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* Page found: Wellenoptik und Beugung (eq math.2137.60)

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Wellenoptik und Beugung Eq: math.2137.60

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TeX (original user input):

\begin{align}
& \Rightarrow \Phi \left( \bar{r}\acute{\ } \right)=C\int_{-d/2}^{d/2}{d{{s}_{1}}}{{e}^{ik\alpha {{s}_{1}}}} \\
& \alpha :=\sin {{\vartheta }_{0}} \\
& \bar{\alpha }\bar{s}={{s}_{1}}\sin {{\vartheta }_{0}} \\
& \Rightarrow \Phi \left( \bar{r}\acute{\ } \right)=\frac{C}{ik\alpha }\left( {{e}^{ik\alpha \frac{d}{2}}}-{{e}^{-ik\alpha \frac{d}{2}}} \right) \\
& \Phi \left( \bar{r}\acute{\ } \right)=Cd\frac{\sin \left( k\alpha \frac{d}{2} \right)}{k\alpha \frac{d}{2}} \\
\end{align}

TeX (checked):

{\begin{aligned}&\Rightarrow \Phi \left({\bar {r}}{\acute {\ }}\right)=C\int _{-d/2}^{d/2}{d{{s}_{1}}}{{e}^{ik\alpha {{s}_{1}}}}\\&\alpha :=\sin {{\vartheta }_{0}}\\&{\bar {\alpha }}{\bar {s}}={{s}_{1}}\sin {{\vartheta }_{0}}\\&\Rightarrow \Phi \left({\bar {r}}{\acute {\ }}\right)={\frac {C}{ik\alpha }}\left({{e}^{ik\alpha {\frac {d}{2}}}}-{{e}^{-ik\alpha {\frac {d}{2}}}}\right)\\&\Phi \left({\bar {r}}{\acute {\ }}\right)=Cd{\frac {\sin \left(k\alpha {\frac {d}{2}}\right)}{k\alpha {\frac {d}{2}}}}\\\end{aligned}}

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\begin{aligned} \Rightarrow Φ (\bar{r} \overset{´}{}) = C \int_{- d / 2}^{d / 2} d s_{1} e^{i k α s_{1}} \\ α : = \sin ϑ_{0} \\ \bar{α} \bar{s} = s_{1} \sin ϑ_{0} \\ \Rightarrow Φ (\bar{r} \overset{´}{}) = \frac{C}{i k α} (e^{i k α \frac{d}{2}} - e^{- i k α \frac{d}{2}}) \\ Φ (\bar{r} \overset{´}{}) = C d \frac{\sin (k α \frac{d}{2})}{k α \frac{d}{2}} \end{aligned}

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Identifiers

$Φ$

$\bar{r}$

$\overset{´}{}$

$C$

$d$

$d$

$s_{1}$

$e$

$i$

$k$

$α$

$s_{1}$

$α$

$ϑ_{0}$

$\bar{α}$

$\bar{s}$

$s_{1}$

$ϑ_{0}$

$Φ$

$\bar{r}$

$\overset{´}{}$

$C$

$i$

$k$

$α$

$e$

$i$

$k$

$α$

$d$

$e$

$i$

$k$

$α$

$d$

$Φ$

$\bar{r}$

$\overset{´}{}$

$C$

$d$

$k$

$α$

$d$

$k$

$α$

$d$

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