[ewald] pokračování
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Marek Nečada
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190
notes/ewald.lyx
190
notes/ewald.lyx
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@ -172,6 +172,11 @@
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\nats}{\mathbb{N}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\reals}{\mathbb{R}}
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\end_inset
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@ -643,7 +648,7 @@ The translation operator
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for compact scatterers in 3d can be expressed as
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\begin_inset Formula
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\[
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S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(\left|\vect r\right|\right)
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S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(k_{0}\left|\vect r\right|\right)
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\]
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\end_inset
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@ -673,14 +678,14 @@ where
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The spherical Hankel functions can be expressed analytically as (REF DLMF
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10.49.6, 10.49.1)
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\begin_inset Formula
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\[
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h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!},
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\]
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\begin{equation}
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h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!},\label{eq:spherical Hankel function series}
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\end{equation}
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\end_inset
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so if we find a way to deal with the radial functions
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\begin_inset Formula $s_{q}(r)=e^{ir}r^{-q}$
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\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
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\end_inset
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,
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@ -699,18 +704,176 @@ h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\righ
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\end_layout
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\begin_layout Standard
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Assume that all scatterers are placed in the plane
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\begin_inset Formula $\vect z=0$
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\end_inset
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, so that the 2d Fourier transform of the long-range part of the translation
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operator in terms of Hankel transforms, according to
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Fourier v. Hankel tf 2d"
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\end_inset
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, reads
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{multline*}
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\uaft{S_{l',m',t'\leftarrow l,m,t}^{\textup{L}}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
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\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\frac{\pi}{2},0\right)e^{i(m'-m)\phi}i^{m'-m}\pht{m'-m}{h_{p}^{(1)\textup{L}}\left(k_{0}\vect{\bullet}\right)}\left(\left|\vect k\right|\right)
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\end{multline*}
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\end_inset
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Here
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\begin_inset Formula $h_{p}^{(1)\textup{L}}=h_{p}^{(1)}-h_{p}^{(1)\textup{S}}$
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\end_inset
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is a long range part of a given spherical Hankel function which has to
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be found and which contains all the terms with far-field (
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\begin_inset Formula $r\to\infty$
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\end_inset
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) asymptotics proportional to
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\begin_inset Formula $\sim e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
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\end_inset
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,
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\begin_inset Formula $q\le Q$
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\end_inset
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where
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\begin_inset Formula $Q$
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\end_inset
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is at least two in order to achieve absolute convergence of the direct-space
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sum, but might be higher in order to speed the convergence up.
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\end_layout
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\begin_layout Standard
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Obviously, all the terms
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\begin_inset Formula $\propto s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
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\end_inset
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,
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\begin_inset Formula $q>Q$
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\end_inset
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of the spherical Hankel function
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:spherical Hankel function series"
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\end_inset
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can be kept untouched as part of
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\begin_inset Formula $h_{p}^{(1)\textup{S}}$
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\end_inset
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, as they decay fast enough.
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\end_layout
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\begin_layout Standard
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The remaining task is therefore to find a suitable decomposition of
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\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
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\end_inset
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,
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\begin_inset Formula $q\le Q$
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\end_inset
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into short-range and long-range parts,
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\begin_inset Formula $s_{q}(r)=s_{q}^{\textup{S}}(r)+s_{q}^{\textup{L}}(r)$
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\end_inset
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, such that
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\begin_inset Formula $s_{q}^{\textup{L}}(r)$
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\end_inset
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contains all the slowly decaying asymptotics and its Hankel transforms
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decay desirably fast as well,
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\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$
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\end_inset
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,
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\begin_inset Formula $z\to\infty$
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\end_inset
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.
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The latter requirement calls for suitable regularisation functions—
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\begin_inset Formula $s_{q}^{\textup{L}}$
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\end_inset
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must be sufficiently smooth in the origin, so that
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\begin_inset Formula
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\begin{equation}
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\pht n{s_{q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement}
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\end{equation}
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\end_inset
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exists and decays fast enough.
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\begin_inset Formula $J_{\nu}(r)\sim\left(r/2\right)^{\nu}/\Gamma\left(\nu+1\right)$
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\end_inset
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(REF DLMF 10.7.3) near the origin, so the regularisation function should
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be
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\begin_inset Formula $\rho(r)=o(r^{q-n-1})$
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\end_inset
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only to make
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\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}$
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\end_inset
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converge.
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The additional decay speed requirement calls for at least
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\begin_inset Formula $\rho(r)=o(r^{q-n+Q-1})$
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\end_inset
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, I guess.
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At the same time,
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\begin_inset Formula $\rho(r)$
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\end_inset
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must converge fast enough to one for
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\begin_inset Formula $r\to\infty$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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The electrostatic Ewald summation uses regularisation with
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\begin_inset Formula $1-e^{-cr^{2}}$
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\end_inset
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.
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However, such choice does not seem to lead to an analytical solution for
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the current problem
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:2d long range regularisation problem statement"
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\end_inset
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.
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But it turns out that the family of functions
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\begin_inset Formula
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\[
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\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\frac{\pi}{2},0\right)e^{i(m'-m)\phi}i^{m'-m}\pht{m'-m}{z_{p}^{(J)}}\left(\left|\vect k\right|\right)
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\rho_{\kappa,c}(r)\equiv\left(1-e^{-cr}\right)^{\text{\kappa}},\quad c>0,\kappa\in\nats
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\]
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\end_inset
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leads to satisfactory results, as will be shown below.
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\end_layout
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\begin_layout Subsubsection
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Hankel transforms of the long-range parts
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\end_layout
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\begin_layout Subsection
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@ -719,9 +882,10 @@ h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\righ
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\begin_layout Standard
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\begin_inset Formula
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\[
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\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect k},\phi_{\vect k}\right)\left(-i\right)^{p}\usht p{z_{p}^{(J)}}\left(\left|\vect k\right|\right)
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\]
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\begin{multline*}
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\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
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\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect k},\phi_{\vect k}\right)\left(-i\right)^{p}\usht p{z_{p}^{(J)}}\left(\left|\vect k\right|\right)
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\end{multline*}
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\end_inset
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@ -777,7 +941,7 @@ where the spherical Hankel transform
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2)
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\begin_inset Formula
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\[
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\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).
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\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).
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\]
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\end_inset
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@ -787,7 +951,7 @@ Using this convention, the inverse spherical Hankel transform is given by
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3)
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\begin_inset Formula
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\[
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g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\, k^{2}\bsht lg(k)j_{l}(k),
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g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\,k^{2}\bsht lg(k)j_{l}(k),
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\]
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\end_inset
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@ -800,7 +964,7 @@ so it is not unitary.
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An unitary convention would look like this:
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\begin_inset Formula
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\begin{equation}
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\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}
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\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}
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\end{equation}
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\end_inset
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@ -854,7 +1018,7 @@ where the Hankel transform of order
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is defined as
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\begin_inset Formula
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\begin{equation}
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\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}
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\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\,g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}
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\end{equation}
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\end_inset
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