From 3bbdee97269280ea011520cb90860df5bf1fb30f Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= <marek@necada.org>
Date: Sun, 11 Nov 2018 07:06:35 +0000
Subject: [PATCH] Some 1D lattice sums

Former-commit-id: 0be7e862f22ebb37523d263ad98856c1dfcc1bff
---
 notes/ewald.lyx | 43 ++++++++++++++++++++++++++++++++++++++-----
 1 file changed, 38 insertions(+), 5 deletions(-)

diff --git a/notes/ewald.lyx b/notes/ewald.lyx
index 6e0fff4..696b1ad 100644
--- a/notes/ewald.lyx
+++ b/notes/ewald.lyx
@@ -3399,6 +3399,39 @@ The only diverging factor here is apparently
 \end_inset
 
 
+\end_layout
+
+\begin_layout Subsection
+1D
+\end_layout
+
+\begin_layout Standard
+One-dimensional lattice sums are provided in [REF LT, sect.
+ 3].
+ However, these are the 
+\begin_inset Quotes eld
+\end_inset
+
+non-shifted
+\begin_inset Quotes erd
+\end_inset
+
+ sums,
+\begin_inset Formula 
+\begin{eqnarray*}
+\ell_{n}\left(\beta\right) & = & \sum_{j\in\ints}^{'}e^{i\beta aj}\mathcal{H}_{n}^{0}\left(aj\hat{\vect z}\right)\\
+ & = & \sum_{j\in\ints}^{'}e^{i\beta aj}h_{n}\left(\left|aj\right|\right)Y_{n}^{0}\\
+ & = & \sqrt{\frac{2n+1}{4\pi}}\sum_{j\in\ints}^{'}P_{n}^{0}\left(\sgn j\right)h_{n}\left(\left|aj\right|\right)e^{i\beta aj}\\
+ & = & \sqrt{\frac{2n+1}{4\pi}}\sum_{j\in\ints}^{'}\left(\sgn j\right)^{n}h_{n}\left(\left|aj\right|\right)e^{i\beta aj},
+\end{eqnarray*}
+
+\end_inset
+
+where we used 
+\begin_inset Formula $P_{n}^{m}\left(\pm1\right)=\left(\pm1\right)^{n}\delta_{m0}$
+\end_inset
+
+
 \end_layout
 
 \begin_layout Section
@@ -3485,7 +3518,7 @@ 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
@@ -3495,7 +3528,7 @@ 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
@@ -3508,7 +3541,7 @@ 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
@@ -3562,8 +3595,8 @@ where the Hankel transform of order
  is defined as
 \begin_inset Formula 
 \begin{eqnarray}
-\pht mg\left(k\right) & = & \int_{0}^{\infty}\ud r\,g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}\\
- & = & \left(-1\right)^{m}\int_{0}^{\infty}\ud r\,g(r)J_{-m}(kr)r
+\pht mg\left(k\right) & = & \int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}\\
+ & = & \left(-1\right)^{m}\int_{0}^{\infty}\ud r\, g(r)J_{-m}(kr)r
 \end{eqnarray}
 
 \end_inset