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ewald.lyx: Fix sign connection formula for cylindrical bessel functions 

Former-commit-id: 6ab112965c8c3060615baeb3d534bda23bf09d44
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Marek Nečada 2018-05-14 10:43:02 +03:00
parent 8bf9a1c54d
commit 08f0332dbb
1 changed files with 5 additions and 5 deletions
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notes/ewald.lyx
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@ -3099,7 +3099,7 @@ where the spherical Hankel transform
2) 2)
\begin_inset Formula \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 \end_inset
@ -3109,7 +3109,7 @@ Using this convention, the inverse spherical Hankel transform is given by
3) 3)
\begin_inset Formula \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 \end_inset
@ -3122,7 +3122,7 @@ so it is not unitary.
An unitary convention would look like this: An unitary convention would look like this:
\begin_inset Formula \begin_inset Formula
\begin{equation} \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{equation}
\end_inset \end_inset
@ -3176,8 +3176,8 @@ where the Hankel transform of order
is defined as is defined as
\begin_inset Formula \begin_inset Formula
\begin{eqnarray} \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}\\ \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 & = & \left(-1\right)^{m}\int_{0}^{\infty}\ud r\, g(r)J_{-m}(kr)r
\end{eqnarray} \end{eqnarray}
\end_inset \end_inset