[ewald] q=2, n=0 solved for large k

Former-commit-id: 8258dec716cda51c321fc5ccba15222e9fad5e24

notes/ewald-calculations-apr1.lyx

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+\pdf_title "Sähköpajan päiväkirja"
+\pdf_author "Marek Nečada"
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+Let
+\end_layout
+
+\begin_layout Paragraph
+
+\lang english
+Large k
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+\begin_inset Formula 
+\begin{eqnarray*}
+\mbox{OK}\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
+\mbox{OK(D15.8.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}(\\
+ &  & \pi\frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{Γ\left(\frac{3-q+n}{2}\right)\text{Γ}\left(1+n-\frac{2-q+n}{2}\right)}\hgfr\left(\begin{array}{c}
+\frac{2-q+n}{2},\frac{2-q+n}{2}-\left(1+n\right)+1\\
+1/2
+\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)\\
+ & - & \pi\frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(1+n-\frac{3-q+n}{2}\right)}\hgfr\left(\begin{array}{c}
+\frac{3-q+n}{2},\frac{3-q+n}{2}-\left(1+n\right)+1\\
+3/2
+\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\
+\mbox{OK20} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi(\\
+ &  & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\hgfr\left(\begin{array}{c}
+\frac{2-q+n}{2},\frac{2-q-n}{2}\\
+1/2
+\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)\\
+ & - & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)}\hgfr\left(\begin{array}{c}
+\frac{3-q+n}{2},\frac{3-q-n}{2}\\
+3/2
+\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\
+\mbox{(D15.2.2)OK3a,b} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi\sum_{s=0}^{\infty}(\\
+ &  & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{1}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s}\\
+ & - & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)}\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s})\\
+\mbox{OK4a} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\
+ &  & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}k^{-2+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{2-q+n}+2s}\\
+ & - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}k^{-3+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{3-q+n}+2s})\\
+\mbox{OK4b} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\
+ &  & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\kor{k^{-2+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{2s}}\\
+ & - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\kor{k^{-3+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{1+2s}})\\
+\mbox{OK4c} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=\kor 0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\\
+ &  & \times\left(\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
+\mbox{OK4d} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)
+\end{eqnarray*}
+
+\end_inset
+
+the fact that the partial sum 
+\begin_inset Formula $\sum_{s=0}^{\left\lceil \kappa/2\right\rceil -1}\ldots$
+\end_inset
+
+ is zero is shown in the old messy notes (or TODO later here) 
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+Using DLMF 5.5.5, which says 
+\begin_inset Formula $Γ(2z)=\pi^{-1/2}2^{2z-1}\text{Γ}(z)\text{Γ}(z+\frac{1}{2})$
+\end_inset
+
+ we have 
+\begin_inset Formula 
+\[
+\text{Γ}\left(2-q+n\right)=\frac{2^{1-q+n}}{\sqrt{\pi}}\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{3-q+n}{2}\right),
+\]
+
+\end_inset
+
+so
+\begin_inset Formula 
+\begin{eqnarray*}
+\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{\text{Γ}\left(2-q+n\right)}}{\kor{2^{n}}k_{0}^{q}}\kor{\pi}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{3-q+n}{2}\right)}\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{2-q+n}{2}\right)}\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
+ & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\koru{2^{1-q}}}{k_{0}^{q}}\koru{\sqrt{\pi}}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(\frac{\kor{\koru{\text{Γ}\left(\frac{2-q+n}{2}\right)}\left(\frac{2-q+n}{2}\right)_{s}}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\kor{\koru{\text{Γ}\left(\frac{3-q+n}{2}\right)}\left(\frac{3-q+n}{2}\right)_{s}}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
+ & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(\frac{\text{Γ}\left(\frac{2-q+n}{2}\right)\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\text{Γ}\left(\frac{3-q+n}{2}\right)\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)
+\end{eqnarray*}
+
+\end_inset
+
+Assuming that 
+\begin_inset Formula $\left\lceil \frac{\kappa}{2}\right\rceil $
+\end_inset
+
+ is large enough so that all the divergent terms are cancelled, either the
+ left or the right part will become finite sums due to the 
+\begin_inset Quotes sld
+\end_inset
+
+extra
+\begin_inset Quotes srd
+\end_inset
+
+ Pochhammer 
+\begin_inset Formula $\left(\frac{3-q-n}{2}\right)_{s}$
+\end_inset
+
+ or 
+\begin_inset Formula $\left(\frac{2-q-n}{2}\right)_{s}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subparagraph
+
+\lang english
+Special case 
+\begin_inset Formula $q=2,n=0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+If 
+\begin_inset Formula $\kappa\ge2$
+\end_inset
+
+, the left part will drop and
+\begin_inset Formula 
+\begin{eqnarray*}
+\mbox{OKSq2n0b}\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\text{Γ}\left(\frac{1}{2}+s\right)\text{Γ}\left(\frac{1}{2}+s\right)}{\text{Γ}\left(\frac{1}{2}\right)\kor{\text{Γ}\left(\frac{3}{2}+s\right)}s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
+ & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\kor{\text{Γ}\left(\frac{1}{2}+s\right)}\text{Γ}\left(\frac{1}{2}+s\right)}{\text{Γ}\left(\frac{1}{2}\right)\koru{\kor{\text{Γ}\left(\frac{1}{2}+s\right)}\left(\frac{1}{2}+s\right)}s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
+ & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\kor{\left\lceil \frac{\kappa}{2}\right\rceil }}^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\text{Γ}\left(\frac{1}{2}\right)\left(\frac{1}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
+\mbox{(explain!)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\koru 0}^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\kor{\text{Γ}\left(\frac{1}{2}\right)}\left(\frac{1}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
+ & = & -\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}\sqrt{\pi}}\frac{\left(\sigma c-ik_{0}\right)}{k}\kor{\sum_{s=0}^{\infty}\left(-1\right)^{s}\left(\frac{\sigma c-ik_{0}}{k}\right)^{2s}\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\left(\frac{1}{2}+s\right)s!}}\\
+ & = & -\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}\sqrt{\pi}}\frac{\left(\sigma c-ik_{0}\right)}{k}\frac{2\sqrt{\pi}\sinh^{-1}\left(\frac{\sigma c-ik_{0}}{k}\right)}{\frac{\sigma c-ik_{0}}{k}}\\
+\mbox{OKSq2n0f} & = & -\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{1}{k_{0}^{2}}\sinh^{-1}\left(\frac{\sigma c-ik_{0}}{k}\right)
+\end{eqnarray*}
+
+\end_inset
+
+where we used (TODO ref)
+\begin_inset Formula 
+\[
+\sum_{s=0}^{\infty}\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\left(\frac{1}{2}+s\right)s!}\left(-x\right)^{s}=\frac{2\sqrt{\pi}\sinh^{-1}\sqrt{x}}{\sqrt{x}}
+\]
+
+\end_inset
+
+The final result has asymptotic behaviour of ...
+ for 
+\begin_inset Formula $k\to\infty$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Small k 
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+\begin_inset Formula 
+\begin{eqnarray*}
+\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
+\mbox{(D15.2.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\kor{Γ\left(2-q+n\right)}}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\sum_{s=0}^{\infty}\frac{\kor{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{3-q+n}{2}\right)_{s}}}{Γ(1+n+s)s!}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{s},\quad\left|\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right|<1
+\end{eqnarray*}
+
+\end_inset
+
+Again we use 
+\begin_inset Formula 
+\[
+\text{Γ}\left(2-q+n\right)=\frac{2^{1-q+n}}{\sqrt{\pi}}\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{3-q+n}{2}\right),
+\]
+
+\end_inset
+
+ so
+\begin_inset Formula 
+\begin{eqnarray*}
+\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \koru{\frac{2^{1-q+n}}{\sqrt{\pi}}}\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\sum_{s=0}^{\infty}\frac{\koru{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\text{Γ}\left(\frac{3-q+n}{2}+s\right)}}{\text{Γ}(1+n+s)s!}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{s}\\
+ & = & \frac{2^{1-q+n}}{\sqrt{\pi}}\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\text{Γ}\left(\frac{3-q+n}{2}+s\right)}{\text{Γ}(1+n+s)s!}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{s}
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

notes/ewald-calculations.lyx

@@ -208,8 +208,8 @@ Let
 \lang english
 \begin_inset Formula 
 \begin{eqnarray*}
-\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
+\mbox{OK}\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
-\mbox{(D15.8.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}(\\
+\mbox{OK(D15.8.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}(\\
  &  & \pi\frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{Γ\left(\frac{3-q+n}{2}\right)\text{Γ}\left(1+n-\frac{2-q+n}{2}\right)}\hgfr\left(\begin{array}{c}
 \frac{2-q+n}{2},\frac{2-q+n}{2}-\left(1+n\right)+1\\
 1/2
@@ -218,7 +218,7 @@ Let
 \frac{3-q+n}{2},\frac{3-q+n}{2}-\left(1+n\right)+1\\
 3/2
 \end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\
- & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi(\\
+\mbox{OK20} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi(\\
  &  & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\hgfr\left(\begin{array}{c}
 \frac{2-q+n}{2},\frac{2-q-n}{2}\\
 1/2
@@ -227,16 +227,16 @@ Let
 \frac{3-q+n}{2},\frac{3-q-n}{2}\\
 3/2
 \end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\
-\mbox{(D15.2.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi\sum_{s=0}^{\infty}(\\
+\mbox{(D15.2.2)OK3a,b} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi\sum_{s=0}^{\infty}(\\
  &  & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{1}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s}\\
  & - & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)}\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s})\\
- & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\
+\mbox{OK4a} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\
  &  & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}k^{-2+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{2-q+n}+2s}\\
  & - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}k^{-3+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{3-q+n}+2s})\\
-\mbox{} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\
+\mbox{OK4b} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\
  &  & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\kor{k^{-2+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{2s}}\\
  & - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\kor{k^{-3+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{1+2s}})\\
- & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\\
+\mbox{OK4c} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\\
  &  & \times\left(\underbrace{\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}}_{\equiv c_{q,n,s}}-\underbrace{\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}}_{č_{q,n,s}}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
  & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\kor{\left(\sigma c-ik_{0}\right)^{2s}}c_{q,n,s}-\frac{\left(\sigma c-ik_{0}\right)^{2s+1}}{k}č_{q,n,s}\right)\\
 \mbox{(binom.)} & = & \kor{\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(c_{q,n,s}\sum_{t=0}^{2s}\binom{2s}{t}\left(\kor{\sigma}c\right)^{t}\left(-ik_{0}\right)^{2s-t}-č_{q,n,s}\sum_{t=0}^{2s+1}\binom{2s+1}{t}\left(\kor{\sigma}c\right)^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)\\
@@ -383,13 +383,13 @@ If
 
 \begin_inset Formula 
 \begin{eqnarray*}
-\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\kor{\hgfr}\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
+\mbox{OK}\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\kor{\hgfr}\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
-\mbox{(D15.1.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)\koru{\text{Γ}(1+n)}}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\koru{\hgf}\left(\frac{2-q+n}{2},\kor{\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}}\right)\\
+\mbox{\ensuremath{\mbox{OK}}(D15.1.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}\koru{\text{Γ}(1+n)}}\koru{\hgf}\left(\frac{2-q+n}{2},\kor{\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}}\right)\\
-\mbox{(D15.8.6)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\koru{\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}}\hgf\left(\begin{array}{c}
+\mbox{(D15.8.6)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}\text{Γ}(1+n)}\koru{\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}}\hgf\left(\begin{array}{c}
 \frac{2-q+n}{2},\koru{\kor{1-\left(1+n\right)+\frac{2-q+n}{2}}}\\
 \koru{\kor{1-\frac{3-q+n}{2}+\frac{2-q+n}{2}}}
 \end{array};\koru{\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}}\right)\\
- & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\koru{k^{q-2}}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\koru{\frac{3}{2}\left(2-q+n\right)}}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\hgf\left(\begin{array}{c}
+\mbox{NOTOK} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\koru{k^{q-2}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\koru{\frac{3}{2}\left(2-q+n\right)}}\text{Γ}(1+n)}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\hgf\left(\begin{array}{c}
 \frac{2-q+n}{2},\koru{\frac{2-q-n}{2}}\\
 \koru{1/2}
 \end{array};\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)}\\

notes/ewald.lyx

@@ -2680,97 +2680,25 @@ Case
 \end_layout
 
 \begin_layout Standard
-[REF 
+As shown in a separate note,
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{russian}
 \end_layout
 
-\end_inset
+\begin_layout Standard
-
-Прудников, том 2
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{russian}
-\end_layout
-
-\end_inset
-
-, 2.12.10.2] provides the following integral
-\begin_inset Formula 
-\begin{multline*}
-\int_{0}^{\infty}\frac{1}{x^{2}}e^{-px-b/x}J_{0}(cx)\,\ud x=2c\left[z_{+}^{-1}J_{1}\left(z_{-}\right)K_{0}\left(z_{+}\right)+z_{-}^{-1}J_{0}\left(z_{-}\right)K_{1}\left(z_{+}\right)\right]\\
-\left[z_{\pm}=\sqrt{2b}\left(\sqrt{p^{2}+c^{2}}\pm p\right)^{1/2};\Re p>\left|\Im c\right|;\Re b>0\right]
-\end{multline*}
-
-\end_inset
-
-where [REF 
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{russian}
-\end_layout
-
-\end_inset
-
-Прудников, том 2
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{russian}
-\end_layout
-
-\end_inset
-
-, p.659] 
-\begin_inset Formula $K_{\nu}(z)$
-\end_inset
-
- is the Macdonald's function (modified Bessel function of 3rd kind) 
 \begin_inset Formula 
 \[
-K_{\nu}\left(z\right)=\frac{\pi\left[I_{-\nu}\left(z\right)-I_{\nu}\left(z\right)\right]}{2\sin\nu\pi}\quad\left[\nu\notin\ints\right],\quad K_{n}\left(z\right)=\lim_{\nu\to n}K_{\nu}\left(z\right)\quad\left[n\in\ints\right],
+\pht 0{s_{2,k_{0}}^{\textup{L}\kappa,c}}\left(k\right)=-\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{1}{k_{0}^{2}}\sinh^{-1}\left(\frac{\sigma c-ik_{0}}{k}\right)
 \]
 
 \end_inset
 
-and 
+for 
-\begin_inset Formula $I_{\nu}\left(z\right)$
+\begin_inset Formula $\kappa\ge?$
 \end_inset
 
- is the modified Bessel function of 1st kind
+, 
-\begin_inset Formula 
+\begin_inset Formula $k>k_{0}?$
-\[
-I_{\nu}\left(z\right)=\frac{1}{\Gamma\left(\nu+1\right)}\left(\frac{z}{2}\right)^{\nu}\ghgf 01\left(\nu+1;\frac{z^{2}}{4}\right)=e^{-\nu\pi i/2}J_{\nu}\left(e^{\pi i/2}z\right).
-\]
-
 \end_inset
 
-The problem of this approach is the insufficiently slow decay 
-\begin_inset Formula $\propto k^{-1}$
-\end_inset
-
-, so it is in fact better to compute the sum in the real space.
- I have to look further.
  
 \end_layout