From 9f9da628ccb5d107bbbccc038d0fc8d80651e999 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Mon, 11 Sep 2017 03:39:50 +0300 Subject: [PATCH] [ewald] Tabulky asymptotik. Former-commit-id: 064d1368911d5d2879407a8ff794c9a483598297 --- notes/ewald.lyx | 1168 ++++++++++++++++++++++++++++++++++++++++++++++- 1 file changed, 1146 insertions(+), 22 deletions(-) diff --git a/notes/ewald.lyx b/notes/ewald.lyx index 8969ba6..320dfd2 100644 --- a/notes/ewald.lyx +++ b/notes/ewald.lyx @@ -29,6 +29,12 @@ \repeat \renewcommand{\lyxmathsym}[1]{#1} + +\usepackage{polyglossia} +\setmainlanguage{english} +\setotherlanguage{russian} +\newfontfamily\russianfont[Script=Cyrillic]{URW Palladio L} +%\newfontfamily\russianfont[Script=Cyrillic]{DejaVu Sans} \end_preamble \use_default_options true \maintain_unincluded_children false @@ -90,9 +96,9 @@ \shortcut idx \color #008000 \end_index -\leftmargin 2cm +\leftmargin 1cm \topmargin 2cm -\rightmargin 2cm +\rightmargin 1cm \bottommargin 2cm \secnumdepth 3 \tocdepth 3 @@ -690,7 +696,7 @@ where \begin_inset Formula $h_{p}^{(1)}$ \end_inset - in the meaningful cases; TODO) and + in all meaningful cases; TODO) and \begin_inset Formula $c_{p}^{l,m,t\leftarrow l',m',t'}$ \end_inset @@ -889,21 +895,71 @@ reference "eq:2d long range regularisation problem statement" . But it turns out that the family of functions \begin_inset Formula -\[ -\rho_{\kappa,c}(r)\equiv\left(1-e^{-cr}\right)^{\text{\kappa}},\quad c>0,\kappa\in\nats -\] +\begin{equation} +\rho_{\kappa,c}(r)\equiv\left(1-e^{-cr}\right)^{\text{\kappa}},\quad c>0,\kappa\in\nats\label{eq:binom regularisation function} +\end{equation} \end_inset -leads to satisfactory results, as will be shown below. +might lead to satisfactory results; see below. + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Another analytically feasible possibility could be +\begin_inset Formula +\begin{equation} +\rho_{p}^{\textup{ig.}}\equiv e^{-p/x^{2}}.\label{eq:inverse gaussian regularisation function} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Nope, propably did not work. +\end_layout + +\end_inset + + \end_layout \begin_layout Subsubsection -Hankel transforms of the long-range parts +Hankel transforms of the long-range parts, „binomial“ regularisation +\begin_inset CommandInset label +LatexCommand label +name "sub:Hankel-transforms-binom-reg" + +\end_inset + + \end_layout \begin_layout Standard -Let +Let +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\rho_{\kappa,c}$ +\end_inset + + from +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:binom regularisation function" + +\end_inset + + serve as the regularisation fuction and +\end_layout + +\end_inset + + \end_layout \begin_layout Standard @@ -963,7 +1019,7 @@ status open \begin_layout Standard \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout \begin_inset Formula @@ -1327,7 +1383,7 @@ zpět \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout \begin_inset Formula @@ -1478,6 +1534,1003 @@ Let's do it. \end_inset +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. + if +\begin_inset Formula $q-n\ge3$ +\end_inset + +; but at least the necessary minimum of +\begin_inset Formula $q=1,2$ +\end_inset + + would be covered this way. + The associated Legendre function can be expressed as a finite +\begin_inset Quotes eld +\end_inset + +polynomial +\begin_inset Quotes erd +\end_inset + + if +\begin_inset Formula $q\ge n$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Float table +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center + +\size footnotesize +\begin_inset Tabular + + + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +\begin_inset Formula $\kappa=0$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +\begin_inset Formula $n$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +0 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +\begin_inset Formula $q$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +x +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +w +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +0 +\end_layout + +\end_inset + + + + +\end_inset + + +\begin_inset space \hspace*{\fill} +\end_inset + + +\begin_inset Tabular + + + + + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +\begin_inset Formula $\kappa=1$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +\begin_inset Formula $n$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +0 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +3 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +4 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +\begin_inset Formula $q$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +w +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +3 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +x +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +w +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + + + +\end_inset + + +\begin_inset space \hspace*{\fill} +\end_inset + + +\begin_inset Tabular + + + + + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +\begin_inset Formula $\kappa=2$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +\begin_inset Formula $n$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +0 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +3 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +4 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +\begin_inset Formula $q$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +0/w +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +3 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +4 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +3 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +3 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +x +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +3 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +2 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\size footnotesize +1 +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Asymptotical behaviour of some +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:2D Hankel transform of regularized outgoing wave, decomposition" + +\end_inset + + obtained by Mathematica for +\begin_inset Formula $k\to\infty$ +\end_inset + +. + The table entries are the +\begin_inset Formula $N$ +\end_inset + + of +\begin_inset Formula $\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right)=o\left(1/k^{N}\right)$ +\end_inset + +. + The special entry +\begin_inset Quotes eld +\end_inset + +x +\begin_inset Quotes erd +\end_inset + + means that Mathematica was not able to calculate the integral, and +\begin_inset Quotes eld +\end_inset + +w +\begin_inset Quotes erd +\end_inset + + denotes that the first returned term was not simply of the kind +\begin_inset Formula $(\ldots)k^{-N-1}$ +\end_inset + +. +\begin_inset CommandInset label +LatexCommand label +name "tab:Asymptotical-behaviour-Mathematica" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + \end_layout @@ -1545,27 +2598,98 @@ More generally, \end_inset -\end_layout +\begin_inset Note Note +status open \begin_layout Subsubsection -Alternative regularisation with +Hankel transforms of the long-range parts, alternative regularisation with + \begin_inset Formula $e^{-p/x^{2}}$ \end_inset +\begin_inset CommandInset label +LatexCommand label +name "sub:Hankel-transforms-ig-reg" + +\end_inset + + \end_layout -\begin_layout Standard -From [REF Прудников, том 2, 2.12.9.14] +\begin_layout Plain Layout +From [REF +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{russian} +\end_layout + +\end_inset + +Прудников, том 2 +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{russian} +\end_layout + +\end_inset + +, 2.12.9.14] \begin_inset Formula -\begin{multline*} +\begin{multline} \int_{0}^{\infty}x^{\alpha-1}e^{-p/x^{2}}J_{\nu}\left(cx\right)\,\ud x=\frac{2^{\alpha-1}}{c^{\alpha}}Γ\begin{bmatrix}\left(\alpha+\nu\right)/2\\ 1+\left(\nu-\alpha\right)/2 \end{bmatrix}{}_{0}F_{2}\left(1-\frac{\nu+\alpha}{2},1+\frac{\nu-\alpha}{2};\frac{c^{2}p}{4}\right)\\ +\frac{c^{\nu}p^{\left(\alpha+\nu\right)/2}}{2^{\nu+1}}\text{Γ}\begin{bmatrix}\left(\alpha+\nu\right)/2\\ \nu+1 -\end{bmatrix}{}_{0}F_{2}\left(1+\frac{\nu+\alpha}{2},\nu+1;\frac{c^{2}p}{4}\right),\qquad[c,\Re p>0;\Re\alpha<3/2]. -\end{multline*} +\end{bmatrix}{}_{0}F_{2}\left(1+\frac{\nu+\alpha}{2},\nu+1;\frac{c^{2}p}{4}\right),\qquad[c,\Re p>0;\Re\alpha<3/2].\label{eq:prudnikov2 eq 2.12.9.14} +\end{multline} + +\end_inset + +Let now +\begin_inset Formula $\rho_{p}^{\textup{ig.}}$ +\end_inset + + from +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:inverse gaussian regularisation function" + +\end_inset + + serve as the regularisation fuction and +\begin_inset Formula +\[ +\pht n{s_{q,k_{0}}^{\textup{L}'p}}\left(k\right)\equiv\int_{0}^{\infty}\frac{e^{ik_{0}r}}{\left(k_{0}r\right)^{q}}J_{n}\left(kr\right)e^{-p/r^{2}}r\,\ud r. +\] + +\end_inset + +And it seems that this is a dead-end, because +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:prudnikov2 eq 2.12.9.14" + +\end_inset + + cannot deal with the +\begin_inset Formula $e^{ik_{0}r}$ +\end_inset + + part. + Damn. +\end_layout \end_inset @@ -1573,7 +2697,7 @@ From [REF Прудников, том 2, 2.12.9.14] \end_layout \begin_layout Subsection -3d +3d (TODO) \end_layout \begin_layout Standard @@ -1795,9 +2919,9 @@ we have and with unitary angular frequency Ft., i.e. \begin_inset Formula -\[ -\uaft f(\vect k)\equiv\frac{1}{\left(2\pi\right)^{n/2}}\int_{\mathbb{R}^{n}}f(\vect x)e^{-i\vect x\cdot\vect k}\ud^{n}\vect x -\] +\begin{equation} +\uaft f(\vect k)\equiv\frac{1}{\left(2\pi\right)^{n/2}}\int_{\mathbb{R}^{n}}f(\vect x)e^{-i\vect x\cdot\vect k}\ud^{n}\vect x\label{eq:Ft unitary angular frequency} +\end{equation} \end_inset