorname{arsinh} x</math>
-	|<math></math>	+	|<math>\frac 1 \sqrt{x^2+1}</math>
-	|<math></math>	+
	|-		|-
-	|<math></math>	+	|<math>\operatorname{arcosh} x</math>
-	|<math></math>	+	|<math>\frac 1 \sqrt{x^2-1}</math>
-	|<math></math>	+
	|-		|-
-	|<math></math>	+	|<math>\operatorname{artanh} x</math>
-	|<math></math>	+	|<math>\frac 1 {1-x^2}</math>
-	|<math></math>	+	|-
		+	|<math>\operatorname{arcoth} x</math>
		+	|<math>\frac 1 {1-x^2}</math>
	|}		|}
		+
		+	siehe [[Integration]]
		+
		+	== Integralfunktionen ==
		+	{|
		+	|Integralsinus
		+	|<math>Si(x) = \int_0^x\frac{\sin\,t}t dt = C+\ln|x|-\frac{x^2}{2\cdot 2!}+\frac{x^4}{4\cdot 4!}-\frac{x^6}{6\cdot 6!} + - \cdots,\ |x|<\infty</math>
		+	|Eulersche Konstante <math>C =0,5772...</math>
		+	|-
		+	|Integralcosinus
		+	|<math>Ci(x) = \int_x^\infty\frac{\cos\,t}t dt = C+\ln|x|+\frac{x}{1\cdot 1!}+\frac{x^2}{2\cdot 2!}+\frac{x^3}{3\cdot 3!} + \cdots,\ |x|<\infty</math>
		+	|-
		+	|Exponentialintegral
		+	|<math>Ei(x) = \int_{-\infty}^x\frac{e^t}t dt = C+\ln|x|+\frac x{1\cdot 1!}+\frac{x^2}{2\cdot 2!}+\frac{x^3}{3\cdot 3!} + \cdots,\ x<0</math>
		+	|-
		+	|Integrallogarithmus
		+	|<math>Li(x) = \int_0^x\frac{dt}{\ln\,t}t = \int_{-\infty}^x\frac{e^t}t dt = C+\ln\left|\ln x\right|+\frac{\ln x}{1\cdot 1!}+\frac{(\ln\,x)^2}{2\cdot 2!}-\frac{(\ln\,x)^3}{3\cdot 3!} + \cdots,\ 0<x<\infty</math>
		+	|-
		+	|Gaußsches Fehlerintegral
		+	|<math>F(x) = \frac 1{\sqrt{2\pi}} \int_{-\infty}^xe^{-\frac{t^2} 2} dt = \frac 1 2 + \frac 1{\sqrt{2\pi}} \left(\frac x 1-\frac{x^3}{2\cdot3\cdot1!}+\frac{x^5}{2^2\cdot5\cdot2!}-\frac{x^7}{2^3\cdot7\cdot3!}+-\cdots\right),\ |x|<\infty</math>
		+	|}
		+
		+
		+	== [http://de.wikipedia.org/wiki/Fourier-Transformation Fourier-Transformation] ==
		+
		+	Bildfunktion <math>F(j\omega) = \mathcal F\{f(t)\} = \int_{-\infty}^\infty f(t)\, e^{-j\omega t} dt</math>
		+
		+
		+	== [http://de.wikipedia.org/wiki/Laplace-Transformation Laplace-Transformation] ==
		+
		+	Originalfunktion, Oberfunktion:
		+
		+	<math>f(t) = \mathcal L^{-1}\{f(t)\} = \frac 1 {2j\pi} \int_{\beta-j\omega}^{\beta+j\omega} F(s)\, e^{s t} ds</math>
		+
		+	Laplace-Transformierte, Unterfunktion, Bildfunktion:
		+
		+	<math>F(s) = \mathcal L\{f(t)\} = \int_0^\infty f(t)\, e^{-s t} dt</math>
		+	[[category:remember]]
	[[category:Mathematik]]		[[category:Mathematik]]

---

Current revision

Contents 1 Ableitungen/Integrale 2 Integralfunktionen 3 Fourier-Transformation 4 Laplace-Transformation

Ableitungen/Integrale

Funktion	Ableitung	Integral

siehe Integration

Integralfunktionen

Integralsinus		Eulersche Konstante
Integralcosinus
Exponentialintegral
Integrallogarithmus
Gaußsches Fehlerintegral

Fourier-Transformation

Bildfunktion

Laplace-Transformation

Originalfunktion, Oberfunktion:

Laplace-Transformierte, Unterfunktion, Bildfunktion: