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* Page found: Bornsche Näherung (eq math.1790.15)

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Bornsche Näherung Eq: math.1790.15

Hash: bd3af09d692ced294b1a736de2d87191

TeX (original user input):

\begin{align}

& f(\vartheta )=-\frac{2m}{{{\hbar }^{2}}}\frac{1}{4\pi }\int_{0}^{\infty }{r{{\acute{\ }}^{2}}dr\acute{\ }}V(\bar{r}\acute{\ })\int_{-1}^{1}{d(\cos \vartheta \acute{\ })}{{e}^{iKr\acute{\ }\cos \vartheta \acute{\ }}}\int_{0}^{2\pi }{d\phi \acute{\ }} \\

& \int_{-1}^{1}{d(\cos \vartheta \acute{\ })}{{e}^{iKr\acute{\ }\cos \vartheta \acute{\ }}}=\frac{1}{iKr\acute{\ }}\left( {{e}^{iKr\acute{\ }}}-{{e}^{-iKr\acute{\ }}} \right)=\frac{2\sin Kr\acute{\ }}{Kr\acute{\ }} \\

\end{align}

TeX (checked):

{\begin{aligned}&f(\vartheta )=-{\frac {2m}{{\hbar }^{2}}}{\frac {1}{4\pi }}\int _{0}^{\infty }{r{{\acute {\ }}^{2}}dr{\acute {\ }}}V({\bar {r}}{\acute {\ }})\int _{-1}^{1}{d(\cos \vartheta {\acute {\ }})}{{e}^{iKr{\acute {\ }}\cos \vartheta {\acute {\ }}}}\int _{0}^{2\pi }{d\phi {\acute {\ }}}\\&\int _{-1}^{1}{d(\cos \vartheta {\acute {\ }})}{{e}^{iKr{\acute {\ }}\cos \vartheta {\acute {\ }}}}={\frac {1}{iKr{\acute {\ }}}}\left({{e}^{iKr{\acute {\ }}}}-{{e}^{-iKr{\acute {\ }}}}\right)={\frac {2\sin Kr{\acute {\ }}}{Kr{\acute {\ }}}}\\\end{aligned}}

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\begin{aligned} f (ϑ) = - \frac{2 m}{ℏ^{2}} \frac{1}{4 π} \int_{0}^{\infty} r {\overset{´}{}}^{2} d r \overset{´}{} V (\bar{r} \overset{´}{}) \int_{- 1}^{1} d (\cos ϑ \overset{´}{}) e^{i K r \overset{´}{} \cos ϑ \overset{´}{}} \int_{0}^{2 π} d ϕ \overset{´}{} \\ \int_{- 1}^{1} d (\cos ϑ \overset{´}{}) e^{i K r \overset{´}{} \cos ϑ \overset{´}{}} = \frac{1}{i K r \overset{´}{}} (e^{i K r \overset{´}{}} - e^{- i K r \overset{´}{}}) = \frac{2 \sin K r \overset{´}{}}{K r \overset{´}{}} \end{aligned}

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Identifiers

$f$

$ϑ$

$m$

$π$

$r$

$\overset{´}{}$

$r$

$\overset{´}{}$

$V$

$\bar{r}$

$\overset{´}{}$

$ϑ$

$\overset{´}{}$

$e$

$i$

$K$

$r$

$\overset{´}{}$

$ϑ$

$\overset{´}{}$

$π$

$ϕ$

$\overset{´}{}$

$ϑ$

$\overset{´}{}$

$e$

$i$

$K$

$r$

$\overset{´}{}$

$ϑ$

$\overset{´}{}$

$i$

$K$

$r$

$\overset{´}{}$

$e$

$i$

$K$

$r$

$\overset{´}{}$

$e$

$i$

$K$

$r$

$\overset{´}{}$

$K$

$r$

$\overset{´}{}$

$K$

$r$

$\overset{´}{}$

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