162 lines
No EOL
3.7 KiB
TeX
162 lines
No EOL
3.7 KiB
TeX
\documentclass[11pt,a4paper]{article}
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\usepackage[cp1251]{inputenc}
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\usepackage[russian]{babel}
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\usepackage{amssymb,amsfonts,amsmath}
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\usepackage{xspace}
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\pagestyle{empty}
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\newcounter{zadacha}
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\setcounter{zadacha}0
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\newcommand*{\nz}{%
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\par\medskip\refstepcounter{zadacha}
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\textbf{\thezadacha.} }
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\let\C\relax
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\newcommand*{\C}{\ensuremath{\mathbb C}\xspace}
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\newcommand*{\R}{\ensuremath{\mathbb R}\xspace}
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\newcommand*{\Z}{\ensuremath{\mathbb Z}\xspace}
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\let\Re\relax
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\let\Im\relax
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\DeclareMathOperator{\Re}{Re}
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\DeclareMathOperator{\Im}{Im}
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\begin{document}
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\begin{center}\large\sc
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Òåîðèÿ ôóíêöèé êîìïëåêñíîãî ïåðåìåííîãî 2025\\ Ëèñòîê 14
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\end{center}
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\noindent
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$\wp$-ôóíêöèÿ Âåéåðøòðàññà ñ ïåðèîäàìè $2\omega_1,
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2\omega_2$ (${\rm Im}\, \omega_2/ \omega_1 >0$):
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$$
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\wp (z)=\frac{1}{z^2}+\sum_{s\neq 0} \Bigl (\frac{1}{(z-s)^2}-
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\frac{1}{s^2}\Bigr ),
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\quad s=2\omega_1 n_1 +2 \omega_2 n_2, \quad n_{1,2}\in \Z .
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$$
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Ðàçëîæåíèå â îêðåñòíîñòè $z=0$:
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$$
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\wp (z)=\frac{1}{z^2}+\frac{g_2}{20}\, z^2 +\frac{g_3}{28}\, z^4+O(z^6),
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$$
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$$
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g_2=60\sum_{s\neq 0} s^{-4}, \quad g_3=140\sum_{s\neq 0} s^{-6}.
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$$
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$\zeta$- è $\sigma$-ôóíêöèè Âåéåðøòðàññà: $\zeta '(z)=-\wp (z)$, $\sigma '(z)/\sigma (z)=
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\zeta (z)$,
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$$
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\zeta (z)=\frac{1}{z}-\int_{0}^{z}\Bigl (\wp (x)-\frac{1}{x^2}\Bigr )dx,
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\quad
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\log (\sigma (z)/z)=\int_{0}^{z}\Bigl (\zeta (x)-\frac{1}{x}\Bigr )dx.
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$$
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$$
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\zeta (z)=\frac{1}{z}+\sum_{s\neq 0} \Bigl (\frac{1}{z-s}+\frac{1}{s}+\frac{z}{s^2}\Bigr ),
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\quad
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\sigma (z)=z\prod_{s\neq 0} \Bigl (1-\frac{z}{s}\Bigr )e^{\frac{z}{s}+\frac{z^2}{2s^2}}.
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$$
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$\zeta (z+2\omega_{\alpha} )=\zeta (z)+2\eta_{\alpha}$,
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$\sigma (z+2\omega_{\alpha})=-e^{2\eta_{\alpha}(z+\omega_{\alpha})}
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\sigma (z)$,
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$\eta_{\alpha}=\zeta (\omega_{\alpha})$.
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\noindent
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Âûðàæåíèå $\sigma$-ôóíêöèè ÷åðåç òýòà-ôóíêöèþ:
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$$
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\sigma (z)=2\omega_1 e^{\eta_1 z^2/(2\omega_1 )}\frac{\theta_1(z/(2\omega_1 ))}{\theta_1'(0)}.
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$$
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\noindent
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Ôóíêöèè ${\rm sn}$, ${\rm cn}$ è ${\rm dn}$:
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$$
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{\rm sn}\, u = \frac{\theta_3(0)}{\theta_2(0)}\,
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\frac{\theta_1 \Bigl (\frac{u}{\pi \theta_3^2(0)}
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\Bigr )}{\theta_4 \Bigl (\frac{u}{\pi \theta_3^2(0)}\Bigr )},
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\;
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{\rm cn}\, u = \frac{\theta_4(0)}{\theta_2(0)}\,
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\frac{\theta_2 \Bigl (\frac{u}{\pi \theta_3^2(0)}
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\Bigr )}{\theta_4 \Bigl (\frac{u}{\pi \theta_3^2(0)}\Bigr )},
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\;
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{\rm dn}\, u = \frac{\theta_4(0)}{\theta_3(0)}\,
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\frac{\theta_3 \Bigl (\frac{u}{\pi \theta_3^2(0)}
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\Bigr )}{\theta_4 \Bigl (\frac{u}{\pi \theta_3^2(0)}\Bigr )}.
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$$
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\nz
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Äîêàæèòå, ÷òî ôóíêöèÿ $\wp (z)$ óäîâëåòâîðÿåò äèôôåðåíöèàëüíîìó óðàâíåíèþ
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$$
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(\wp '(z))^2 =4\wp ^3(z)-g_2\wp (z)-g_3.
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$$
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\nz
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Äîêàæèòå òîæäåñòâî
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$
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\wp '''(z)=12\wp (z)\wp '(z).
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$
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\nz
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Äîêàæèòå ñîîòíîøåíèå $2\eta_1 \omega_2 -2\eta_2 \omega_1 =\pi i$.
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%\nz
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%Äîêàæèòå òîæäåñòâî
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%$$
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%\wp (z+y)-\wp (z-y)=-\frac{\wp '(z)\wp '(y)}{(\wp (z)-\wp (y))^2}.
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%$$
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\nz
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Äîêàæèòå, ÷òî ïðè $x+y+z=0$ ñïðàâåäëèâî òîæäåñòâî
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$$
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\left | \begin{array}{lll}
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1 & \wp (x) & \wp '(x)
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\\
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1 & \wp (y) & \wp '(y)
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\\
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1 & \wp (z) & \wp '(z)
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\end{array}
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\right |=0.
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$$
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\nz
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Âûðàçèòå
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$\displaystyle{
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\left | \begin{array}{lll}
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1 & \wp (x) & \wp '(x)
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\\
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1 & \wp (y) & \wp '(y)
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\\
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1 & \wp (z) & \wp '(z)
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\end{array}
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\right |}
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$
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ïðè ïðîèçâîëüíûõ $x,y,z \in \C$ ÷åðåç $\sigma$-ôóíêöèè.
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\nz
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Äîêàæèòå, ÷òî ïðè $x+y+z=0$ ñïðàâåäëèâî òîæäåñòâî
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$$
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\Bigl (\zeta (x)+\zeta (y) +\zeta (z)\Bigr )^2=
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\wp (x)+\wp (y)+\wp (z).
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$$
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\nz
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Äîêàæèòå òîæäåñòâî
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$$
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\zeta (x-a_1)-\zeta (x-a_2)+\zeta (a_1-a_2)=-\frac{1}{2}\,
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\frac{\wp '(x-a_1)+\wp '(x-a_2)}{\wp (x-a_1)-\wp (x-a_2)}.
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$$
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\nz
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Äîêàæèòå òîæäåñòâà
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$$
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{\rm sn}^2\, u +{\rm cn}^2\, u =1, \quad
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k^2{\rm sn}^2\, u +{\rm dn}^2\, u =1,
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$$
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ãäå $k^2=\theta_2^4(0)/\theta_3^4(0)$.
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\end{document} |