Differentialrechnung

From SkyWiki

(Difference between revisions)
Jump to: navigation, search
m
Current revision (16:52, 20 May 2011) (view source)
m (Reverted edits by SallyJo (Talk) to last version by Martin)
 
(51 intermediate revisions not shown.)
Line 1: Line 1:
 +
== Ableitungen/Integrale ==
 +
{|border=1
{|border=1
|'''Funktion <math>f(x)</math>'''
|'''Funktion <math>f(x)</math>'''
Line 14: Line 16:
|<math>x^n</math>
|<math>x^n</math>
|<math>n \cdot x^{n-1}</math>
|<math>n \cdot x^{n-1}</math>
-
|<math>\frac 1{n+1} \cdot x^{n+1}+C</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>
Line 22: Line 32:
|<math>a^x</math>
|<math>a^x</math>
|<math>(\ln\,a) \cdot 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>\ln x</math>
|<math>\frac 1 x</math>
|<math>\frac 1 x</math>
 +
|<math>-x + x\,\ln\,x+C</math>
|-
|-
|<math>\log_a x</math>
|<math>\log_a x</math>
Line 39: 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>\arccos x</math>
|<math>\arccos x</math>
-
|<math>-\frac 1 {1-x^2}</math>
+
|<math>-\frac 1 {\sqrt{1-x^2}}</math>
|-
|-
|<math>\arctan x</math>
|<math>\arctan x</math>
Line 65: Line 79:
|<math>\tanh x</math>
|<math>\tanh x</math>
|<math>\frac 1 {\cosh^2 x} = 1-\tanh^2 x</math>
|<math>\frac 1 {\cosh^2 x} = 1-\tanh^2 x</math>
 +
|<math>\ln\,\cosh x+C</math>
|-
|-
|<math>\coth x</math>
|<math>\coth x</math>
|<math>-\frac 1 {sinh^2 x} = 1-coth^2 x</math>
|<math>-\frac 1 {sinh^2 x} = 1-coth^2 x</math>
 +
|<math>\ln\,\left|\sinh x\right|+C</math>
|-
|-
|<math>\operatorname{arsinh} x</math>
|<math>\operatorname{arsinh} x</math>
Line 80: Line 96:
|<math>\operatorname{arcoth} x</math>
|<math>\operatorname{arcoth} x</math>
|<math>\frac 1 {1-x^2}</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>
|-
|-
-
|<math></math>
+
|Integralcosinus
-
|<math></math>
+
|<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>
|-
|-
-
|<math></math>
+
|Exponentialintegral
-
|<math></math>
+
|<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>
-
|<math></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

Ableitungen/Integrale

Funktion MathImage Ableitung MathImage Integral MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage
MathImage MathImage
MathImage MathImage
MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage
MathImage MathImage
MathImage MathImage
MathImage MathImage

siehe Integration

Integralfunktionen

Integralsinus MathImage Eulersche Konstante MathImage
Integralcosinus MathImage
Exponentialintegral MathImage
Integrallogarithmus MathImage
Gaußsches Fehlerintegral MathImage


Fourier-Transformation

Bildfunktion MathImage


Laplace-Transformation

Originalfunktion, Oberfunktion:

MathImage

Laplace-Transformierte, Unterfunktion, Bildfunktion:

MathImage

Personal tools