From aae12f71454a9d5416e4c0cb03caeae32caf2fbb Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Tue, 12 Sep 2017 22:15:06 +0000 Subject: [PATCH] =?UTF-8?q?[=C3=ABwald]=20dusp=C3=A1t?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Former-commit-id: abbd48658c237c0e4ad4cfff912a4b05532959aa --- notes/ewald-calculations.lyx | 95 ++++++++++++++++++++++++++++++++---- notes/ewald.lyx | 59 ++++++++++++++++++++-- 2 files changed, 140 insertions(+), 14 deletions(-) diff --git a/notes/ewald-calculations.lyx b/notes/ewald-calculations.lyx index 94695df..9611803 100644 --- a/notes/ewald-calculations.lyx +++ b/notes/ewald-calculations.lyx @@ -455,16 +455,16 @@ now & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\kor{\left(1+n\right)_{-\frac{2-q+n}{2}}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\koru{\text{Γ}\left(2-q+n+s\right)}\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\ & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}}\frac{\koru{\text{Γ}\left(1+n\right)}\text{Γ}\left(\frac{3-q-n}{2}\right)}{\koru{\text{Γ}\left(\frac{q+n}{2}\right)}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\kor{\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\ & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{Γ\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\koru{\left(ik\right)^{-r}}\koru{\kor{\left(\sigma c-ik_{0}\right)^{r-\frac{3}{2}\left(2-q+n\right)}}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\ -(bionm) & = & \kor{\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\koru{\sum_{w=0}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)}2^{r-s}\left(\left(-1\right)^{r}+1\right)\label{eq:ugliness withous singularities}\\ +(bionm) & = & \kor{\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\koru{\sum_{w=0}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}\right)^{r-\frac{3}{2}\left(2-q+n\right)-w}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\label{eq:ugliness withous singularities}\\ & = & \koru{\kappa!\left(-1\right)^{\kappa}}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=\kor 0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=\kor 0}^{s}\binom{\kor s}{\kor r}\left(ik\right)^{-r}\sum_{w=\kor 0}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{\kor w}\koru{\kor{\begin{Bmatrix}w\\ \kappa -\end{Bmatrix}}}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\ +\end{Bmatrix}}}c^{w}\left(-ik_{0}\right)^{r-\frac{3}{2}\left(2-q+n\right)-w}2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\ & = & \kappa!\left(-1\right)^{\kappa}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=\koru{\kappa}}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=\koru{\kappa}}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=\koru{\kappa}}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\begin{Bmatrix}w\\ \kappa -\end{Bmatrix}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\ +\end{Bmatrix}c^{w}\left(-ik_{0}\right)^{r-\frac{3}{2}\left(2-q+n\right)-w}2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\ & = & \kappa!\left(-1\right)^{\kappa}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=\kappa}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=\kappa}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=\kappa}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\begin{Bmatrix}w\\ \kappa -\end{Bmatrix}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber +\end{Bmatrix}c^{w}\left(-ik_{0}\right)^{r-\frac{3}{2}\left(2-q+n\right)-w}2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \end{eqnarray} \end_inset @@ -596,8 +596,8 @@ reference "eq:ugliness withous singularities" \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^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=0}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)\\ -\pht 0{s_{2+\epsilon,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(s-\epsilon\right)\left(-\epsilon\right)_{s}}{\left(\epsilon+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=0}^{\infty}\binom{r+\frac{3}{2}\epsilon}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}^{r+\frac{3}{2}\epsilon-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right) +\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^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=0}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}\right)^{r-\frac{3}{2}\left(2-q+n\right)-w}2^{r-s}\left(\left(-1\right)^{r}+1\right)\\ +\pht 0{s_{2+\epsilon,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(s-\epsilon\right)\left(-\epsilon\right)_{s}}{\left(\epsilon+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=0}^{\infty}\binom{r+\frac{3}{2}\epsilon}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}\right)^{r+\frac{3}{2}\epsilon-w}2^{r-s}\left(\left(-1\right)^{r}+1\right) \end{eqnarray*} \end_inset @@ -618,9 +618,9 @@ There is one problematic factor on the previous line, Let us analyse the problematic term. \begin_inset Formula \begin{eqnarray*} -\mbox{problem} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\frac{\text{Γ}\left(-\epsilon\right)\kor{\left(-\epsilon\right)_{0}}}{\kor{\left(\epsilon+\frac{1}{2}\right)_{0}0!}}\kor{\sum_{r=0}^{0}\binom{0}{r}\left(ik\right)^{-r}}\sum_{w=0}^{\infty}\binom{\kor r+\frac{3}{2}\epsilon}{w}\sigma^{w}c^{w}\left(-ik_{0}^{\kor r+\frac{3}{2}\epsilon-w}\right)2^{\kor r-0}\kor{\left(\left(-1\right)^{r}+1\right)}\\ - & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\text{Γ}\left(-\epsilon\right)\sum_{w=0}^{\infty}\kor{\binom{\frac{3}{2}\epsilon}{w}}\sigma^{w}c^{w}\left(-ik_{0}^{\frac{3}{2}\epsilon-w}\right)2\\ - & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\text{Γ}\left(-\epsilon\right)\sum_{w=0}^{\infty}\frac{\Gamma\left(1+\frac{3}{2}\epsilon\right)}{\Gamma\left(w+1\right)\Gamma\left(1+\frac{3}{2}\epsilon-w\right)}\sigma^{w}c^{w}\left(-ik_{0}^{\frac{3}{2}\epsilon-w}\right)2. +\mbox{problem} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1)}{k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\frac{\text{Γ}\left(-\epsilon\right)\kor{\left(-\epsilon\right)_{0}}}{\kor{\left(\epsilon+\frac{1}{2}\right)_{0}0!}}\kor{\sum_{r=0}^{0}\binom{0}{r}\left(ik\right)^{-r}}\sum_{w=0}^{\infty}\binom{\kor r+\frac{3}{2}\epsilon}{w}\sigma^{w}c^{w}\left(-ik_{0}\right)^{\kor r+\frac{3}{2}\epsilon-w}2^{\kor r-0}\kor{\left(\left(-1\right)^{r}+1\right)}\\ + & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1)}{k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\text{Γ}\left(-\epsilon\right)\sum_{w=0}^{\infty}\kor{\binom{\frac{3}{2}\epsilon}{w}}\sigma^{w}c^{w}\left(-ik_{0}\right)^{\frac{3}{2}\epsilon-w}2\\ + & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1)}{k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\text{Γ}\left(-\epsilon\right)\sum_{w=0}^{\infty}\frac{\Gamma\left(1+\frac{3}{2}\epsilon\right)}{\Gamma\left(w+1\right)\Gamma\left(1+\frac{3}{2}\epsilon-w\right)}\sigma^{w}c^{w}\left(-ik_{0}\right)^{\frac{3}{2}\epsilon-w}2. \end{eqnarray*} \end_inset @@ -645,11 +645,86 @@ In the last sum, the divisor -regularisation, \begin_inset Formula \[ -\mbox{problem}=\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\text{Γ}\left(-\epsilon\right)\sum_{w=\koru{\kappa}}^{\infty}\frac{\Gamma\left(1+\frac{3}{2}\epsilon\right)}{\Gamma\left(w+1\right)\Gamma\left(1+\frac{3}{2}\epsilon-w\right)}\sigma^{w}c^{w}\left(-ik_{0}^{\frac{3}{2}\epsilon-w}\right)2 +\mbox{problem}=\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\kor{\text{Γ}\left(-\epsilon\right)}\sum_{w=\koru{\kappa}}^{\infty}\frac{\Gamma\left(1+\frac{3}{2}\epsilon\right)}{\Gamma\left(w+1\right)\kor{\Gamma\left(1+\frac{3}{2}\epsilon-w\right)}}\sigma^{w}c^{w}\left(-ik_{0}\right)^{\frac{3}{2}\epsilon-w}2. \] \end_inset +6amma function has simple poles for non-positive integer arguments with + +\begin_inset Formula $\mathrm{Res}\left(Γ,-n\right)=\left(-1\right)^{n}/n!$ +\end_inset + +, so writing the Laurent series for the underlined factors gives +\begin_inset Formula +\[ +\lim_{\epsilon\to0}\frac{Γ\left(-\epsilon\right)}{Γ\left(1+\frac{3}{2}\epsilon-w\right)}=\lim_{\epsilon\to0}\frac{\left(-\epsilon\right)^{-1}+\sum_{n=0}^{\infty}\dots\epsilon^{n}}{\left(-1\right)^{w-1}\left(\frac{3}{2}\epsilon\left(w-1\right)!\right)^{-1}+\sum_{n=0}^{\infty}\dots\epsilon^{n}}=\lim_{\epsilon\to0}\frac{-1+\epsilon\sum_{n=0}^{\infty}\dots\epsilon^{n}}{\left(-1\right)^{w-1}\frac{2}{3}/\left(w-1\right)!+\epsilon\sum_{n=0}^{\infty}\dots\epsilon^{n}}=\frac{3}{2}\left(-1\right)^{w}\left(w-1\right)! +\] + +\end_inset + + and the rest is obviously continuous with regard to +\begin_inset Formula $\epsilon$ +\end_inset + +. + Therefore, +\begin_inset Formula +\begin{eqnarray*} +\lim_{\epsilon\to0}\mbox{problem} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\sum_{w=\kappa}^{\infty}\frac{1}{\kor{\Gamma\left(w+1\right)}}\frac{3}{2}\left(-1\right)^{w}\kor{\left(w-1\right)!}\sigma^{w}c^{w}\left(-ik_{0}\right)^{-w}\\ + & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{3}{2}\sum_{w=\kappa}^{\infty}\frac{\left(-1\right)^{w}\sigma^{w}c^{w}\left(-ik_{0}\right)^{-w}}{\koru w}\\ + & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{3}{2}\sum_{w=\kappa}^{\infty}\frac{1}{w}\left(-i\frac{\sigma c}{k_{0}}\right)^{w} +\end{eqnarray*} + +\end_inset + +and if +\begin_inset Formula $\left|\sigma c/k_{0}\right|<1$ +\end_inset + +, the last expression is (almost) the well known power series +\begin_inset Formula $\log\left(1+x\right)=-\sum_{n=1}^{\infty}\left(-x\right)^{n}/n$ +\end_inset + +, so +\begin_inset Formula +\[ +\lim_{\epsilon\to0}\mbox{problem}=-\kor{\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}}\frac{3}{2}\left[\log\left(1+i\frac{\sigma c}{k_{0}}\right)+\sum_{w=1}^{\kappa-1}\frac{1}{w}\kor{\left(-i\frac{\sigma c}{k_{0}}\right)^{w}}\right] +\] + +\end_inset + +and the +\begin_inset Formula $\kappa$ +\end_inset + +-regularisation makes the last term identically zero, providing even simpler + expression +\begin_inset Formula +\[ +\lim_{\epsilon\to0}\mbox{problem}=-\frac{3}{2}\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\log\left(1+i\frac{\sigma c}{k_{0}}\right). +\] + +\end_inset + + +\emph on +What does this mean w.r.t. + the limit +\begin_inset Formula $k\to\infty$ +\end_inset + +? +\emph default + Back to the whole Bessel transform, +\begin_inset Formula +\begin{eqnarray*} +\pht 0{s_{2+\epsilon,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & - & \mbox{problem}\\ + & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\sum_{s=1}^{\infty}\frac{\text{Γ}\left(s-\epsilon\right)\left(-\epsilon\right)_{s}}{\left(\epsilon+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=0}^{\infty}\binom{r+\frac{3}{2}\epsilon}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}\right)^{r+\frac{3}{2}\epsilon-w}2^{r-s}\left(\left(-1\right)^{r}+1\right) +\end{eqnarray*} + +\end_inset + \end_layout diff --git a/notes/ewald.lyx b/notes/ewald.lyx index 7d50ed5..58271c4 100644 --- a/notes/ewald.lyx +++ b/notes/ewald.lyx @@ -37,6 +37,9 @@ %\newfontfamily\russianfont[Script=Cyrillic]{DejaVu Sans} \end_preamble \use_default_options true +\begin_modules +theorems-starred +\end_modules \maintain_unincluded_children false \language english \language_package default @@ -2618,6 +2621,54 @@ name "tab:Asymptotical-behaviour-Mathematica" \end_inset +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{russian} +\end_layout + +\end_inset + +Градштейн и Рыжик +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{russian} +\end_layout + +\end_inset + + 6.512.1 has expression for +\begin_inset Formula $\int_{0}^{\infty}J_{\mu}\left(ax\right)J_{\nu}\left(bx\right)\ud x$ +\end_inset + +, +\begin_inset Formula $\Re\left(\mu+\nu\right)>-1$ +\end_inset + + in terms of hypergeometric function. + Unfortunately, no corresponding and general enough expression for +\begin_inset Formula $\int_{0}^{\infty}J_{\mu}\left(ax\right)Y_{\nu}\left(bx\right)\ud x$ +\end_inset + +. +\end_layout + +\end_inset + + \end_layout \begin_layout Paragraph @@ -2985,7 +3036,7 @@ where the spherical Hankel transform 2) \begin_inset Formula \[ -\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right). +\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right). \] \end_inset @@ -2995,7 +3046,7 @@ Using this convention, the inverse spherical Hankel transform is given by 3) \begin_inset Formula \[ -g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\,k^{2}\bsht lg(k)j_{l}(k), +g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\, k^{2}\bsht lg(k)j_{l}(k), \] \end_inset @@ -3008,7 +3059,7 @@ so it is not unitary. An unitary convention would look like this: \begin_inset Formula \begin{equation} -\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition} +\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition} \end{equation} \end_inset @@ -3062,7 +3113,7 @@ where the Hankel transform of order is defined as \begin_inset Formula \begin{equation} -\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\,g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition} +\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition} \end{equation} \end_inset