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Display information for equation id:math.1444.228 on revision:1444

* Page found: Materie in elektrischen und magnetischen Feldern (eq math.1444.228)

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\begin{align}
& \frac{1}{2\pi }\int_{-\infty }^{\infty }{{}}d\omega {{\varepsilon }_{0}}\hat{\chi }\left( \omega  \right)\int_{-\infty }^{\infty }{{}}dt\acute{\ }{{e}^{+i\omega \left( t\acute{\ }-t \right)}}:=\frac{{{\varepsilon }_{0}}}{\sqrt{2\pi }}\chi \left( t-t\acute{\ } \right) \\
& \Rightarrow \bar{P}\left( \bar{r},t \right)=\frac{1}{2\pi }\int_{-\infty }^{\infty }{{}}d\omega {{\varepsilon }_{0}}\hat{\chi }\left( \omega  \right)\int_{-\infty }^{\infty }{{}}dt\acute{\ }\bar{E}\left( \bar{r},t\acute{\ } \right){{e}^{+i\omega \left( t\acute{\ }-t \right)}}=\frac{{{\varepsilon }_{0}}}{\sqrt{2\pi }}\int_{-\infty }^{t}{{}}dt\acute{\ }\chi \left( t-t\acute{\ } \right)\bar{E}\left( \bar{r},t\acute{\ } \right) \\
\end{align}

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1 2 ⁢ π ⁢ ∫ - ∞ ∞ 𝑑 ω ⁢ ε 0 ⁢ χ ^ ⁢ ( ω ) ⁢ ∫ - ∞ ∞ 𝑑 t ⁢ ´ ⁢ e + i ⁢ ω ⁢ ( t ⁢ ´ - t ) := ε 0 2 ⁢ π ⁢ χ ⁢ ( t - t ⁢ ´ ) ⇒ P ¯ ⁢ ( r ¯ , t ) = 1 2 ⁢ π ⁢ ∫ - ∞ ∞ 𝑑 ω ⁢ ε 0 ⁢ χ ^ ⁢ ( ω ) ⁢ ∫ - ∞ ∞ 𝑑 t ⁢ ´ ⁢ E ¯ ⁢ ( r ¯ , t ⁢ ´ ) ⁢ e + i ⁢ ω ⁢ ( t ⁢ ´ - t ) = ε 0 2 ⁢ π ⁢ ∫ - ∞ t 𝑑 t ⁢ ´ ⁢ χ ⁢ ( t - t ⁢ ´ ) ⁢ E ¯ ⁢ ( r ¯ , t ⁢ ´ ) missing-subexpression assign 1 2 𝜋 superscript subscript differential-d 𝜔 subscript 𝜀 0 ^ 𝜒 𝜔 superscript subscript differential-d 𝑡 ´ absent superscript 𝑒 𝑖 𝜔 𝑡 ´ absent 𝑡 subscript 𝜀 0 2 𝜋 𝜒 𝑡 𝑡 ´ absent missing-subexpression ⇒ absent ¯ 𝑃 ¯ 𝑟 𝑡 1 2 𝜋 superscript subscript differential-d 𝜔 subscript 𝜀 0 ^ 𝜒 𝜔 superscript subscript differential-d 𝑡 ´ absent ¯ 𝐸 ¯ 𝑟 𝑡 ´ absent superscript 𝑒 𝑖 𝜔 𝑡 ´ absent 𝑡 subscript 𝜀 0 2 𝜋 superscript subscript 𝑡 differential-d 𝑡 ´ absent 𝜒 𝑡 𝑡 ´ absent ¯ 𝐸 ¯ 𝑟 𝑡 ´ absent {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\frac{1}{2\pi}\int_{% -\infty}^{\infty}{{}}d\omega{{\varepsilon}_{0}}\hat{\chi}\left(\omega\right)% \int_{-\infty}^{\infty}{{}}dt\acute{\ }{{e}^{+i\omega\left(t\acute{\ }-t\right% )}}:=\frac{{{\varepsilon}_{0}}}{\sqrt{2\pi}}\chi\left(t-t\acute{\ }\right)\\ &\displaystyle\Rightarrow\bar{P}\left(\bar{r},t\right)=\frac{1}{2\pi}\int_{-% \infty}^{\infty}{{}}d\omega{{\varepsilon}_{0}}\hat{\chi}\left(\omega\right)% \int_{-\infty}^{\infty}{{}}dt\acute{\ }\bar{E}\left(\bar{r},t\acute{\ }\right)% {{e}^{+i\omega\left(t\acute{\ }-t\right)}}=\frac{{{\varepsilon}_{0}}}{\sqrt{2% \pi}}\int_{-\infty}^{t}{{}}dt\acute{\ }\chi\left(t-t\acute{\ }\right)\bar{E}% \left(\bar{r},t\acute{\ }\right)\\ \end{aligned}}}

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12π∫−∞∞dωε0χ^(ω)∫−∞∞dt´e+iω(t´−t):=ε02πχ(t−t´)⇒P¯(r¯,t)=12π∫−∞∞dωε0χ^(ω)∫−∞∞dt´E¯(r¯,t´)e+iω(t´−t)=ε02π∫−∞tdt´χ(t−t´)E¯(r¯,t´)

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