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3.5 Logarithmus, Hauptwert, $\delta_{\pm }^{}$

Für den komplexen Logarithmus ist

$\displaystyle \lim_{\varepsilon\rightarrow 0}^{}$ ln(x $\displaystyle \pm$ i $\displaystyle \varepsilon$ ) = ln| x| $\displaystyle \pm$ i $\displaystyle \pi$ $\displaystyle \Theta$ (x)

(3.17)

Durch diffenenzieren folgt:

$\displaystyle \lim_{\varepsilon\rightarrow 0}^{}$ $\displaystyle {1 \over {x \pm i\varepsilon}}$ = P $\displaystyle {1 \over x}$ $\displaystyle \mp$ i $\displaystyle \pi$ $\displaystyle \delta$ (x) = : $\displaystyle \mp$ 2 $\displaystyle \pi$ i $\displaystyle \delta_{\pm}^{}$ (x)

(3.18)

Dabei bedeutet P den Cauchy'schen Hauptwert:

(P $\displaystyle {1 \over x}$ , $\displaystyle \varphi$ (x)) = $\displaystyle \lim_{\varepsilon\rightarrow 0}^{}$ $\displaystyle \left(\vphantom{\int_{-\infty}^{-\varepsilon} + \int_{\varepsilon}^{\infty}}\right.$ $\displaystyle \int_{-\infty}^{-\varepsilon}$ + $\displaystyle \int_{\varepsilon}^{\infty}$ $\displaystyle \left.\vphantom{\int_{-\infty}^{-\varepsilon} + \int_{\varepsilon}^{\infty}}\right)$ $\displaystyle {\varphi(x) \over x}$ dx

(3.19)

$\displaystyle \delta_{\pm}^{}$ (- x) = $\displaystyle \delta_{\mp}^{}$ (x)

(3.20)

x $\displaystyle \delta_{\pm}^{}$ (x) = $\displaystyle \mp$ $\displaystyle {1 \over {2 \pi i}}$

(3.21)

$\displaystyle \delta_{+}^{}$ (x) + $\displaystyle \delta_{-}^{}$ (x) = $\displaystyle {1 \over {2 \pi i}}$ $\displaystyle \lim_{\varepsilon\rightarrow 0}^{}$ ( $\displaystyle {1 \over {x-i \varepsilon}}$ - $\displaystyle {1 \over {x+i \varepsilon}}$ ) = $\displaystyle \delta$ (x)

(3.22)

$\displaystyle \delta_{+}^{}$ (x) - $\displaystyle \delta_{-}^{}$ (x) = - $\displaystyle {1 \over {2 \pi i}}$ $\displaystyle \lim_{\varepsilon\rightarrow 0}^{}$ ( $\displaystyle {1 \over {x-i \varepsilon}}$ + $\displaystyle {1 \over {x+i \varepsilon}}$ ) = $\displaystyle {i \over \pi}$ P $\displaystyle {1 \over x}$

(3.23)

P $\displaystyle {1 \over x}$ = $\displaystyle \lim_{\varepsilon\rightarrow 0}^{}$ $\displaystyle {x \over {x^2+\varepsilon^2}}$

(3.24)

Alexander Wagner
2000-04-14