382 lines
18 KiB
Plaintext
382 lines
18 KiB
Plaintext
#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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Let
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\begin_layout Paragraph
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\lang english
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Large k
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\begin_layout Standard
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\lang english
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\begin_inset Formula
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\begin{eqnarray*}
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\mbox{OK}\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)\\
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\mbox{OK(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}}(\\
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& & \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}
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\frac{2-q+n}{2},\frac{2-q+n}{2}-\left(1+n\right)+1\\
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1/2
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\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)\\
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& - & \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}
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\frac{3-q+n}{2},\frac{3-q+n}{2}-\left(1+n\right)+1\\
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3/2
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\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\
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\mbox{OK20} & = & \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(\\
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& & \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}
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\frac{2-q+n}{2},\frac{2-q-n}{2}\\
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1/2
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\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right)\\
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& - & \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}
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\frac{3-q+n}{2},\frac{3-q-n}{2}\\
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3/2
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\end{array};-\frac{\left(\sigma c-ik_{0}\right)^{2}}{k^{2}}\right))\\
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\mbox{(D15.2.2)OK3a,b} & = & \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}(\\
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& & \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}\\
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& - & \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})\\
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\mbox{OK4a} & = & \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}(\\
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& & \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}\\
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& - & \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})\\
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\mbox{OK4b} & = & \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}(\\
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& & \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}}\\
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& - & \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}})\\
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\mbox{OK4c} & = & \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=\kor 0}^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\\
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& & \times\left(\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!}-\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!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
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\mbox{OK4d} & = & \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=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(\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!}-\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!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)
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\end{eqnarray*}
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\end_inset
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the fact that the partial sum
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\begin_inset Formula $\sum_{s=0}^{\left\lceil \kappa/2\right\rceil -1}\ldots$
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\end_inset
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is zero is shown in the old messy notes (or TODO later here)
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\end_layout
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\begin_layout Standard
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\lang english
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Using DLMF 5.5.5, which says
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\begin_inset Formula $Γ(2z)=\pi^{-1/2}2^{2z-1}\text{Γ}(z)\text{Γ}(z+\frac{1}{2})$
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\end_inset
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we have
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\begin_inset Formula
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\[
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\text{Γ}\left(2-q+n\right)=\frac{2^{1-q+n}}{\sqrt{\pi}}\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{3-q+n}{2}\right),
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\]
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\end_inset
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so
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\begin_inset Formula
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\begin{eqnarray*}
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\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=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\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!}-\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!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
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& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{\koru{2^{1-q}}}{k_{0}^{q}}\koru{\sqrt{\pi}}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(\frac{\kor{\koru{\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{\kor{\koru{\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)\\
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& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{1-q}}{k_{0}^{q}}\sqrt{\pi}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2+q-2s}\left(\sigma c-ik_{0}\right)^{2s}\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)
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\end{eqnarray*}
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\end_inset
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Assuming that
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\begin_inset Formula $\left\lceil \frac{\kappa}{2}\right\rceil $
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\end_inset
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is large enough so that all the divergent terms are cancelled, either the
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left or the right part will become finite sums due to the
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\begin_inset Quotes sld
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\end_inset
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extra
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\begin_inset Quotes srd
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\end_inset
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Pochhammer
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\begin_inset Formula $\left(\frac{3-q-n}{2}\right)_{s}$
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\end_inset
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or
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\begin_inset Formula $\left(\frac{2-q-n}{2}\right)_{s}$
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\end_inset
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.
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\end_layout
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\begin_layout Subparagraph
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\lang english
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Special case
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\begin_inset Formula $q=2,n=0$
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\end_inset
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\end_layout
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\begin_layout Standard
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\lang english
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If
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\begin_inset Formula $\kappa\ge2$
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\end_inset
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, the left part will drop and
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\begin_inset Formula
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\begin{eqnarray*}
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\mbox{OKSq2n0b}\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{2^{-1}}{k_{0}^{2}}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\text{Γ}\left(\frac{1}{2}+s\right)\text{Γ}\left(\frac{1}{2}+s\right)}{\text{Γ}\left(\frac{1}{2}\right)\kor{\text{Γ}\left(\frac{3}{2}+s\right)}s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
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& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\left\lceil \frac{\kappa}{2}\right\rceil }^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\kor{\text{Γ}\left(\frac{1}{2}+s\right)}\text{Γ}\left(\frac{1}{2}+s\right)}{\text{Γ}\left(\frac{1}{2}\right)\koru{\kor{\text{Γ}\left(\frac{1}{2}+s\right)}\left(\frac{1}{2}+s\right)}s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
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& = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\kor{\left\lceil \frac{\kappa}{2}\right\rceil }}^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\text{Γ}\left(\frac{1}{2}\right)\left(\frac{1}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
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\mbox{(explain!)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}}\sum_{s=\koru 0}^{\infty}\left(-1\right)^{s}k^{-2s}\left(\sigma c-ik_{0}\right)^{2s}\left(-\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\kor{\text{Γ}\left(\frac{1}{2}\right)}\left(\frac{1}{2}+s\right)s!}\frac{\left(\sigma c-ik_{0}\right)}{k}\right)\\
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& = & -\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}\sqrt{\pi}}\frac{\left(\sigma c-ik_{0}\right)}{k}\kor{\sum_{s=0}^{\infty}\left(-1\right)^{s}\left(\frac{\sigma c-ik_{0}}{k}\right)^{2s}\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\left(\frac{1}{2}+s\right)s!}}\\
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& = & -\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{2^{-1}}{k_{0}^{2}\sqrt{\pi}}\frac{\left(\sigma c-ik_{0}\right)}{k}\frac{2\sqrt{\pi}\sinh^{-1}\left(\frac{\sigma c-ik_{0}}{k}\right)}{\frac{\sigma c-ik_{0}}{k}}\\
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\mbox{OKSq2n0f} & = & -\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)
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\end{eqnarray*}
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\end_inset
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where we used (TODO ref)
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\begin_inset Formula
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\[
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\sum_{s=0}^{\infty}\frac{\text{Γ}\left(\frac{1}{2}+s\right)}{\left(\frac{1}{2}+s\right)s!}\left(-x\right)^{s}=\frac{2\sqrt{\pi}\sinh^{-1}\sqrt{x}}{\sqrt{x}}
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\]
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\end_inset
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The final result has asymptotic behaviour of ...
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for
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\begin_inset Formula $k\to\infty$
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\end_inset
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.
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\end_layout
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\begin_layout Paragraph
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Small k
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\end_layout
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\begin_layout Standard
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\lang english
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\begin_inset Formula
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\begin{eqnarray*}
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\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)\\
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\mbox{(D15.2.2)} & = & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}\kor{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{\kor{\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
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\end{eqnarray*}
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\end_inset
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Again we use
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\begin_inset Formula
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\[
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\text{Γ}\left(2-q+n\right)=\frac{2^{1-q+n}}{\sqrt{\pi}}\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{3-q+n}{2}\right),
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\]
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\end_inset
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so
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\begin_inset Formula
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\begin{eqnarray*}
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\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & = & \koru{\frac{2^{1-q\kor{+n}}}{\sqrt{\pi}}}\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}}{\kor{2^{n}}k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\sum_{s=0}^{\infty}\frac{\koru{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\text{Γ}\left(\frac{3-q+n}{2}+s\right)}}{\text{Γ}(1+n+s)s!}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{s}\\
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\mbox{OKShort} & = & \frac{2^{1-q}}{\sqrt{\pi}}\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\kor{\sum_{s=0}^{\infty}\frac{\text{Γ}\left(\frac{2-q+n}{2}+s\right)\text{Γ}\left(\frac{3-q+n}{2}+s\right)}{\text{Γ}(1+n+s)s!}\left(\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)^{s}}\\
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\mbox{(D15.2.1)} & = & \frac{2^{1-q}}{\sqrt{\pi}}\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{k^{n}}{k_{0}^{q}\left(\sigma c-ik_{0}\right)^{2-q+n}}\koru{\frac{\text{Γ}\left(1+n\right)}{\text{Γ}\left(\frac{2-q+n}{2}\right)\text{Γ}\left(\frac{3-q+n}{2}\right)}\kor{\hgf\left(\begin{array}{c}
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\frac{2-q+n}{2},\frac{3-q+n}{2}\\
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1+n
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\end{array};\frac{-k^{2}}{\left(\sigma c-ik_{0}\right)^{2}}\right)}}
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\end{eqnarray*}
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\end_inset
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\end_layout
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\end_body
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\end_document
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