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WIP Ewald 1D in 3D notes: partial index fix etc. 

Former-commit-id: 23b253c8179a8f62e05675fdf2fef26dc484790d
This commit is contained in:
Marek Nečada 2020-06-15 13:12:26 +03:00
parent 0c442ba745
commit 827499c3ff
1 changed files with 40 additions and 4 deletions
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44
notes/ewald_1D_in_3D.lyx
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@ -491,7 +491,7 @@ Let's do the polar integration next:
\begin_inset Formula \begin_inset Formula
\[ \[
B_{l}^{m}\equiv\int_{0}^{\pi}\sin\theta\ud\theta\,P_{l}^{-m}\left(\cos\theta\right)P_{l}^{0}\left(\cos\theta\right)e^{-\left(\sin\theta\right)^{2}r^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\left(-\sin\theta\,rs_{\bot}\kappa^{2}\gamma_{\vect K}^{2}/4\tau\right)^{2k-m} B_{l'}^{m'}\equiv\int_{0}^{\pi}\sin\theta\ud\theta\,P_{l'}^{-m'}\left(\cos\theta\right)P_{l}^{0}\left(\cos\theta\right)e^{-\left(\sin\theta\right)^{2}r^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\left(-\sin\theta\,rs_{\bot}\kappa^{2}\gamma_{\vect K}^{2}/4\tau\right)^{2k-m'}
\] \]
\end_inset \end_inset
@ -503,13 +503,49 @@ Label
; then ; then
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
B_{l}^{m} & =\int_{0}^{\pi}\sin\theta\ud\theta\,P_{l}^{-m}\left(\cos\theta\right)P_{l}^{0}\left(\cos\theta\right)e^{-u\left(\sin\theta\right)^{2}}\left(-v\sin\theta\right)^{2k-m}\\ B_{l'}^{m'} & =\int_{0}^{\pi}\sin\theta\ud\theta\,P_{l'}^{-m'}\left(\cos\theta\right)P_{l}^{0}\left(\cos\theta\right)e^{-u\left(\sin\theta\right)^{2}}\left(-v\sin\theta\right)^{2k-m'}\\
& =\int_{0}^{\pi}\sin\theta\ud\theta\,P_{l}^{-m}\left(\cos\theta\right)P_{l}^{0}\left(\cos\theta\right)\left(-v\sin\theta\right)^{2k-m}\sum_{a=0}^{\infty}\frac{\left(-u\right)^{a}}{a!}\left(\sin\theta\right)^{2a}\\ & =\int_{0}^{\pi}\sin\theta\ud\theta\,P_{l'}^{-m'}\left(\cos\theta\right)P_{l}^{0}\left(\cos\theta\right)\left(-v\sin\theta\right)^{2k-m'}\sum_{a=0}^{\infty}\frac{\left(-u\right)^{a}}{a!}\left(\sin\theta\right)^{2a}\\
& =\left(-v\right)^{2k-m}\sum_{a=0}^{\infty}\frac{\left(-u\right)^{a}}{a!}\int_{0}^{\pi}\sin\theta\ud\theta\,P_{l}^{-m}\left(\cos\theta\right)P_{l}^{0}\left(\cos\theta\right)\left(\sin\theta\right)^{2a+2k-m} & =\left(-v\right)^{2k-m'}\sum_{a=0}^{\infty}\frac{\left(-u\right)^{a}}{a!}\int_{0}^{\pi}\sin\theta\ud\theta\,P_{l'}^{-m'}\left(\cos\theta\right)P_{l}^{0}\left(\cos\theta\right)\left(\sin\theta\right)^{2a+2k-m'}
\end{align*} \end{align*}
\end_inset \end_inset
If we now perform the limit
\begin_inset Formula $r\to0$
\end_inset
and compare the radial parts (incl.
those in
\begin_inset Formula $u,v$
\end_inset
) powers, the leading term indices will have
\begin_inset Formula
\[
l'\sim l+2a+2k-m'
\]
\end_inset
so we can fix
\begin_inset Formula $2a+2k-m'=l'-l$
\end_inset
and get
\begin_inset Formula
\[
\int_{0}^{\pi}\sin\theta\ud\theta\,P_{l'}^{-m'}\left(\cos\theta\right)P_{l}^{0}\left(\cos\theta\right)\left(\sin\theta\right)^{l'-l}=\begin{cases}
0 & l'-l+m'\text{ odd}\\
? & l'-l+m'\text{ even}
\end{cases}
\]
\end_inset
\begin_inset Formula $ $
\end_inset
\end_layout \end_layout