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dudopráce 

Former-commit-id: 02b6635802e642e43dc546e506bbef7fd443bebf
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Marek Nečada 2017-08-07 07:13:36 +00:00
parent 995920e447
commit e708c353e9
1 changed files with 21 additions and 3 deletions
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24
notes/ewald.lyx
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@ -406,9 +406,11 @@ where changed the sign of
.
Fourier transform of product is convolution of Fourier transforms, so
\begin_inset Formula
\[
W_{\alpha\beta}(\vect k)=\left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)
\]
\begin{eqnarray*}
W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\\
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\rec{\basis u}}^{(d)}\left(\frac{1}{2\pi}\vect{\circ}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\quad\mbox{(re-check facs)}\\
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}
\end{eqnarray*}
\end_inset
@ -755,6 +757,22 @@ And consequently, for unitary/angular frequency it is
\end_inset
Using my own
\begin_inset Quotes eld
\end_inset
basis notation
\begin_inset Quotes erd
\end_inset
TODO
\begin_inset Formula
\[
\uoft{\dc{\basis u}}\left(\vect{\xi}\right)=\left|\det\rec{\basis u}\right|\dc{}^{(d)}\left(A^{-T}\vect{\xi}\right).
\]
\end_inset
\end_layout