3441 lines
94 KiB
Plaintext
3441 lines
94 KiB
Plaintext
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\begin_layout Title
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Accelerating lattice mode calculations with
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\begin_inset Formula $T$
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\end_inset
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-matrix method
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\end_layout
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\begin_layout Author
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Marek Nečada
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\end_layout
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\begin_layout Abstract
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The
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\begin_inset Formula $T$
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\end_inset
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-matrix approach is the method of choice for simulating optical response
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of a reasonably small system of compact linear scatterers on isotropic
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background.
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However, its direct utilisation for problems with infinite lattices is
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problematic due to slowly converging sums over the lattice.
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Here I develop a way to compute the problematic sums in the reciprocal
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space, making the
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\begin_inset Formula $T$
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\end_inset
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-matrix method very suitable for infinite periodic systems as well.
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\end_layout
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\begin_layout Section
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Formulation of the problem
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\end_layout
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\begin_layout Standard
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Assume a system of compact EM scatterers in otherwise homogeneous and isotropic
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medium, and assume that the system, i.e.
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both the medium and the scatterers, have linear response.
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A scattering problem in such system can be written as
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\begin_inset Formula
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\[
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A_{α}=T_{α}P_{α}=T_{α}(\sum_{β}S_{α\leftarrowβ}A_{β}+P_{0α})
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\]
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\end_inset
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where
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\begin_inset Formula $T_{α}$
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\end_inset
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is the
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\begin_inset Formula $T$
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\end_inset
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-matrix for scatterer α,
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\begin_inset Formula $A_{α}$
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\end_inset
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is its vector of the scattered wave expansion coefficient (the multipole
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indices are not explicitely indicated here) and
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\begin_inset Formula $P_{α}$
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\end_inset
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is the local expansion of the incoming sources.
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\begin_inset Formula $S_{α\leftarrowβ}$
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\end_inset
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is ...
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and ...
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is ...
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\end_layout
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\begin_layout Standard
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...
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\sum_{β}(\delta_{αβ}-T_{α}S_{α\leftarrowβ})A_{β}=T_{α}P_{0α}.
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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Now suppose that the scatterers constitute an infinite lattice
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{\vect aα}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=T_{\vect aα}P_{0\vect aα}.
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\]
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\end_inset
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Due to the periodicity, we can write
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\begin_inset Formula $S_{\vect aα\leftarrow\vect bβ}=S_{α\leftarrowβ}(\vect b-\vect a)$
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\end_inset
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and
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\begin_inset Formula $T_{\vect aα}=T_{\alpha}$
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\end_inset
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.
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In order to find lattice modes, we search for solutions with zero RHS
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\begin_inset Formula
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\[
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=0
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\]
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\end_inset
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and we assume periodic solution
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\begin_inset Formula $A_{\vect b\beta}(\vect k)=A_{\vect a\beta}e^{i\vect k\cdot\vect r_{\vect b-\vect a}}$
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\end_inset
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, yielding
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\begin_inset Formula
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\begin{eqnarray*}
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
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\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect 0α\leftarrow\vect bβ})A_{\vect 0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
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\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect 0\beta}\left(\vect k\right) & = & 0,\\
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A_{\vect 0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect 0\beta}\left(\vect k\right) & = & 0.
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\end{eqnarray*}
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\end_inset
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Therefore, in order to solve the modes, we need to compute the
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\begin_inset Quotes eld
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\end_inset
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lattice Fourier transform
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\begin_inset Quotes erd
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\end_inset
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of the translation operator,
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\begin_inset Formula
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\begin{equation}
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W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Section
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Computing the Fourier sum of the translation operator
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\end_layout
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\begin_layout Standard
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The problem evaluating
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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\end_inset
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is the asymptotic behaviour of the translation operator,
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\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
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\end_inset
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that makes the convergence of the sum quite problematic for any
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\begin_inset Formula $d>1$
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\end_inset
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-dimensional lattice.
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\begin_inset Foot
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status open
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\begin_layout Plain Layout
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Note that
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\begin_inset Formula $d$
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\end_inset
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here is dimensionality of the lattice, not the space it lies in, which
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I for certain reasons assume to be three.
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(TODO few notes on integration and reciprocal lattices in some appendix)
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\end_layout
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\end_inset
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In electrostatics, one can solve this problem with Ewald summation.
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Its basic idea is that if what asymptoticaly decays poorly in the direct
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space, will perhaps decay fast in the Fourier space.
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I use the same idea here, but everything will be somehow harder than in
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electrostatics.
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\end_layout
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\begin_layout Standard
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Let us re-express the sum in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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\end_inset
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in terms of integral with a delta comb
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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{equation}
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W_{\alpha\beta}(\vect k)=\int\ud^{d}\vect r\dc{\basis u}(\vect r)S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})e^{i\vect k\cdot\vect r}.\label{eq:W integral}
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\end{equation}
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\end_inset
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The translation operator
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\begin_inset Formula $S$
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\end_inset
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is now a function defined in the whole 3d space;
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\begin_inset Formula $\vect r_{\alpha},\vect r_{\beta}$
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\end_inset
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||
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are the displacements of scatterers
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\begin_inset Formula $\alpha$
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\end_inset
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and
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\begin_inset Formula $\beta$
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\end_inset
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||
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in a unit cell.
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The arrow notation
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||
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$
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\end_inset
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||
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means
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\begin_inset Quotes eld
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\end_inset
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||
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translation operator for spherical waves originating in
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\begin_inset Formula $\vect r+\vect r_{\beta}$
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||
\end_inset
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||
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evaluated in
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\begin_inset Formula $\vect r_{\alpha}$
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\end_inset
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||
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||
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\begin_inset Quotes erd
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\end_inset
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||
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and obviously
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||
\begin_inset Formula $S$
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||
\end_inset
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||
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||
is in fact a function of a single 3d argument,
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||
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
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||
\end_inset
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||
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||
.
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||
Expression
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\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:W integral"
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||
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||
\end_inset
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||
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||
can be rewritten as
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||
\begin_inset Formula
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||
\[
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||
W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
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\]
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||
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\end_inset
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||
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where changed the sign of
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||
\begin_inset Formula $\vect r/\vect{\bullet}$
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||
\end_inset
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||
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||
has been swapped under integration, utilising evenness of
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||
\begin_inset Formula $\dc{\basis u}$
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||
\end_inset
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||
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||
.
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||
Fourier transform of product is convolution of Fourier transforms, so (using
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||
formula
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||
\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:Dirac comb uaFt"
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||
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||
\end_inset
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||
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||
for the Fourier transform of Dirac comb)
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||
\begin_inset Formula
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||
\begin{eqnarray}
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||
W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\
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& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
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& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right).\label{eq:W sum in reciprocal space}
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\end{eqnarray}
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\end_inset
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\begin_inset Note Note
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||
status open
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||
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||
\begin_layout Plain Layout
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||
Factor
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||
\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$
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||
\end_inset
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||
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||
cancels out with the
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||
\begin_inset Formula $\left(2\pi\right)^{-\frac{d}{2}}$
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||
\end_inset
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||
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||
factor appearing in the convolution/product formula in the unitary angular
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||
momentum convention.
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||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
As such, this is not extremely helpful because the the
|
||
\emph on
|
||
whole
|
||
\emph default
|
||
translation operator
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
has singularities in origin, hence its Fourier transform
|
||
\begin_inset Formula $\uaft S$
|
||
\end_inset
|
||
|
||
will decay poorly.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
However, Fourier transform is linear, so we can in principle separate
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
in two parts,
|
||
\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset Formula $S^{\textup{S}}$
|
||
\end_inset
|
||
|
||
is a short-range part that decays sufficiently fast with distance so that
|
||
its direct-space lattice sum converges well;
|
||
\begin_inset Formula $S^{\textup{S}}$
|
||
\end_inset
|
||
|
||
must as well contain all the singularities of
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
in the origin.
|
||
The other part,
|
||
\begin_inset Formula $S^{\textup{L}}$
|
||
\end_inset
|
||
|
||
, will retain all the slowly decaying terms of
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
but it also has to be smooth enough in the origin, so that its Fourier
|
||
transform
|
||
\begin_inset Formula $\uaft{S^{\textup{L}}}$
|
||
\end_inset
|
||
|
||
decays fast enough.
|
||
(The same idea lies behind the Ewald summation in electrostatics.) Using
|
||
the linearity of Fourier transform and formulae
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W definition"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W sum in reciprocal space"
|
||
|
||
\end_inset
|
||
|
||
, the operator
|
||
\begin_inset Formula $W_{\alpha\beta}$
|
||
\end_inset
|
||
|
||
can then be re-expressed as
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
|
||
W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
|
||
W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
where both sums should converge nicely.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Finding a good decomposition
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The remaining challenge is therefore finding a suitable decomposition
|
||
\begin_inset Formula $S^{\textup{L}}+S^{\textup{S}}$
|
||
\end_inset
|
||
|
||
such that both
|
||
\begin_inset Formula $S^{\textup{S}}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\uaft{S^{\textup{L}}}$
|
||
\end_inset
|
||
|
||
decay fast enough with distance and are expressable analytically.
|
||
With these requirements, I do not expect to find gaussian asymptotics as
|
||
in the electrostatic Ewald formula—having
|
||
\begin_inset Formula $\sim x^{-t}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $t>d$
|
||
\end_inset
|
||
|
||
asymptotics would be nice, making the sums in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W Short definition"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W Long definition"
|
||
|
||
\end_inset
|
||
|
||
absolutely convergent.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The translation operator
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
for compact scatterers in 3d can be expressed as
|
||
\begin_inset Formula
|
||
\[
|
||
S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(k_{0}\left|\vect r\right|\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $Y_{l,m}\left(\theta,\phi\right)$
|
||
\end_inset
|
||
|
||
are the spherical harmonics,
|
||
\begin_inset Formula $z_{p}^{(J)}\left(r\right)$
|
||
\end_inset
|
||
|
||
some of the Bessel or Hankel functions (probably
|
||
\begin_inset Formula $h_{p}^{(1)}$
|
||
\end_inset
|
||
|
||
in all meaningful cases; TODO) and
|
||
\begin_inset Formula $c_{p}^{l,m,t\leftarrow l',m',t'}$
|
||
\end_inset
|
||
|
||
are some ugly but known coefficients (REF Xu 1996, eqs.
|
||
76,77).
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The spherical Hankel functions can be expressed analytically as (REF DLMF
|
||
10.49.6, 10.49.1)
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
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}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
so if we find a way to deal with the radial functions
|
||
\begin_inset Formula $s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $q=1,2$
|
||
\end_inset
|
||
|
||
in 2d case or
|
||
\begin_inset Formula $q=1,2,3$
|
||
\end_inset
|
||
|
||
in 3d case, we get absolutely convergent summations in the direct space.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
2d
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Assume that all scatterers are placed in the plane
|
||
\begin_inset Formula $\vect z=0$
|
||
\end_inset
|
||
|
||
, so that the 2d Fourier transform of the long-range part of the translation
|
||
operator in terms of Hankel transforms, according to
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Fourier v. Hankel tf 2d"
|
||
|
||
\end_inset
|
||
|
||
, reads
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{multline*}
|
||
\uaft{S_{l',m',t'\leftarrow l,m,t}^{\textup{L}}\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}{h_{p}^{(1)\textup{L}}\left(k_{0}\vect{\bullet}\right)}\left(\left|\vect k\right|\right)
|
||
\end{multline*}
|
||
|
||
\end_inset
|
||
|
||
Here
|
||
\begin_inset Formula $h_{p}^{(1)\textup{L}}=h_{p}^{(1)}-h_{p}^{(1)\textup{S}}$
|
||
\end_inset
|
||
|
||
is a long range part of a given spherical Hankel function which has to
|
||
be found and which contains all the terms with far-field (
|
||
\begin_inset Formula $r\to\infty$
|
||
\end_inset
|
||
|
||
) asymptotics proportional to
|
||
\begin_inset Formula $\sim e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $q\le Q$
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $Q$
|
||
\end_inset
|
||
|
||
is at least two in order to achieve absolute convergence of the direct-space
|
||
sum, but might be higher in order to speed the convergence up.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Obviously, all the terms
|
||
\begin_inset Formula $\propto s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $q>Q$
|
||
\end_inset
|
||
|
||
of the spherical Hankel function
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:spherical Hankel function series"
|
||
|
||
\end_inset
|
||
|
||
can be kept untouched as part of
|
||
\begin_inset Formula $h_{p}^{(1)\textup{S}}$
|
||
\end_inset
|
||
|
||
, as they decay fast enough.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The remaining task is therefore to find a suitable decomposition of
|
||
\begin_inset Formula $s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $q\le Q$
|
||
\end_inset
|
||
|
||
into short-range and long-range parts,
|
||
\begin_inset Formula $s_{k_{0},q}(r)=s_{k_{0},q}^{\textup{S}}(r)+s_{k_{0},q}^{\textup{L}}(r)$
|
||
\end_inset
|
||
|
||
, such that
|
||
\begin_inset Formula $s_{k_{0},q}^{\textup{L}}(r)$
|
||
\end_inset
|
||
|
||
contains all the slowly decaying asymptotics and its Hankel transforms
|
||
decay desirably fast as well,
|
||
\begin_inset Formula $\pht n{s_{k_{0},q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $z\to\infty$
|
||
\end_inset
|
||
|
||
.
|
||
The latter requirement calls for suitable regularisation functions—
|
||
\begin_inset Formula $s_{q}^{\textup{L}}$
|
||
\end_inset
|
||
|
||
must be sufficiently smooth in the origin, so that
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\pht n{s_{k_{0},q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{k_{0},q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{k_{0},q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
exists and decays fast enough.
|
||
|
||
\begin_inset Formula $J_{\nu}(r)\sim\left(r/2\right)^{\nu}/\Gamma\left(\nu+1\right)$
|
||
\end_inset
|
||
|
||
(REF DLMF 10.7.3) near the origin, so the regularisation function should
|
||
be
|
||
\begin_inset Formula $\rho(r)=o(r^{q-n-1})$
|
||
\end_inset
|
||
|
||
only to make
|
||
\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}$
|
||
\end_inset
|
||
|
||
converge.
|
||
The additional decay speed requirement calls for at least
|
||
\begin_inset Formula $\rho(r)=o(r^{q-n+Q-1})$
|
||
\end_inset
|
||
|
||
, I guess.
|
||
At the same time,
|
||
\begin_inset Formula $\rho(r)$
|
||
\end_inset
|
||
|
||
must converge fast enough to one for
|
||
\begin_inset Formula $r\to\infty$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The electrostatic Ewald summation uses regularisation with
|
||
\begin_inset Formula $1-e^{-cr^{2}}$
|
||
\end_inset
|
||
|
||
.
|
||
However, such choice does not seem to lead to an analytical solution (really?
|
||
could not something be dug out of DLMF 10.22.54?) for the current problem
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:2d long range regularisation problem statement"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
But it turns out that the family of functions
|
||
\begin_inset Formula
|
||
\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
|
||
|
||
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, „binomial“ regularisation
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sub:Hankel-transforms-binom-reg"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
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
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & \equiv & \int_{0}^{\infty}\frac{e^{ik_{0}r}}{\left(k_{0}r\right)^{q}}J_{n}\left(kr\right)\left(1-e^{-cr}\right)^{\kappa}r\,\ud r\nonumber \\
|
||
& = & k_{0}^{-q}\int_{0}^{\infty}r^{1-q}J_{n}\left(kr\right)\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}e^{-(\sigma c-ik_{0})r}\ud r\nonumber \\
|
||
& \underset{\equiv}{\textup{form.}} & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\pht n{s_{q,k_{0}}^{\textup{L}1,\sigma c}}\left(k\right).\label{eq:2D Hankel transform of regularized outgoing wave, decomposition}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
From [REF DLMF 10.22.49] one digs
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\mu & \leftarrow & 2-q\\
|
||
\nu & \leftarrow & n\\
|
||
b & \leftarrow & k\\
|
||
a & \leftarrow & c-ik_{0}
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{multline}
|
||
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}Γ\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(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(c-ik_{0}\right)^{2}}\right),\\
|
||
\Re\left(2-q+n\right)>0,\Re(c-ik_{0}\pm k)\ge0,\label{eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1}
|
||
\end{multline}
|
||
|
||
\end_inset
|
||
|
||
and using [REF DLMF 15.9.17] and
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $P_{\nu}^{\mu}=P_{-\nu-1}^{\mu}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
[REF DLMF 14.9.5]
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\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 Plain Layout
|
||
\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 Plain Layout
|
||
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
|
||
\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
|
||
|
||
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 Plain Layout
|
||
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 Plain Layout
|
||
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 Plain Layout
|
||
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)?
|
||
\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)}\\
|
||
\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)}\\
|
||
& = & \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)\\
|
||
& = & \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)}\\
|
||
\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)}\\
|
||
\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)}\\
|
||
& = & \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)\\
|
||
& = & \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)\\
|
||
& = & \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)\\
|
||
(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)\\
|
||
& = & \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)\\
|
||
& = & \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)\\
|
||
& = & \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)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
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 Plain Layout
|
||
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 Plain Layout
|
||
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 Plain Layout
|
||
\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
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
a & \leftarrow & \frac{2-q+n}{2}\\
|
||
c & \leftarrow & 1+n\\
|
||
z & \leftarrow & \frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right) & = & \frac{k^{n}Γ\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}2^{n}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1-\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right)^{-\frac{2-q+n}{2}+\frac{n}{2}}P_{2-q+n-(1+n)}^{1-(1+n)}\left(\frac{1}{\sqrt{1-\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)}}\right)\\
|
||
& = & \frac{k^{n}Γ\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{1-q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
\left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|<\pi,\quad\left|\ph\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right|<\pi
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
in other words, neither
|
||
\begin_inset Formula $-k^{2}/\left(c-ik_{0}\right)^{2}$
|
||
\end_inset
|
||
|
||
nor
|
||
\begin_inset Formula $1+k^{2}/\left(c-ik_{0}\right)^{2}$
|
||
\end_inset
|
||
|
||
can be non-positive real number.
|
||
For assumed positive
|
||
\begin_inset Formula $k_{0}$
|
||
\end_inset
|
||
|
||
and non-negative
|
||
\begin_inset Formula $c$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $k$
|
||
\end_inset
|
||
|
||
, the former case can happen only if
|
||
\begin_inset Formula $k=0$
|
||
\end_inset
|
||
|
||
and the latter only if
|
||
\begin_inset Formula $c=0\wedge k_{0}=k$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|<\pi & \Leftrightarrow & \left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|\neq\pi\\
|
||
\varphi & \equiv & \ph\left(c-ik_{0}\right)<0,\\
|
||
\ph k & \equiv & 0\\
|
||
\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}} & = & 2\varphi\\
|
||
\rightsquigarrow\left|\varphi\right| & \neq & \pi/2\\
|
||
\rightsquigarrow c & \neq & k_{0}\\
|
||
\left|\ph\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right| & = & \left|-2\varphi+\ph\left(\left(c-ik_{0}\right)^{2}+k^{2}\right)\right|
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
Finally, swapping the first two arguments of
|
||
\begin_inset Formula $\hgfr$
|
||
\end_inset
|
||
|
||
in the hypergeometric represenation [REF DLMF 14.3.6] (note [REF DLMF §14.21(iii)]
|
||
that this also holds for complex arguments) of Legendre functions gives
|
||
|
||
\begin_inset Formula $P_{\nu}^{\mu}=P_{-\nu-1}^{\mu}$
|
||
\end_inset
|
||
|
||
, so the above result can be written
|
||
\begin_inset Formula
|
||
\[
|
||
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}\text{Γ}\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
Let's polish it a bit more
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right) & = & \frac{Γ\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q}}\left(-1\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right)\\
|
||
& = & \frac{\text{Γ}\left(2-q+n\right)}{k_{0}^{q}}\left(-1\right)^{-\frac{n}{2}}\left(\left(c-ik_{0}\right)^{2}+k^{2}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right).
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\size footnotesize
|
||
|
||
\begin_inset Formula
|
||
\begin{multline}
|
||
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}Γ\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right),\\
|
||
k>0\wedge k_{0}>0\wedge c\ge0\wedge\lnot\left(c=0\wedge k_{0}=k\right)\label{eq:2D Hankel transform of exponentially suppressed outgoing wave expanded}
|
||
\end{multline}
|
||
|
||
\end_inset
|
||
|
||
|
||
\size default
|
||
with principal branches of the hypergeometric functions, associated Legendre
|
||
functions, and fractional powers.
|
||
The conditions from
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1"
|
||
|
||
\end_inset
|
||
|
||
should hold, but we will use
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:2D Hankel transform of exponentially suppressed outgoing wave expanded"
|
||
|
||
\end_inset
|
||
|
||
formally even if they are violated, with the hope that the divergences
|
||
eventually cancel in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:2D Hankel transform of regularized outgoing wave, decomposition"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
Let's do it.
|
||
\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)}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}}}\right)\\
|
||
& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\text{Γ}\left(2-q+n\right)}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}}}\right)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\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 argument is zero or negative integer, i.e.
|
||
if
|
||
\begin_inset Formula $q-n\ge2$
|
||
\end_inset
|
||
|
||
; which is painful especially because of the case
|
||
\begin_inset Formula $q=2,n=0$
|
||
\end_inset
|
||
|
||
.
|
||
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
|
||
|
||
.
|
||
In other cases, different expressions can be obtained from
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1"
|
||
|
||
\end_inset
|
||
|
||
using various transformation formulae from either DLMF or
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
begin{russian}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
Прудников
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
end{russian}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In fact, Mathematica is usually able to calculate the transforms for specific
|
||
values of
|
||
\begin_inset Formula $\kappa,q,n$
|
||
\end_inset
|
||
|
||
, but it did not find any general formula for me.
|
||
The resulting expressions are finite sums of algebraic functions, Table
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "tab:Asymptotical-behaviour-Mathematica"
|
||
|
||
\end_inset
|
||
|
||
shows how fast they decay with growing
|
||
\begin_inset Formula $k$
|
||
\end_inset
|
||
|
||
for some parameters.
|
||
One particular case where Mathematica did not help at all is
|
||
\begin_inset Formula $q=2,n=0$
|
||
\end_inset
|
||
|
||
, which is unfortunately important.
|
||
|
||
\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
|
||
<lyxtabular version="3" rows="4" columns="5">
|
||
<features rotate="0" tabularvalignment="middle">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<row>
|
||
<cell multicolumn="1" alignment="center" valignment="top" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
\begin_inset Formula $\kappa=0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell multicolumn="2" alignment="center" valignment="top" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell multicolumn="1" alignment="center" valignment="top" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
0
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
1
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell multirow="3" alignment="center" valignment="middle" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
\begin_inset Formula $q$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
1
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
1
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
1
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell multirow="4" alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
x
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
w
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
0
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset space \hspace*{\fill}
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="4" columns="7">
|
||
<features rotate="0" tabularvalignment="middle">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<row>
|
||
<cell multicolumn="1" alignment="center" valignment="top" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
\begin_inset Formula $\kappa=1$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell multicolumn="2" alignment="center" valignment="top" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell multicolumn="1" alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\size footnotesize
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
|
||
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|
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|
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|
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|
||
\begin_layout Plain Layout
|
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Asymptotical behaviour of some
|
||
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|
||
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
|
||
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|
||
\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"
|
||
|
||
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|
||
|
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||
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|
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|
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status open
|
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|
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|
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|
||
status open
|
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|
||
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|
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|
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|
||
\backslash
|
||
begin{russian}
|
||
\end_layout
|
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|
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|
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|
||
Градштейн и Рыжик
|
||
\begin_inset ERT
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
|
||
end{russian}
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
6.512.1 has expression for
|
||
\begin_inset Formula $\int_{0}^{\infty}J_{\mu}\left(ax\right)J_{\nu}\left(bx\right)\ud x$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\Re\left(\mu+\nu\right)>-1$
|
||
\end_inset
|
||
|
||
in terms of hypergeometric function.
|
||
Unfortunately, no corresponding and general enough expression for
|
||
\begin_inset Formula $\int_{0}^{\infty}J_{\mu}\left(ax\right)Y_{\nu}\left(bx\right)\ud x$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Paragraph
|
||
Case
|
||
\begin_inset Formula $n=0,q=2$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
As shown in a separate note,
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\pht 0{s_{2,k_{0}}^{\textup{L}\kappa,c}}\left(k\right)=-\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{1}{k_{0}^{2}}\sinh^{-1}\left(\frac{\sigma c-ik_{0}}{k}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
for
|
||
\begin_inset Formula $\kappa\ge?$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $k>k_{0}?$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
From Wikipedia page on binomial coefficient, eq.
|
||
(10) and around:
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
When
|
||
\begin_inset Formula $P(x)$
|
||
\end_inset
|
||
|
||
is of degree less than or equal to
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula
|
||
\[
|
||
\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}P(n-j)=n!a_{n}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $a_{n}$
|
||
\end_inset
|
||
|
||
is the coefficient of degree
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
in
|
||
\begin_inset Formula $P(x)$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
More generally,
|
||
\begin_inset Formula
|
||
\[
|
||
\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}P(m+(n-j)d)=d^{n}n!a_{n}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $m$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
are complex numbers.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Subsubsection
|
||
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 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}
|
||
\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].\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
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
3d (TODO)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{multline*}
|
||
\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)
|
||
\end{multline*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Major TODOs and open questions
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Check if
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:2D Hankel transform of exponentially suppressed outgoing wave expanded"
|
||
|
||
\end_inset
|
||
|
||
gives a satisfactory result for the case
|
||
\begin_inset Formula $q=2,n=0$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Analyse the behaviour
|
||
\begin_inset Formula $k\to k_{0}$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Find a general algorithm for generating the expressions of the Hankel transforms.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Three-dimensional case.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
(Appendix) Fourier vs.
|
||
Hankel transform
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Three dimensions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Given a nice enough function
|
||
\begin_inset Formula $f$
|
||
\end_inset
|
||
|
||
of a real 3d variable, assume its factorisation into radial and angular
|
||
parts
|
||
\begin_inset Formula
|
||
\[
|
||
f(\vect r)=\sum_{l,m}f_{l,m}(\left|\vect r\right|)\ush lm\left(\theta_{\vect r},\phi_{\vect r}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
Acording to (REF Baddour 2010, eqs.
|
||
13, 16), its Fourier transform can then be expressed in terms of Hankel
|
||
transforms (CHECK normalisation of
|
||
\begin_inset Formula $j_{n}$
|
||
\end_inset
|
||
|
||
, REF Baddour (1))
|
||
\begin_inset Formula
|
||
\[
|
||
\uaft f(\vect k)=\frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sum_{l,m}\left(-i\right)^{l}\left(\bsht{f_{l,m}}{}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where the spherical Hankel transform
|
||
\begin_inset Formula $\bsht l{}$
|
||
\end_inset
|
||
|
||
of degree
|
||
\begin_inset Formula $l$
|
||
\end_inset
|
||
|
||
is defined as (REF Baddour eq.
|
||
2)
|
||
\begin_inset Formula
|
||
\[
|
||
\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
Using this convention, the inverse spherical Hankel transform is given by
|
||
(REF Baddour eq.
|
||
3)
|
||
\begin_inset Formula
|
||
\[
|
||
g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\,k^{2}\bsht lg(k)j_{l}(k),
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
so it is not unitary.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
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}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Then
|
||
\begin_inset Formula $\usht l{}^{-1}=\usht l{}$
|
||
\end_inset
|
||
|
||
and the unitary, angular-momentum Fourier transform reads
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\uaft f(\vect k) & = & \frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sqrt{\frac{\pi}{2}}\sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)\nonumber \\
|
||
& = & \sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right).\label{eq:Fourier v. Hankel tf 3d}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
Cool.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Two dimensions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Similarly in 2d, let the expansion of
|
||
\begin_inset Formula $f$
|
||
\end_inset
|
||
|
||
be
|
||
\begin_inset Formula
|
||
\[
|
||
f\left(\vect r\right)=\sum_{m}f_{m}\left(\left|\vect r\right|\right)e^{im\phi_{\vect r}},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
its Fourier transform is then (CHECK this, it is taken from the Wikipedia
|
||
article on Hankel transform)
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\uaft f\left(\vect k\right)=\sum_{m}i^{m}e^{im\phi_{\vect k}}\pht mf_{m}\left(\left|\vect k\right|\right)\label{eq:Fourier v. Hankel tf 2d}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where the Hankel transform of order
|
||
\begin_inset Formula $m$
|
||
\end_inset
|
||
|
||
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}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
which is already self-inverse,
|
||
\begin_inset Formula $\pht m{}^{-1}=\pht m{}$
|
||
\end_inset
|
||
|
||
(hence also unitary).
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
(Appendix) Multidimensional Dirac comb
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
1D
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This is all from Wikipedia
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Definitions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
Ш(t) & \equiv & \sum_{k=-\infty}^{\infty}\delta(t-k)\\
|
||
Ш_{T}(t) & \equiv & \sum_{k=-\infty}^{\infty}\delta(t-kT)=\frac{1}{T}Ш\left(\frac{t}{T}\right)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Fourier series representation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Ш_{T}(t)=\sum_{n=-\infty}^{\infty}e^{2\pi int/T}\label{eq:1D Dirac comb Fourier series}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Fourier transform
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
With unitary ordinary frequency Ft., i.e.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\uoft f(\vect{\xi})\equiv\int_{\mathbb{R}^{n}}f(\vect x)e^{-2\pi i\vect x\cdot\vect{\xi}}\ud^{n}\vect x
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
we have
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\uoft{Ш_{T}}(f)=\frac{1}{T}Ш_{\frac{1}{T}}(f)=\sum_{n=-\infty}^{\infty}e^{-i2\pi fnT}\label{eq:1D Dirac comb Ft ordinary freq}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and with unitary angular frequency Ft., i.e.
|
||
\begin_inset Formula
|
||
\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
|
||
|
||
we have (CHECK)
|
||
\begin_inset Formula
|
||
\[
|
||
\uaft{Ш_{T}}(\omega)=\frac{\sqrt{2\pi}}{T}Ш_{\frac{2\pi}{T}}(\omega)=\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-i\omega nT}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Dirac comb for multidimensional lattices
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Definitions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Let
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
be the dimensionality of the real vector space in question, and let
|
||
\begin_inset Formula $\basis u\equiv\left\{ \vect u_{i}\right\} _{i=1}^{d}$
|
||
\end_inset
|
||
|
||
denote a basis for some lattice in that space.
|
||
Let the corresponding lattice delta comb be
|
||
\begin_inset Formula
|
||
\[
|
||
\dc{\basis u}\left(\vect x\right)\equiv\sum_{n_{1}=-\infty}^{\infty}\ldots\sum_{n_{d}=-\infty}^{\infty}\delta\left(\vect x-\sum_{i=1}^{d}n_{i}\vect u_{i}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Furthemore, let
|
||
\begin_inset Formula $\rec{\basis u}\equiv\left\{ \rec{\vect u}_{i}\right\} _{i=1}^{d}$
|
||
\end_inset
|
||
|
||
be the reciprocal lattice basis, that is the basis satisfying
|
||
\begin_inset Formula $\vect u_{i}\cdot\rec{\vect u_{j}}=\delta_{ij}$
|
||
\end_inset
|
||
|
||
.
|
||
This slightly differs from the usual definition of a reciprocal basis,
|
||
here denoted
|
||
\begin_inset Formula $\recb{\basis u}\equiv\left\{ \recb{\vect u_{i}}\right\} _{i=1}^{d}$
|
||
\end_inset
|
||
|
||
, which satisfies
|
||
\begin_inset Formula $\vect u_{i}\cdot\recb{\vect u_{j}}=2\pi\delta_{ij}$
|
||
\end_inset
|
||
|
||
instead.
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Factorisation of a multidimensional lattice delta comb
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
By simple drawing, it can be seen that
|
||
\begin_inset Formula
|
||
\[
|
||
\dc{\basis u}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $c_{\basis u}$
|
||
\end_inset
|
||
|
||
is some numerical volume factor.
|
||
In order to determine
|
||
\begin_inset Formula $c_{\basis u}$
|
||
\end_inset
|
||
|
||
, let us consider only the
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
zero tooth
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
of the comb, leading to
|
||
\begin_inset Formula
|
||
\[
|
||
\delta^{d}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\delta\left(\vect x\cdot\rec{\vect u_{i}}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
From the scaling property of delta function,
|
||
\begin_inset Formula $\delta(ax)=\left|a\right|^{-1}\delta(x)$
|
||
\end_inset
|
||
|
||
, we get
|
||
\begin_inset Formula
|
||
\[
|
||
\delta^{d}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert ^{-1}\delta\left(\vect x\cdot\frac{\rec{\vect u_{i}}}{\left\Vert \rec{\vect u_{i}}\right\Vert }\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From the Osgood's book (p.
|
||
375):
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\dc A(\vect x)=\frac{1}{\left|\det A\right|}\dc{}^{(d)}\left(A^{-1}\vect x\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Applying both sides to a test function that is one at the origin, we get
|
||
|
||
\begin_inset Formula $c_{\basis u}=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert $
|
||
\end_inset
|
||
|
||
SRSLY?, and hence
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\dc{\basis u}(\vect x)=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert \dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right).\label{eq:Dirac comb factorisation}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Fourier series representation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Utilising the Fourier series for 1D Dirac comb
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:1D Dirac comb Fourier series"
|
||
|
||
\end_inset
|
||
|
||
and the factorisation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Dirac comb factorisation"
|
||
|
||
\end_inset
|
||
|
||
, we get
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\dc{\basis u}(\vect x) & = & \prod_{j=1}^{d}\left\Vert \rec{\vect u_{j}}\right\Vert \sum_{n_{j}=-\infty}^{\infty}e^{2\pi in_{i}\vect x\cdot\rec{\vect u_{i}}}\\
|
||
& = & \left(\prod_{j=1}^{d}\left\Vert \rec{\vect u_{j}}\right\Vert \right)\sum_{\vect n\in\mathbb{Z}^{d}}e^{2\pi i\vect x\cdot\sum_{k=1}^{d}n_{k}\rec{\vect u_{k}}}.
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Fourier transform (OK)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From the Osgood's book https://see.stanford.edu/materials/lsoftaee261/chap8.pdf,
|
||
p.
|
||
379
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\uoft{\dc{\basis u}}\left(\vect{\xi}\right)=\left|\det\rec{\basis u}\right|\dc{\rec{\basis u}}^{(d)}\left(\vect{\xi}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
And consequently, for unitary/angular frequency it is
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\uaft{\dc{\basis u}}\left(\vect k\right) & = & \frac{1}{\left(2\pi\right)^{\frac{d}{2}}}\uoft{\dc{\basis u}}\left(\frac{\vect k}{2\pi}\right)\nonumber \\
|
||
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{\rec{\basis u}}^{(d)}\left(\frac{\vect k}{2\pi}\right)\nonumber \\
|
||
& = & \left(2\pi\right)^{\frac{d}{2}}\left|\det\rec{\basis u}\right|\dc{\recb{\basis u}}\left(\vect k\right)\nonumber \\
|
||
& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{\recb{\basis u}}\left(\vect k\right).\label{eq:Dirac comb uaFt}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
On the third line, we used the stretch theorem, getting
|
||
\begin_inset Formula
|
||
\[
|
||
\dc{\recb{\basis u}}\left(\vect k\right)=\dc{2\pi\rec{\basis u}}\left(\vect k\right)=\left(2\pi\right)^{-d}\dc{\rec{\basis u}}\left(\frac{\vect k}{2\pi}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Convolution
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\left(f\ast\dc{\basis u}\right)(\vect x)=\sum_{\vect t\in\basis u\ints^{d}}f(\vect x-\vect t)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
So, from the stretch theorem
|
||
\begin_inset Formula $\uoft{(f(A\vect x))}=\frac{1}{\left|\det A\right|}\uoft{f\left(A^{-T}\vect{\xi}\right)}=\left|\det A^{-T}\right|\uoft{f\left(A^{-T}\vect{\xi}\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
From
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Dirac comb factorisation"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:1D Dirac comb Ft ordinary freq"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
\uoft{\dc{\basis u}}(\vect{\xi})=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert \dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|