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[ewald] chybička 

Former-commit-id: 8db8771c6e111256d0a933996f351960fcb799df
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Marek Nečada 2017-09-11 16:04:01 +03:00
parent 814ea36415
commit 6016fcd55d
1 changed files with 10 additions and 17 deletions
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27
notes/ewald.lyx
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@ -1551,16 +1551,16 @@ One problematic element here is the gamma function
\begin_inset Formula $\text{Γ}\left(2-q+n\right)$
\end_inset
which is singular if the arguments are negative integers, i.e.
which is singular if the argument is zero or negative integer, i.e.
if
\begin_inset Formula $q-n\ge3$
\begin_inset Formula $q-n\ge2$
\end_inset
; but at least the necessary minimum of
\begin_inset Formula $q=1,2$
; which is painful especially because of the case
\begin_inset Formula $q=2,n=0$
\end_inset
would be covered this way.
.
The associated Legendre function can be expressed as a finite
\begin_inset Quotes eld
\end_inset
@ -1636,14 +1636,7 @@ reference "tab:Asymptotical-behaviour-Mathematica"
\end_inset
, which is unfortunately important.
But if I have not made some mistake, the expression
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2D Hankel transform of exponentially suppressed outgoing wave expanded"
\end_inset
is applicable for this case.
\end_layout
\begin_layout Standard
@ -2879,7 +2872,7 @@ 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
@ -2889,7 +2882,7 @@ 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
@ -2902,7 +2895,7 @@ 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
@ -2956,7 +2949,7 @@ 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