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Display information for equation id:math.1199.783 on revision:1199

* Page found: Elektrodynamik Schöll (eq math.1199.783)

Occurrences on the following pages:

Wellenoptik und Beugung Eq: math.2137.12

Hash: 0cad955e51b2393e659bbf2d88da91ec

TeX (original user input):

\begin{align}
& \Phi \left( \bar{r},t \right)=\int_{{}}^{{}}{{{d}^{3}}r\acute{\ }\int_{-\infty }^{t}{dt\acute{\ }}}\frac{\rho \left( \bar{r}\acute{\ },t\acute{\ } \right)}{{{\varepsilon }_{0}}}G\left( \bar{r}-\bar{r}\acute{\ },t-t\acute{\ } \right)=\int_{{}}^{{}}{{{d}^{3}}r\acute{\ }\int_{-\infty }^{t}{dt\acute{\ }}}\frac{\rho \left( \bar{r}\acute{\ } \right)}{{{\varepsilon }_{0}}}{{e}^{-i\omega t\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ },t-t\acute{\ } \right) \\
& t-t\acute{\ }:=\tau  \\
& \Rightarrow \int_{-\infty }^{t}{dt\acute{\ }}{{e}^{-i\omega t\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ },t-t\acute{\ } \right)=\int_{-\infty }^{t}{dt\acute{\ }}{{e}^{-i\omega t\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ },\tau  \right) \\
& =\left[ \int_{0}^{\infty }{d\tau }{{e}^{i\omega \tau }}G\left( \bar{r}-\bar{r}\acute{\ },\tau  \right) \right]{{e}^{-i\omega t}}:=\tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right){{e}^{-i\omega t}} \\
& \int_{0}^{\infty }{d\tau }{{e}^{i\omega \tau }}G\left( \bar{r}-\bar{r}\acute{\ },\tau  \right):=\tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right) \\
\end{align}

TeX (checked):

{\begin{aligned}&\Phi \left({\bar {r}},t\right)=\int _{}^{}{{{d}^{3}}r{\acute {\ }}\int _{-\infty }^{t}{dt{\acute {\ }}}}{\frac {\rho \left({\bar {r}}{\acute {\ }},t{\acute {\ }}\right)}{{\varepsilon }_{0}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},t-t{\acute {\ }}\right)=\int _{}^{}{{{d}^{3}}r{\acute {\ }}\int _{-\infty }^{t}{dt{\acute {\ }}}}{\frac {\rho \left({\bar {r}}{\acute {\ }}\right)}{{\varepsilon }_{0}}}{{e}^{-i\omega t{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},t-t{\acute {\ }}\right)\\&t-t{\acute {\ }}:=\tau \\&\Rightarrow \int _{-\infty }^{t}{dt{\acute {\ }}}{{e}^{-i\omega t{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},t-t{\acute {\ }}\right)=\int _{-\infty }^{t}{dt{\acute {\ }}}{{e}^{-i\omega t{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},\tau \right)\\&=\left[\int _{0}^{\infty }{d\tau }{{e}^{i\omega \tau }}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},\tau \right)\right]{{e}^{-i\omega t}}:={\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right){{e}^{-i\omega t}}\\&\int _{0}^{\infty }{d\tau }{{e}^{i\omega \tau }}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},\tau \right):={\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\\\end{aligned}}

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\begin{aligned} Φ (\bar{r}, t) = \int_{}^{} d^{3} r \overset{´}{} \int_{- \infty}^{t} d t \overset{´}{} \frac{ρ (\bar{r} \overset{´}{}, t \overset{´}{})}{ε_{0}} G (\bar{r} - \bar{r} \overset{´}{}, t - t \overset{´}{}) = \int_{}^{} d^{3} r \overset{´}{} \int_{- \infty}^{t} d t \overset{´}{} \frac{ρ (\bar{r} \overset{´}{})}{ε_{0}} e^{- i ω t \overset{´}{}} G (\bar{r} - \bar{r} \overset{´}{}, t - t \overset{´}{}) \\ t - t \overset{´}{} : = τ \\ \Rightarrow \int_{- \infty}^{t} d t \overset{´}{} e^{- i ω t \overset{´}{}} G (\bar{r} - \bar{r} \overset{´}{}, t - t \overset{´}{}) = \int_{- \infty}^{t} d t \overset{´}{} e^{- i ω t \overset{´}{}} G (\bar{r} - \bar{r} \overset{´}{}, τ) \\ = [\int_{0}^{\infty} d τ e^{i ω τ} G (\bar{r} - \bar{r} \overset{´}{}, τ)] e^{- i ω t} : = \tilde{G} (\bar{r} - \bar{r} \overset{´}{}) e^{- i ω t} \\ \int_{0}^{\infty} d τ e^{i ω τ} G (\bar{r} - \bar{r} \overset{´}{}, τ) : = \tilde{G} (\bar{r} - \bar{r} \overset{´}{}) \end{aligned}

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Identifiers

$Φ$

$\bar{r}$

$t$

$r$

$\overset{´}{}$

$t$

$t$

$\overset{´}{}$

$ρ$

$\bar{r}$

$\overset{´}{}$

$t$

$\overset{´}{}$

$ε_{0}$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$t$

$t$

$\overset{´}{}$

$r$

$\overset{´}{}$

$t$

$t$

$\overset{´}{}$

$ρ$

$\bar{r}$

$\overset{´}{}$

$ε_{0}$

$e$

$i$

$ω$

$t$

$\overset{´}{}$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$t$

$t$

$\overset{´}{}$

$t$

$t$

$\overset{´}{}$

$τ$

$t$

$t$

$\overset{´}{}$

$e$

$i$

$ω$

$t$

$\overset{´}{}$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$t$

$t$

$\overset{´}{}$

$t$

$t$

$\overset{´}{}$

$e$

$i$

$ω$

$t$

$\overset{´}{}$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$τ$

$τ$

$e$

$i$

$ω$

$τ$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$τ$

$e$

$i$

$ω$

$t$

$\tilde{G}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$e$

$i$

$ω$

$t$

$τ$

$e$

$i$

$ω$

$τ$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$τ$

$\tilde{G}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

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