From 93cde82a9a6d0cbc35aa368d853b5dc9d98798a8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Tue, 26 May 2020 16:00:20 +0300 Subject: [PATCH] =?UTF-8?q?Notes=20on=20evaluating=20=CE=94=5Fn=20factor?= =?UTF-8?q?=20in=20the=20lattice=20sums.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Former-commit-id: f6bf3c3167f8c3f48b552e271b99d3b28b4bfc46 --- notes/Kambe_Delta_n.lyx | 266 ++++++++++++++++++++++++++++++++++++++++ 1 file changed, 266 insertions(+) create mode 100644 notes/Kambe_Delta_n.lyx diff --git a/notes/Kambe_Delta_n.lyx b/notes/Kambe_Delta_n.lyx new file mode 100644 index 0000000..9b879d0 --- /dev/null +++ b/notes/Kambe_Delta_n.lyx @@ -0,0 +1,266 @@ +#LyX 2.4 created this file. For more info see https://www.lyx.org/ +\lyxformat 584 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\use_default_options true +\maintain_unincluded_children false +\language finnish +\language_package default +\inputencoding utf8 +\fontencoding auto +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_roman_osf false +\font_sans_osf false +\font_typewriter_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\use_lineno 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style english +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tablestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +\begin_inset FormulaMacro +\newcommand{\ud}{\mathrm{d}} +\end_inset + + +\begin_inset Formula +\begin{equation} +\Delta_{n}(x,z)\equiv\int_{x}^{\infty}t^{-\frac{1}{2}-n}e^{-t+\frac{z^{2}}{4t}}\ud t\label{eq:Delta definition} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Integration per partes: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int t^{-\frac{1}{2}-n}\ud t=\frac{t^{\frac{1}{2}-n}}{\frac{1}{2}-n}; +\] + +\end_inset + + +\begin_inset Formula +\[ +\frac{\ud}{\ud t}e^{-t+\frac{z^{2}}{4t}}=\left(-1-\frac{z^{2}}{4t^{2}}\right)e^{-t+\frac{z^{2}}{4t}} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align*} +\left(\frac{1}{2}-n\right)\Delta_{n} & =-x^{\frac{1}{2}-n}e^{-x+\frac{z^{2}}{4x}}+\int_{x}^{\infty}t^{\frac{1}{2}-n}e^{-t+\frac{z^{2}}{4t}}\ud t+\frac{z^{2}}{4}\int_{x}^{\infty}t^{\frac{-3}{2}-n}e^{-t+\frac{z^{2}}{4t}}\ud t\\ + & =-x^{\frac{1}{2}-n}e^{-x+\frac{z^{2}}{4x}}+\Delta_{n-1}+\frac{z^{2}}{4}\Delta_{n+1}, +\end{align*} + +\end_inset + + +\begin_inset Formula +\begin{equation} +\Delta_{n+1}=\frac{4}{z^{2}}\left(\left(\frac{1}{2}-n\right)\Delta_{n}-\Delta_{n-1}+x^{\frac{1}{2}-n}e^{-x+\frac{z^{2}}{4x}}\right).\label{eq:Delta recurrence} +\end{equation} + +\end_inset + +There are obviously wrong signs in Kambe II, (A 3.3). +\end_layout + +\begin_layout Standard +Eq. + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Delta recurrence" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + is obviously unsuitable for numerical computation when +\begin_inset Formula $z$ +\end_inset + +approaches 0. + However, the definition +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Delta definition" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + suggests that the function should be analytical around +\begin_inset Formula $z=0$ +\end_inset + +. + If +\begin_inset Formula $z=0$ +\end_inset + +, one has (by definition of incomplete Г function) +\begin_inset Formula +\begin{equation} +\Delta_{n}(x,0)=\Gamma\left(\frac{1}{2}-n,x\right).\label{eq:Delta:z = 0} +\end{equation} + +\end_inset + +For convenience, label +\begin_inset Formula $w=z^{2}/4$ +\end_inset + + and +\begin_inset Formula +\[ +\Delta'_{n}\left(x,w\right)\equiv\int_{x}^{\infty}t^{-\frac{1}{2}-n}e^{-t+\frac{w}{t}}\ud t. +\] + +\end_inset + + Differentiating by parameter +\begin_inset Formula $w$ +\end_inset + + (which should be fine as long as the integration contour does not go through + zero) gives +\begin_inset Formula +\[ +\frac{\partial\Delta'_{n}\left(x,w\right)}{\partial w}=\Delta'_{n+1}\left(x,w\right), +\] + +\end_inset + +so by recurrence +\begin_inset Formula +\[ +\frac{\partial^{k}}{\partial w^{k}}\Delta'_{n}\left(x,w\right)=\Delta'_{n+k}\left(x,w\right). +\] + +\end_inset + +Together with +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Delta:z = 0" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, this gives an expansion around +\begin_inset Formula $w=0$ +\end_inset + +: +\begin_inset Formula +\[ +\Delta_{n}'\left(x,w\right)=\sum_{k=0}^{\infty}\Gamma\left(\frac{1}{2}-n-k,x\right)\frac{w^{k}}{k!}, +\] + +\end_inset + + +\begin_inset Formula +\[ +\Delta_{n}\left(x,z\right)=\sum_{k=0}^{\infty}\Gamma\left(\frac{1}{2}-n-k,x\right)\frac{\left(z/2\right)^{2k}}{k!}. +\] + +\end_inset + +The big negative first arguments in incomplete +\begin_inset Formula $\Gamma$ +\end_inset + + functions should be good (at least I think so, CHECKME), as well as the + +\begin_inset Formula $1/k!$ +\end_inset + + factor (of course). + I am not sure what the convergence radius is, but for +\begin_inset Formula $\left|z\right|<2$ +\end_inset + + there seems to be absolutely no problem in using this formula. +\end_layout + +\end_body +\end_document