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Next: 3.3 Formeln mit Up: 3. Die Dirac'sche -Funktion Previous: 3.1 Limesdarstellungen   Inhalt

3.2 Die $ \delta$-Funktion als Ableitung

Stufenfunktionen:

$\displaystyle \Theta$(x) = $\displaystyle \left\{\vphantom{
\begin{array}{ll}
1 & x>0 \\
0 & x<0 \\
\end{array}}\right.$$\displaystyle \begin{array}{ll}
1 & x>0 \\
0 & x<0 \\
\end{array}$ $\displaystyle \left.\vphantom{
\begin{array}{ll}
1 & x>0 \\
0 & x<0 \\
\end{array}}\right.$ (3.6)

$\displaystyle \varepsilon$(x) = $\displaystyle \left\{\vphantom{
\begin{array}{ll}
1 & x>0 \\
-1 & x<0 \\
\end{array}}\right.$$\displaystyle \begin{array}{ll}
1 & x>0 \\
-1 & x<0 \\
\end{array}$ $\displaystyle \left.\vphantom{
\begin{array}{ll}
1 & x>0 \\
-1 & x<0 \\
\end{array}}\right.$ (3.7)

Verschiedene Darstellungen der $ \delta$-Funktion:

$\displaystyle \varepsilon$(x) = $\displaystyle \Theta$(x) - $\displaystyle \Theta$(- x) = $\displaystyle {d \over dx}$| x| (3.8)

$\displaystyle \Theta$(x) = $\displaystyle {1 \over 2}$(1 + $\displaystyle \varepsilon$(x)) (3.9)

$\displaystyle \delta$(x)     =    $\displaystyle {d \over dx}$$\displaystyle \Theta$(x)     =    $\displaystyle {1 \over 2}$$\displaystyle {d \over dx}$$\displaystyle \varepsilon$(x)     =    $\displaystyle {1 \over 2}$$\displaystyle {d^2 \over dx^2}$| x| (3.10)



Alexander Wagner
2000-04-14