Differentialrechnung
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| + | == 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> | + | |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> |
|- | |- | ||
| - | |<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 | Ableitung | Integral |
siehe Integration
Integralfunktionen
| Integralsinus | Eulersche Konstante | |
| Integralcosinus | ||
| Exponentialintegral | ||
| Integrallogarithmus | ||
| Gaußsches Fehlerintegral |
Fourier-Transformation
Bildfunktion
Laplace-Transformation
Originalfunktion, Oberfunktion:
Laplace-Transformierte, Unterfunktion, Bildfunktion:
