From 08f0332dbb565f8d0cce928f6bcf8a498f8fdeff Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Mon, 14 May 2018 10:43:02 +0300 Subject: [PATCH] ewald.lyx: Fix sign connection formula for cylindrical bessel functions Former-commit-id: 6ab112965c8c3060615baeb3d534bda23bf09d44 --- notes/ewald.lyx | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/notes/ewald.lyx b/notes/ewald.lyx index 4ee8591..ade0969 100644 --- a/notes/ewald.lyx +++ b/notes/ewald.lyx @@ -3099,7 +3099,7 @@ 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 @@ -3109,7 +3109,7 @@ 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 @@ -3122,7 +3122,7 @@ 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 @@ -3176,8 +3176,8 @@ 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_{\left|m\right|}(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