next up previous contents
Next: 4. Orthogonale Funktionensysteme Up: 3. Die Dirac'sche -Funktion Previous: 3.5 Logarithmus, Hauptwert,   Inhalt

3.6 Fourierintegrale

$\displaystyle \delta$(x) = $\displaystyle {1 \over {2 \pi}}$$\displaystyle \int_{-\infty}^{+\infty}$eikxdk (3.25)

$\displaystyle \delta_{\pm}^{}$(x) = $\displaystyle {1 \over {2 \pi}}$$\displaystyle \int_{-\infty}^{+\infty}$eikx$\displaystyle \Theta$($\displaystyle \pm$k)dk (3.26)

P$\displaystyle {1
\over x}$ = $\displaystyle {1 \over {2 i}}$$\displaystyle \int_{-\infty}^{+\infty}$eikx$\displaystyle \varepsilon$(k)dk (3.27)

$\displaystyle \Theta$($\displaystyle \pm$x) = $\displaystyle \pm$$\displaystyle {1 \over {2 \pi i}}$$\displaystyle \int_{-\infty}^{+\infty}$$\displaystyle {{e^{i k x}} \over {k \mp i \varepsilon}}$, wobei $\displaystyle \varepsilon$(x) = $\displaystyle {1 \over {i \pi}}$$\displaystyle \int_{-\infty}^{+\infty}$eikxP$\displaystyle {1 \over k}$dk (3.28)

$\displaystyle \delta{^\prime}$(x) = $\displaystyle \int_{-\infty}^{+\infty}$ikeikxdk etc. (3.29)

$\displaystyle \delta$($\displaystyle \vec{x}\,$) = $\displaystyle {1 \over {(2\pi)^3}}$$\displaystyle \int$ei$\scriptstyle \vec{k}\,$$\scriptstyle \vec{x}\,$d3k (3.30)

$\displaystyle \delta$(r - a) = $\displaystyle {a \over {2 \pi^2}}$$\displaystyle \int$ei$\scriptstyle \vec{k}\,$$\scriptstyle \vec{x}\,$$\displaystyle {\sin(a k) \over k}$d3k mit k = $\displaystyle \sqrt{\vec{k}^2}$ (3.31)



Alexander Wagner
2000-04-14