84 lines
No EOL
2.6 KiB
TeX
84 lines
No EOL
2.6 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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Теория функций комплексного переменного -- 2025\\ листок 8
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\end{center}
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\nz
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Вычислите интегралы Френеля
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$\displaystyle{
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\int_{0}^{\infty}\! \cos x^2 dx, \;\; \int_{0}^{\infty}\! \sin x^2 dx}
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$.
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\nz
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Вычислите интеграл
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$\displaystyle{\int_{0}^{\infty}\frac{\cos ax -\cos bx}{x^2}\, dx}$.
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\nz
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Вычислите интегралы
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а) $\displaystyle{\int_{0}^{\infty}\frac{\sin x}{x}\, dx}$,
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б) $\displaystyle{\int_{0}^{\infty}\frac{\sin x}{x}\,
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\frac{\sin (x/3)}{x/3}\, dx}$.
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\nz
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Вычислите интеграл
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$\displaystyle{\int_{0}^{\infty}\cos x\, \frac{\sin x}{x}\, dx}$.
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\nz
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Вычислите определенные интегралы:
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а) $\displaystyle{\int_{0}^{2\pi}\!\!\frac{d\varphi}{a+b\cos \varphi}}$ ($a>b>0$),\\
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б) $\displaystyle{\int_{0}^{2\pi}\!\!
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e^{\cos \varphi}\cos (n\varphi -\sin \varphi )d\varphi}$,
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в) $\displaystyle{\int_{0}^{\infty}\frac{x^2+1}{x^4+1}\, dx}$,
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г) $\displaystyle{\int_{0}^{\infty}\frac{dx}{x^n+1}}$,\\
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д) $\displaystyle{\int_{0}^{\infty}\frac{\cos ax}{x^2+b^2}\, dx}$,
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е) $\displaystyle{\int_{0}^{\infty}\frac{x\sin ax}{x^2+b^2}\, dx}$,
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ж) $\displaystyle{\int_{0}^{\infty}\frac{\sin ax}{x(x^2+b^2)}\, dx}$,
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з) $\displaystyle{\int_{-\infty}^{\infty}\frac{e^{ax}dx}{1+e^x}}\;\;$ ($0<a<1$),\\
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и) $\displaystyle{\int_{-\infty}^{\infty}\frac{\cos kx\, dx}{\cosh x}}$.
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\nz
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Вычислите определенные интегралы
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а) $\displaystyle{\int_{-\infty}^{\infty}\frac{\cos kx\, dx}{\cosh x +\cosh a}}\;\;
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(k, a>0)$,
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(б) $\displaystyle{\int_{0}^{\infty}e^{-\pi x}\, \frac{\sin ax}{\sinh \pi x}\, dx}$,
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(в) $\displaystyle{\int_{0}^{\infty}\frac{x-\sin x}{x^3}\, dx}$,
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г) $\displaystyle{\int_{-\infty}^{\infty}\frac{e^{ax}-e^{bx}}{1-e^x}\, dx}\;\;
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(0<a,b<1)$.
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(д) $\displaystyle{\int_{0}^{\infty}\frac{x^2 dx}{\cosh x}\,} $,
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(е) $\displaystyle{\int_{0}^{\infty}\frac{xdx}{\sinh x}}$,
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(ж) $\displaystyle{\int_{0}^{\infty}\frac{dx}{(x^2+\pi^2)\cosh x}\,} $.
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\end{document} |