[ewald] Jdu domů
Former-commit-id: 9d73be0abec970cbb8054c4e07c129418e4f512e
This commit is contained in:
Marek Nečada
parent
6016fcd55d
commit
34a6d1a764
|
|
@ -0,0 +1,657 @@
|
|||
#LyX 2.1 created this file. For more info see http://www.lyx.org/
|
||||
\lyxformat 474
|
||||
\begin_document
|
||||
\begin_header
|
||||
\textclass article
|
||||
\use_default_options true
|
||||
\maintain_unincluded_children false
|
||||
\language finnish
|
||||
\language_package default
|
||||
\inputencoding auto
|
||||
\fontencoding global
|
||||
\font_roman TeX Gyre Pagella
|
||||
\font_sans default
|
||||
\font_typewriter default
|
||||
\font_math auto
|
||||
\font_default_family default
|
||||
\use_non_tex_fonts true
|
||||
\font_sc false
|
||||
\font_osf true
|
||||
\font_sf_scale 100
|
||||
\font_tt_scale 100
|
||||
\graphics default
|
||||
\default_output_format pdf4
|
||||
\output_sync 0
|
||||
\bibtex_command default
|
||||
\index_command default
|
||||
\paperfontsize 10
|
||||
\spacing single
|
||||
\use_hyperref true
|
||||
\pdf_title "Sähköpajan päiväkirja"
|
||||
\pdf_author "Marek Nečada"
|
||||
\pdf_bookmarks true
|
||||
\pdf_bookmarksnumbered false
|
||||
\pdf_bookmarksopen false
|
||||
\pdf_bookmarksopenlevel 1
|
||||
\pdf_breaklinks false
|
||||
\pdf_pdfborder false
|
||||
\pdf_colorlinks false
|
||||
\pdf_backref false
|
||||
\pdf_pdfusetitle true
|
||||
\papersize a3paper
|
||||
\use_geometry true
|
||||
\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
|
||||
\biblio_style plain
|
||||
\use_bibtopic false
|
||||
\use_indices false
|
||||
\paperorientation portrait
|
||||
\suppress_date false
|
||||
\justification true
|
||||
\use_refstyle 1
|
||||
\index Index
|
||||
\shortcut idx
|
||||
\color #008000
|
||||
\end_index
|
||||
\leftmargin 1cm
|
||||
\topmargin 5mm
|
||||
\rightmargin 1cm
|
||||
\bottommargin 1cm
|
||||
\secnumdepth 3
|
||||
\tocdepth 3
|
||||
\paragraph_separation indent
|
||||
\paragraph_indentation default
|
||||
\quotes_language swedish
|
||||
\papercolumns 1
|
||||
\papersides 1
|
||||
\paperpagestyle 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
|
||||
|
||||
\lang english
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\uoft}[1]{\mathfrak{F}#1}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\uaft}[1]{\mathfrak{\mathbb{F}}#1}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\usht}[2]{\mathbb{S}_{#1}#2}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\bsht}[2]{\mathrm{S}_{#1}#2}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\pht}[2]{\mathfrak{\mathbb{H}}_{#1}#2}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\vect}[1]{\mathbf{#1}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\ud}{\mathrm{d}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\basis}[1]{\mathfrak{#1}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\dc}[1]{Ш_{#1}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\rec}[1]{#1^{-1}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\recb}[1]{#1^{\widehat{-1}}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\ints}{\mathbb{Z}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\nats}{\mathbb{N}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\reals}{\mathbb{R}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\ush}[2]{Y_{#1,#2}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\hgfr}{\mathbf{F}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\ph}{\mathrm{ph}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\kor}[1]{\underline{#1}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\koru}[1]{\overline{#1}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\hgf}{F}
|
||||
\end_inset
|
||||
|
||||
Let
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
|
||||
\mbox{(D15.2.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}Γ\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\sum_{s=0}^{\infty}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{3-q+n}{2}\right)_{s}}{Γ(1+n+s)s!}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{s},\quad\left|\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right|<1\\
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
|
||||
\mbox{(D15.8.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}(\\
|
||||
& & \pi\frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{Γ\left(\frac{3-q+n}{2}\right)\text{Γ}\left(1+n-\frac{2-q+n}{2}\right)}\hgfr\left(\begin{array}{c}
|
||||
\frac{2-q+n}{2},\frac{2-q+n}{2}-\left(1+n\right)+1\\
|
||||
1/2
|
||||
\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)\\
|
||||
& - & \pi\frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(1+n-\frac{3-q+n}{2}\right)}\hgfr\left(\begin{array}{c}
|
||||
\frac{3-q+n}{2},\frac{3-q+n}{2}-\left(1+n\right)+1\\
|
||||
3/2
|
||||
\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi(\\
|
||||
& & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\hgfr\left(\begin{array}{c}
|
||||
\frac{2-q+n}{2},\frac{2-q-n}{2}\\
|
||||
1/2
|
||||
\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)\\
|
||||
& - & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)}\hgfr\left(\begin{array}{c}
|
||||
\frac{3-q+n}{2},\frac{3-q-n}{2}\\
|
||||
3/2
|
||||
\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\
|
||||
\mbox{(D15.2.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\pi\sum_{s=0}^{\infty}(\\
|
||||
& & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{1}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s}\\
|
||||
& - & \frac{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{3-q+n}{2}}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)}\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3}{2}+s\right)s!}\left(-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)^{s})\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\
|
||||
& & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}k^{-2+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{2-q+n}+2s}\\
|
||||
& - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}k^{-3+q\kor{-n}-2s}\left(\sigma c-ik_{0}\right)^{\kor{3-q+n}+2s})\\
|
||||
\mbox{} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}(\\
|
||||
& & \frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\kor{k^{-2+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{2s}}\\
|
||||
& - & \frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\kor{k^{-3+q-2s}}\kor{\left(\sigma c-ik_{0}\right)^{1+2s}})\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\\
|
||||
& & \times\left(\underbrace{\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{3-q+n}{2}\right)\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}}_{\equiv c_{q,n,s}}-\underbrace{\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}}_{č_{q,n,s}}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\kor{\left(\sigma c-ik_{0}\right)^{2s}}c_{q,n,s}-\frac{\left(\sigma c-ik_{0}\right)^{2s+1}}{k}č_{q,n,s}\right)\\
|
||||
\mbox{(binom.)} & = & \kor{\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}}\frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(c_{q,n,s}\sum_{t=0}^{2s}\binom{2s}{t}\left(\kor{\sigma}c\right)^{t}\left(-ik_{0}\right)^{2s-t}-č_{q,n,s}\sum_{t=0}^{2s+1}\binom{2s+1}{t}\left(\kor{\sigma}c\right)^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)\\
|
||||
\mbox{(conds?)} & = & \frac{\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}}\pi\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(c_{q,n,s}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-č_{q,n,s}\sum_{t=0}^{2s+1}\binom{2s+1}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
now the Stirling number of the 2nd kind
|
||||
\begin_inset Formula $\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}=0$
|
||||
\end_inset
|
||||
|
||||
if
|
||||
\begin_inset Formula $\kappa>t$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
What about the gamma fn on the left? Using DLMF 5.5.5, which says
|
||||
\begin_inset Formula $Γ(2z)=\pi^{-1/2}2^{2z-1}\text{Γ}(z)\text{Γ}(z+\frac{1}{2})$
|
||||
\end_inset
|
||||
|
||||
we have
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\text{Γ}\left(2-q+n\right)=\frac{2^{1-q+n}}{\sqrt{\pi}}\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{3-q+n}{2}\right),
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
so
|
||||
\size footnotesize
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \frac{\kor{\text{Γ}\left(2-q+n\right)}}{\kor{2^{n}}k_{0}^{q}}\kor{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{3-q+n}{2}\right)}\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{2-q+n}{2}\right)}\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s+1}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)\\
|
||||
& = & \frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(\frac{\kor{\text{Γ}\left(\frac{2-q+n}{2}\right)\left(\frac{2-q+n}{2}\right)_{s}}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-\frac{\kor{\text{Γ}\left(\frac{3-q+n}{2}\right)\left(\frac{3-q+n}{2}\right)_{s}}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s+1}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)\\
|
||||
\mbox{(D5.2.5)} & = & \frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(\frac{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-\frac{\text{Γ}\left(\frac{3-q+n}{2}+s\right)\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s+1}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\size default
|
||||
The two terms have to be treated fifferently depending on whether q
|
||||
\begin_inset Formula $q+n$
|
||||
\end_inset
|
||||
|
||||
is even or odd.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
First, assume that
|
||||
\begin_inset Formula $q+n$
|
||||
\end_inset
|
||||
|
||||
is even, so the left term has gamma functions and pochhammer symbols with
|
||||
integer arguments, while the right one has half-integer arguments.
|
||||
As
|
||||
\begin_inset Formula $n$
|
||||
\end_inset
|
||||
|
||||
is non-negative and
|
||||
\begin_inset Formula $q$
|
||||
\end_inset
|
||||
|
||||
is positive,
|
||||
\begin_inset Formula $\frac{q+n}{2}$
|
||||
\end_inset
|
||||
|
||||
is positive, and the Pochhammer symbol
|
||||
\begin_inset Formula $\left(\frac{2-q-n}{2}\right)_{s}=0$
|
||||
\end_inset
|
||||
|
||||
if
|
||||
\begin_inset Formula $s\ge\frac{q+n}{2}$
|
||||
\end_inset
|
||||
|
||||
, which transforms the sum over
|
||||
\begin_inset Formula $s$
|
||||
\end_inset
|
||||
|
||||
to a finite sum for the left term.
|
||||
However, there still remain divergent terms if
|
||||
\begin_inset Formula $\frac{2-q+n}{2}+s\le0$
|
||||
\end_inset
|
||||
|
||||
(let's handle this later; maybe D15.8.6–7 may be then be useful)! Now we
|
||||
need to perform some transformations of variables to make the other sum
|
||||
finite as well
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
Pár kroků zpět:
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{\text{Γ}\left(2-q+n\right)}}{\kor{2^{n}}k_{0}^{q}}\kor{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\times\left(\underbrace{\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{3-q+n}{2}\right)}\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}}_{\equiv c_{q,n,s}}-\underbrace{\frac{\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\kor{\text{Γ}\left(\frac{2-q+n}{2}\right)}\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}}_{č_{q,n,s}}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\times\left(\frac{\text{Γ}\left(\frac{2-q+n}{2}\right)\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}-\frac{\text{Γ}\left(\frac{3-q+n}{2}\right)\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
If
|
||||
\begin_inset Formula $q+n$
|
||||
\end_inset
|
||||
|
||||
is even and
|
||||
\begin_inset Formula $2-q+n\le0$
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\kor{\hgfr}\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)\\
|
||||
\mbox{(D15.1.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)\koru{\text{Γ}(1+n)}}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\koru{\hgf}\left(\frac{2-q+n}{2},\kor{\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}}\right)\\
|
||||
\mbox{(D15.8.6)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\kor{k^{n}}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{2-q+n}}}\koru{\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\left(\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{2-q+n}{2}}}}\hgf\left(\begin{array}{c}
|
||||
\frac{2-q+n}{2},\koru{\kor{1-\left(1+n\right)+\frac{2-q+n}{2}}}\\
|
||||
\koru{\kor{1-\frac{3-q+n}{2}+\frac{2-q+n}{2}}}
|
||||
\end{array};\koru{\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}}\right)\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\koru{k^{q-2}}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\koru{\frac{3}{2}\left(2-q+n\right)}}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\hgf\left(\begin{array}{c}
|
||||
\frac{2-q+n}{2},\koru{\frac{2-q-n}{2}}\\
|
||||
\koru{1/2}
|
||||
\end{array};\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)}\\
|
||||
\mbox{(D15.2.1)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\kor{\text{Γ}\left(2-q+n\right)}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\koru{\sum_{s=0}^{\infty}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}}\\
|
||||
\mbox{(D5.5.5)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{\kor{2^{n}}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\koru{\frac{2^{1-q\kor{+n}}}{\sqrt{\pi}}\kor{\text{Γ}\left(\frac{2-q+n}{2}\right)}\text{Γ}\left(\frac{3-q+n}{2}\right)}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\infty}\frac{\kor{\left(\frac{2-q+n}{2}\right)_{s}}\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}\\
|
||||
\mbox{(D5.2.5)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\koru{2^{1-q}}}{\sqrt{\pi}}\text{Γ}\left(\frac{3-q+n}{2}\right)\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\infty}\frac{\koru{\text{Γ}\left(\frac{2-q+n}{2}+s\right)}\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{2^{1-q}}{\sqrt{\pi}}\text{Γ}\left(\frac{3-q+n}{2}\right)\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\frac{q+n}{2}}\frac{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{2^{1-q}}{\sqrt{\pi}}\text{Γ}\left(\frac{3-q+n}{2}\right)\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\frac{q+n}{2}}\frac{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
now
|
||||
\begin_inset Formula $\left(\frac{2-q-n}{2}\right)_{s}=0$
|
||||
\end_inset
|
||||
|
||||
whenever
|
||||
\begin_inset Formula $s\ge\frac{q+n}{2}$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $\text{Γ}\left(\frac{2-q+n}{2}+s\right)$
|
||||
\end_inset
|
||||
|
||||
is singular whenever
|
||||
\begin_inset Formula $s\le-\frac{2-q+n}{2}$
|
||||
\end_inset
|
||||
|
||||
, so we are no less fucked than before.
|
||||
Maybe let's try the other variable transformation.
|
||||
Or what about (D15.8.27)?
|
||||
\size footnotesize
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray}
|
||||
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\kor{\hgf\left(\begin{array}{c}
|
||||
\frac{2-q+n}{2},\frac{2-q-n}{2}\\
|
||||
1/2
|
||||
\end{array};\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)}\nonumber \\
|
||||
\mbox{(D15.8.27)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\kor{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\koru{\frac{\kor{Γ\left(\frac{3-q+n}{2}\right)}Γ\left(\frac{3-q-n}{2}\right)}{2Γ\left(\frac{1}{2}\right)Γ\left(2-q+\frac{1}{2}\right)}\left(\hgf\left(\begin{array}{c}
|
||||
2-q+n,2-q-n\\
|
||||
2-q+\frac{1}{2}
|
||||
\end{array};\frac{1}{2}-\frac{\sigma c-ik_{0}}{ik}\right)+\hgf\left(\begin{array}{c}
|
||||
2-q+n,2-q-n\\
|
||||
2-q+\frac{1}{2}
|
||||
\end{array};\frac{1}{2}+\frac{\sigma c-ik_{0}}{ik}\right)\right)}\nonumber \\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\kor{\text{Γ}\koru{\left(\frac{3-q+n}{2}-\frac{2-q+n}{2}\right)}}\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\kor{\text{Γ}\left(\frac{1}{2}\right)}\text{Γ}\left(2-q+\frac{1}{2}\right)}\left(\hgf\left(\begin{array}{c}
|
||||
2-q+n,2-q-n\\
|
||||
2-q+\frac{1}{2}
|
||||
\end{array};\frac{1}{2}-\frac{\sigma c-ik_{0}}{ik}\right)+\hgf\left(\begin{array}{c}
|
||||
2-q+n,2-q-n\\
|
||||
2-q+\frac{1}{2}
|
||||
\end{array};\frac{1}{2}+\frac{\sigma c-ik_{0}}{ik}\right)\right)\nonumber \\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\kor{\left(\hgf\left(\begin{array}{c}
|
||||
2-q+n,2-q-n\\
|
||||
2-q+\frac{1}{2}
|
||||
\end{array};\frac{1}{2}-\frac{\sigma c-ik_{0}}{ik}\right)+\hgf\left(\begin{array}{c}
|
||||
2-q+n,2-q-n\\
|
||||
2-q+\frac{1}{2}
|
||||
\end{array};\frac{1}{2}+\frac{\sigma c-ik_{0}}{ik}\right)\right)}\nonumber \\
|
||||
\mbox{(D15.2.1)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\koru{\sum_{s=0}^{\infty}\left(\frac{\left(2-q+n\right)_{s}\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\kor{\left(\left(\frac{1}{2}-\frac{\sigma c-ik_{0}}{ik}\right)^{s}+\left(\frac{1}{2}+\frac{\sigma c-ik_{0}}{ik}\right)^{s}\right)}\right)}\nonumber \\
|
||||
\mbox{(binom)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\kor{\left(2-q+n\right)_{s}}\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\koru{\sum_{r=0}^{s}\binom{s}{r}\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}2^{r-s}\left(\left(-1\right)^{r}+1\right)}\nonumber \\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\kor{\left(1+n\right)_{-\frac{2-q+n}{2}}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\koru{\text{Γ}\left(2-q+n+s\right)}\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\kor{\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}}\frac{\koru{\text{Γ}\left(1+n\right)}\text{Γ}\left(\frac{3-q-n}{2}\right)}{\koru{\text{Γ}\left(\frac{q+n}{2}\right)}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\kor{\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{Γ\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\koru{\left(ik\right)^{-r}}\koru{\kor{\left(\sigma c-ik_{0}\right)^{r-\frac{3}{2}\left(2-q+n\right)}}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\
|
||||
(bionm) & = & \kor{\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\koru{\sum_{w=0}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)}2^{r-s}\left(\left(-1\right)^{r}+1\right)\label{eq:ugliness withous singularities}\\
|
||||
& = & \koru{\kappa!\left(-1\right)^{\kappa}}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=\kor 0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=\kor 0}^{s}\binom{\kor s}{\kor r}\left(ik\right)^{-r}\sum_{w=\kor 0}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{\kor w}\koru{\kor{\begin{Bmatrix}w\\
|
||||
\kappa
|
||||
\end{Bmatrix}}}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\
|
||||
& = & \kappa!\left(-1\right)^{\kappa}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=\koru{\kappa}}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=\koru{\kappa}}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=\koru{\kappa}}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\begin{Bmatrix}w\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber \\
|
||||
& = & \kappa!\left(-1\right)^{\kappa}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=\kappa}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=\kappa}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=\kappa}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\begin{Bmatrix}w\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)\nonumber
|
||||
\end{eqnarray}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
The previous things are valid only if
|
||||
\begin_inset Formula $q$
|
||||
\end_inset
|
||||
|
||||
has a small non-integer part,
|
||||
\begin_inset Formula $q=q'+\varepsilon$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
They might still play a role in the series (especially in the infinite
|
||||
ones) when taking the limit
|
||||
\begin_inset Formula $\varepsilon\to0$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
However, we got rid of the singularities in
|
||||
\begin_inset Formula $\text{Γ}\left(2-q+n+s\right)$
|
||||
\end_inset
|
||||
|
||||
if
|
||||
\begin_inset Formula $\kappa$
|
||||
\end_inset
|
||||
|
||||
is large enough.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
and we get same shit as before due to the singular
|
||||
\begin_inset Formula $\text{Γ}\left(2-q+n+s\right)$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
However,
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
(...) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}2^{r-s}\kor{\left(\left(-1\right)^{r}+1\right)}\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{\koru{floor(s/2)}}\binom{s}{\koru{2r}}\left(\frac{\sigma c-ik_{0}}{ik}\right)^{\koru{2r}}2^{\koru{2r}-s}\left(\left(-1\right)^{\koru{2r}}+1\right)
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
(...) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\kor{\left(\frac{\sigma c-ik_{0}}{ik}\right)^{r}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
|
||||
binom & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\text{Γ}\left(\frac{3-q-n}{2}\right)}{\left(1+n\right)_{-\frac{2-q+n}{2}}2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\koru{\left(ik\right)^{-r}\sum_{b=0}^{r}\binom{r}{b}\sigma^{b}c^{b}\left(-ik_{0}\right)^{r-b}}2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
|
||||
& =
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
aaah.
|
||||
Let's assume that
|
||||
\begin_inset Formula $q$
|
||||
\end_inset
|
||||
|
||||
is not exactly
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\kor{\text{Γ}\left(2-q+n\right)}\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\infty}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}\left(2-q+n\right)\text{Γ}(1+n)}{2^{n}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{\frac{3}{2}\left(2-q+n\right)}}\frac{\left(\frac{3-q+n}{2}\right)_{-\frac{2-q+n}{2}}}{\left(1+n\right)_{-\frac{2-q+n}{2}}}\sum_{s=0}^{\infty}k^{-2s}\frac{\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\left(\frac{1}{2}\right)_{s}s!}\left(\frac{\left(\sigma c-ik_{0}\right)^{2}}{-k^{2}}\right)^{s}
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
zpět
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
& = & \frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(\frac{\text{Γ}\left(\frac{2-q+n}{2}\right)\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)s!}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-\frac{\text{Γ}\left(\frac{3-q+n}{2}\right)\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)s!}\sum_{t=0}^{2s+1}\binom{2s+1}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)\\
|
||||
& = & \frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\kappa!\left(-1\right)^{\kappa}\left(\frac{\text{Γ}\left(\frac{2-q+n}{2}\right)\left(\frac{2-q+n}{2}\right)_{s}\left(\frac{2-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n}{2}\right)\text{Γ}\left(\frac{1}{2}+s\right)\text{Γ}\left(1+s\right)}\sum_{t=0}^{2s}\binom{2s}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s-t}-\frac{\text{Γ}\left(\frac{3-q+n}{2}\right)\left(\frac{3-q+n}{2}\right)_{s}\left(\frac{3-q-n}{2}\right)_{s}}{\text{Γ}\left(\frac{q+n-1}{2}\right)\text{Γ}\left(\frac{3}{2}+s\right)\text{Γ}\left(1+s\right)}\sum_{t=0}^{2s+1}\binom{2s+1}{t}\begin{Bmatrix}t\\
|
||||
\kappa
|
||||
\end{Bmatrix}c^{t}\left(-ik_{0}\right)^{2s+1-t}k^{-1}\right)
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Paragraph*
|
||||
|
||||
\lang english
|
||||
Special case
|
||||
\begin_inset Formula $n=0,q=2$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
Take
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:ugliness withous singularities"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{q-2}\text{Γ}(1+n)}{2^{n}k_{0}^{q}}\frac{\text{Γ}\left(1+n\right)\text{Γ}\left(\frac{3-q-n}{2}\right)}{\text{Γ}\left(\frac{q+n}{2}\right)2\text{Γ}\left(2-q+\frac{1}{2}\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(2-q+n+s\right)\left(2-q-n\right)_{s}}{\left(2-q+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=0}^{\infty|r-\frac{3}{2}\left(2-q+n\right)}\binom{r-\frac{3}{2}\left(2-q+n\right)}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}^{r-\frac{3}{2}\left(2-q+n\right)-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)\\
|
||||
\pht 0{s_{2+\epsilon,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\sum_{s=0}^{\infty}\frac{\text{Γ}\left(s-\epsilon\right)\left(-\epsilon\right)_{s}}{\left(\epsilon+\frac{1}{2}\right)_{s}s!}\sum_{r=0}^{s}\binom{s}{r}\left(ik\right)^{-r}\sum_{w=0}^{\infty}\binom{r+\frac{3}{2}\epsilon}{w}\kor{\sigma^{w}}c^{w}\left(-ik_{0}^{r+\frac{3}{2}\epsilon-w}\right)2^{r-s}\left(\left(-1\right)^{r}+1\right)
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
There is one problematic factor on the previous line,
|
||||
\begin_inset Formula $\Gamma(s-\epsilon)$
|
||||
\end_inset
|
||||
|
||||
for
|
||||
\begin_inset Formula $s=0$
|
||||
\end_inset
|
||||
|
||||
; the other elementary summands are finite in the limit
|
||||
\begin_inset Formula $\epsilon\to0$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Let us analyse the problematic term.
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
\mbox{problem} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\frac{\text{Γ}\left(-\epsilon\right)\kor{\left(-\epsilon\right)_{0}}}{\kor{\left(\epsilon+\frac{1}{2}\right)_{0}0!}}\kor{\sum_{r=0}^{0}\binom{0}{r}\left(ik\right)^{-r}}\sum_{w=0}^{\infty}\binom{\kor r+\frac{3}{2}\epsilon}{w}\sigma^{w}c^{w}\left(-ik_{0}^{\kor r+\frac{3}{2}\epsilon-w}\right)2^{\kor r-0}\kor{\left(\left(-1\right)^{r}+1\right)}\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\text{Γ}\left(-\epsilon\right)\sum_{w=0}^{\infty}\kor{\binom{\frac{3}{2}\epsilon}{w}}\sigma^{w}c^{w}\left(-ik_{0}^{\frac{3}{2}\epsilon-w}\right)2\\
|
||||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\text{Γ}\left(-\epsilon\right)\sum_{w=0}^{\infty}\frac{\Gamma\left(1+\frac{3}{2}\epsilon\right)}{\Gamma\left(w+1\right)\Gamma\left(1+\frac{3}{2}\epsilon-w\right)}\sigma^{w}c^{w}\left(-ik_{0}^{\frac{3}{2}\epsilon-w}\right)2.
|
||||
\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
In the last sum, the divisor
|
||||
\begin_inset Formula $\Gamma\left(1+\frac{3}{2}\epsilon-w\right)$
|
||||
\end_inset
|
||||
|
||||
counters the
|
||||
\begin_inset Formula $\epsilon\to0$
|
||||
\end_inset
|
||||
|
||||
divergence for all summands except for the case
|
||||
\begin_inset Formula $w=0$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
However, that divergence gets canceled by the
|
||||
\begin_inset Formula $\kappa$
|
||||
\end_inset
|
||||
|
||||
-regularisation,
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\mbox{problem}=\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{\epsilon}\text{Γ}(1+n)}{2^{n}k_{0}^{\epsilon}}\frac{\text{Γ}\left(1\right)\text{Γ}\left(\frac{1-\epsilon}{2}\right)}{\text{Γ}\left(\frac{2+\epsilon}{2}\right)2\text{Γ}\left(\frac{1}{2}-\epsilon\right)}\text{Γ}\left(-\epsilon\right)\sum_{w=\koru{\kappa}}^{\infty}\frac{\Gamma\left(1+\frac{3}{2}\epsilon\right)}{\Gamma\left(w+1\right)\Gamma\left(1+\frac{3}{2}\epsilon-w\right)}\sigma^{w}c^{w}\left(-ik_{0}^{\frac{3}{2}\epsilon-w}\right)2
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_body
|
||||
\end_document
|
||||
115
notes/ewald.lyx
115
notes/ewald.lyx
|
|
@ -203,6 +203,16 @@
|
|||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\ghgf}[2]{\mbox{}_{#1}F_{#2}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\ghgfr}[2]{\mbox{}_{#1}\mathbf{F}_{#2}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\ph}{\mathrm{ph}}
|
||||
\end_inset
|
||||
|
|
@ -1631,7 +1641,7 @@ reference "tab:Asymptotical-behaviour-Mathematica"
|
|||
\end_inset
|
||||
|
||||
for some parameters.
|
||||
The only case where Mathematica did not help at all is
|
||||
One particular case where Mathematica did not help at all is
|
||||
\begin_inset Formula $q=2,n=0$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -2608,6 +2618,109 @@ name "tab:Asymptotical-behaviour-Mathematica"
|
|||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Paragraph
|
||||
Case
|
||||
\begin_inset Formula $n=0,q=2$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
[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
|
||||
|
||||
, 2.12.10.2] provides the following integral
|
||||
\begin_inset Formula
|
||||
\begin{multline*}
|
||||
\int_{0}^{\infty}\frac{1}{x^{2}}e^{-px-b/x}J_{0}(cx)\,\ud x=2c\left[z_{+}^{-1}J_{1}\left(z_{-}\right)K_{0}\left(z_{+}\right)+z_{-}^{-1}J_{0}\left(z_{-}\right)K_{1}\left(z_{+}\right)\right]\\
|
||||
\left[z_{\pm}=\sqrt{2b}\left(\sqrt{p^{2}+c^{2}}\pm p\right)^{1/2};\Re p>\left|\Im c\right|;\Re b>0\right]
|
||||
\end{multline*}
|
||||
|
||||
\end_inset
|
||||
|
||||
where [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
|
||||
|
||||
, p.659]
|
||||
\begin_inset Formula $K_{\nu}(z)$
|
||||
\end_inset
|
||||
|
||||
is the Macdonald's function (modified Bessel function of 3rd kind)
|
||||
\begin_inset Formula
|
||||
\[
|
||||
K_{\nu}\left(z\right)=\frac{\pi\left[I_{-\nu}\left(z\right)-I_{\nu}\left(z\right)\right]}{2\sin\nu\pi}\quad\left[\nu\notin\ints\right],\quad K_{n}\left(z\right)=\lim_{\nu\to n}K_{\nu}\left(z\right)\quad\left[n\in\ints\right],
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $I_{\nu}\left(z\right)$
|
||||
\end_inset
|
||||
|
||||
is the modified Bessel function of 1st kind
|
||||
\begin_inset Formula
|
||||
\[
|
||||
I_{\nu}\left(z\right)=\frac{1}{\Gamma\left(\nu+1\right)}\left(\frac{z}{2}\right)^{\nu}\ghgf 01\left(\nu+1;\frac{z^{2}}{4}\right)=e^{-\nu\pi i/2}J_{\nu}\left(e^{\pi i/2}z\right).
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
The problem of this approach is the insufficiently slow decay
|
||||
\begin_inset Formula $\propto k^{-1}$
|
||||
\end_inset
|
||||
|
||||
, so it is in fact better to compute the sum in the real space.
|
||||
I have to look further.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
|
|||
Loading…
Reference in New Issue