[Ewald] ...
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Marek Nečada
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notes/ewald.lyx
126
notes/ewald.lyx
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@ -1489,6 +1489,8 @@ Let's polish it a bit more
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\end_inset
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\size footnotesize
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\begin_inset Formula
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\begin{multline}
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\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),\\
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@ -1497,6 +1499,8 @@ k>0\wedge k_{0}>0\wedge c\ge0\wedge\lnot\left(c=0\wedge k_{0}=k\right)\label{eq:
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\end_inset
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\size default
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with principal branches of the hypergeometric functions, associated Legendre
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functions, and fractional powers.
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The conditions from
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@ -1525,6 +1529,10 @@ reference "eq:2D Hankel transform of regularized outgoing wave, decomposition"
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\end_layout
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\begin_layout Standard
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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Let's do it.
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\begin_inset Formula
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\begin{eqnarray*}
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@ -1534,6 +1542,11 @@ Let's do it.
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\end_inset
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\end_layout
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\end_inset
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One problematic element here is the gamma function
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\begin_inset Formula $\text{Γ}\left(2-q+n\right)$
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\end_inset
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@ -1561,6 +1574,76 @@ polynomial
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\end_inset
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.
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In other cases, different expressions can be obtained from
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1"
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\end_inset
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using various transformation formulae from either DLMF or
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\begin_inset ERT
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status open
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\begin_layout Plain Layout
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\backslash
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begin{russian}
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\end_layout
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\end_inset
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Прудников
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\begin_inset ERT
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status open
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\begin_layout Plain Layout
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\backslash
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end{russian}
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\end_layout
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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In fact, Mathematica is usually able to calculate the transforms for specific
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values of
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\begin_inset Formula $\kappa,q,n$
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\end_inset
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, but it did not find any general formula for me.
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The resulting expressions are finite sums of algebraic functions, Table
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "tab:Asymptotical-behaviour-Mathematica"
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\end_inset
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shows how fast they decay with growing
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\begin_inset Formula $k$
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\end_inset
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for some parameters.
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The only case where Mathematica did not help at all is
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\begin_inset Formula $q=2,n=0$
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\end_inset
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, which is unfortunately important.
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But if I have not made some mistake, the expression
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:2D Hankel transform of exponentially suppressed outgoing wave expanded"
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\end_inset
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is applicable for this case.
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\end_layout
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\begin_layout Standard
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@ -2712,6 +2795,41 @@ reference "eq:prudnikov2 eq 2.12.9.14"
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\end_layout
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\begin_layout Section
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Major TODOs and open questions
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\end_layout
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\begin_layout Itemize
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Check if
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:2D Hankel transform of exponentially suppressed outgoing wave expanded"
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\end_inset
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gives a satisfactory result for the case
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\begin_inset Formula $q=2,n=0$
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\end_inset
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.
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\end_layout
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\begin_layout Itemize
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Analyse the behaviour
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\begin_inset Formula $k\to k_{0}$
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\end_inset
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.
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\end_layout
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\begin_layout Itemize
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Find a general algorithm for generating the expressions of the Hankel transforms.
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\end_layout
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\begin_layout Itemize
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Three-dimensional case.
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\end_layout
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\begin_layout Section
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(Appendix) Fourier vs.
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Hankel transform
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@ -2761,7 +2879,7 @@ where the spherical Hankel transform
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2)
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\begin_inset Formula
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\[
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\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).
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\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).
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\]
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\end_inset
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@ -2771,7 +2889,7 @@ Using this convention, the inverse spherical Hankel transform is given by
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3)
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\begin_inset Formula
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\[
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g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\,k^{2}\bsht lg(k)j_{l}(k),
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g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\, k^{2}\bsht lg(k)j_{l}(k),
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\]
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\end_inset
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@ -2784,7 +2902,7 @@ so it is not unitary.
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An unitary convention would look like this:
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\begin_inset Formula
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\begin{equation}
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\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition}
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\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition}
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\end{equation}
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\end_inset
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@ -2838,7 +2956,7 @@ where the Hankel transform of order
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is defined as
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\begin_inset Formula
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\begin{equation}
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\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\,g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}
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\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}
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\end{equation}
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\end_inset
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