106 lines
No EOL
2.8 KiB
TeX
106 lines
No EOL
2.8 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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\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 12
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\end{center}
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\nz
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Ïóñòü ðåãóëÿðíàÿ âåòâü $g(z)$ ôóíêöèè $\sqrt{z^2-4}$ îïðåäåëåíà â îáëàñòè $D$,
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ïðåäñòàâëÿþùåé ñîáîé êîìïëåêñíóþ ïëîñêîñòü ñ ðàçðåçîì ïî ïîëóîêðóæíîñòè
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$|z|=2$, ${\rm Im} z\geq 0$, ïðè÷åì ãëàâíàÿ ÷àñòü ðÿäà Ëîðàíà ôóíêöèè $g(z)$
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â îêðåñòíîñòè $\infty$ ðàâíà $z$. Âû÷èñëèòå èíòåãðàë
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$$
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I=\oint _{|z|=1}\frac{dz}{g(z)-3z}.
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$$
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\nz
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Âû÷èñëèòå èíòåãðàë
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$$
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\oint_{|z|=\frac{1}{2}}\frac{dz}{(2+g(z))\sin z},
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$$
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ãäå $g(z)$ -- îäíîçíà÷íàÿ âåòâü ôóíêöèè $\sqrt{z-1}$ â êðóãå
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$|z|<\frac{1}{2}$, òàêàÿ, ÷òî $g(0)=i$.
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\nz
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Ïóñòü $f(z)$ -- ðåãóëÿðíàÿ âåòâü ôóíêöèè $\sqrt[3]{2z-8}$ â êîìïëåêñíîé ïëîñêîñòè
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ñ ðàçðåçîì ïî ëó÷ó $[4, 4-i\infty ]$ òàêàÿ, ÷òî $f(8)=-1-i\sqrt{3}$. Âû÷èñëèòå èíòåãðàë
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$$
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I=\oint _{|z-2|=3/2}\frac{dz}{f(z)-z+2}.
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$$
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\nz
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Âû÷èñëèòå èíòåãðàëû ñ ïîìîùüþ âû÷åòîâ:
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à) $\displaystyle{\int\limits_{-1}^{1}
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\frac{dx}{\sqrt{1-x^2}}}\,\,$;
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á) $\displaystyle{\int\limits_{0}^{1}
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\sqrt{\frac{1-x}{x}}\, dx} \,\,$.
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\nz
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Âû÷èñëèòå èíòåãðàë ñ ïîìîùüþ âû÷åòîâ:
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$\displaystyle{\int\limits_{0}^{\infty}
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\frac{x^{\alpha -1}}{x+1}\,dx} \,\,$ ($0<\alpha <1$).
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\nz
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Âû÷èñëèòå èíòåãðàëû ñ ïîìîùüþ âû÷åòîâ:
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à) $\displaystyle{\int\limits_{0}^{1}\left (\frac{x}{1-x}\right )^{\alpha}
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\frac{dx}{x+1}\,\,}$ ($-1<\alpha <1$);
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á) $\displaystyle{\int\limits_{1}^{2}\sqrt[5]{\frac{(2-x)^3}{(x-1)^3}}\, dx}$;\\
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â) $\displaystyle{\int\limits_{-2}^{2}\frac{dx}{\sqrt[4]{(2+x)^2 (4-x^2)}}}\,$.
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\nz
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Âû÷èñëèòå èíòåãðàëû ñ ïîìîùüþ âû÷åòîâ:
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à) $\displaystyle{\int\limits_{0}^{\infty}\frac{dx}{x^3+1}}\,$;
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á) $\displaystyle{\int\limits_{0}^{\infty}\frac{\log x \, dx}{x^2+a^2} \,\,\, (a>0)}$;
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â) $\displaystyle{\int\limits_{0}^{\infty}\frac{\log x \, dx}{(x+1)(x+2)^2}\,}$;\\
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ã) $\displaystyle{\int\limits_{0}^{\infty}\frac{\log x \, dx}{x^2-1}}\,$.
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\nz
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Âû÷èñëèòå èíòåãðàëû ñ ïîìîùüþ âû÷åòîâ:
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à) $\displaystyle{\int\limits_{0}^{\infty}\left (\frac{\log x}{x-1}\right )^2 dx}$;
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á) $\displaystyle{\int\limits_{0}^{\infty}\frac{\sqrt{x}\, \log x dx}{x^2+1}}$;
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â) $\displaystyle{\int\limits_{-\infty}^{\infty}\frac{\log |x^2-1|}{x^2+1}\, dx}$;\\
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ã) $\displaystyle{\int\limits_{0}^{\infty}
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\frac{\log (1+x^2)}{x^2}\, \frac{dx}{1+x^2}}$.
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\end{document} |