From 69f2cbd749aa02e7b0926dc6c10dab66b1515ce6 Mon Sep 17 00:00:00 2001 From: mikl Date: Wed, 15 Jul 2026 05:07:40 +0300 Subject: [PATCH] =?UTF-8?q?=D0=A2=D0=A4=D0=9A=D0=9F:=20problem=20list,=20l?= =?UTF-8?q?istki,=20lecture,=20README=20with=20topic=20checklist?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .gitignore | 17 + ForStudents/CA-prob.pdf | Bin 0 -> 396915 bytes ForStudents/CA-prob.tex | 1732 +++++++++++++++++++++ ForStudents/bd.png | Bin 0 -> 56682 bytes ForStudents/лекция 13.03.2026.pdf | Bin 0 -> 1956799 bytes ForStudents/лекция 13.03.2026.pptx | Bin 0 -> 3681808 bytes ForStudents/листки Забродина/listok1.tex | 144 ++ ForStudents/листки Забродина/listok10.tex | 109 ++ ForStudents/листки Забродина/listok11.tex | 111 ++ ForStudents/листки Забродина/listok12.tex | 106 ++ ForStudents/листки Забродина/listok13.tex | 148 ++ ForStudents/листки Забродина/listok14.tex | 162 ++ ForStudents/листки Забродина/listok2.tex | 171 ++ ForStudents/листки Забродина/listok3.tex | 134 ++ ForStudents/листки Забродина/listok4.tex | 121 ++ ForStudents/листки Забродина/listok5.tex | 99 ++ ForStudents/листки Забродина/listok6.tex | 134 ++ ForStudents/листки Забродина/listok7.tex | 129 ++ ForStudents/листки Забродина/listok8.tex | 84 + ForStudents/листки Забродина/listok9.pdf | Bin 0 -> 124557 bytes ForStudents/листки Забродина/prob1.txt | 0 README.md | 68 + problem-list.pdf | Bin 0 -> 396915 bytes 23 files changed, 3469 insertions(+) create mode 100644 .gitignore create mode 100644 ForStudents/CA-prob.pdf create mode 100644 ForStudents/CA-prob.tex create mode 100644 ForStudents/bd.png create mode 100644 ForStudents/лекция 13.03.2026.pdf create mode 100644 ForStudents/лекция 13.03.2026.pptx create mode 100644 ForStudents/листки Забродина/listok1.tex create mode 100644 ForStudents/листки Забродина/listok10.tex create mode 100644 ForStudents/листки Забродина/listok11.tex create mode 100644 ForStudents/листки Забродина/listok12.tex create mode 100644 ForStudents/листки Забродина/listok13.tex create mode 100644 ForStudents/листки Забродина/listok14.tex create mode 100644 ForStudents/листки Забродина/listok2.tex create mode 100644 ForStudents/листки Забродина/listok3.tex create mode 100644 ForStudents/листки Забродина/listok4.tex create mode 100644 ForStudents/листки Забродина/listok5.tex create mode 100644 ForStudents/листки Забродина/listok6.tex create mode 100644 ForStudents/листки Забродина/listok7.tex create mode 100644 ForStudents/листки Забродина/listok8.tex create mode 100644 ForStudents/листки Забродина/listok9.pdf create mode 100644 ForStudents/листки Забродина/prob1.txt create mode 100644 README.md create mode 100644 problem-list.pdf diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..da85b9f --- /dev/null +++ b/.gitignore @@ -0,0 +1,17 @@ +# build artifacts +ForStudents/CA-prob.aux +ForStudents/CA-prob.bbl +ForStudents/CA-prob.blg +ForStudents/CA-prob.log 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+\usepackage{amsmath,amsthm,amsfonts,amssymb,graphicx} +\usepackage{pstricks,pst-node,verbatim} +\usepackage[utf8]{inputenc} +\usepackage[russian]{babel} + +\newtheorem{theorem}{Теорема} +\newtheorem{lemma}{Лемма} +\newtheorem{remark}{Замечание} +\newtheorem{corollary}{Следствие} +\newtheorem{proposition}{Предложение} +\newtheorem{definition}{Определение} +\newtheorem{exercise}{Упражнение} +\theoremstyle{definition} +\newtheorem{p}{} +\newtheorem{example}{Пример} + +\def\N{\mathbb N} +\def\Z{\mathbb Z} +\def\Q{\mathbb Q} +\def\R{\mathbb R} +\def\C{\mathbb C} +\def\F{\mathbb F} +\def\Pc{\mathcal{P}} +\def\0{\varnothing} +\def\sm{\setminus} +\renewcommand\le{\leqslant} +\renewcommand\ge{\geqslant} + +\def\Re{\mathrm{Re}} +\def\Im{\mathrm{Im}} + +\begin{document} + +\tableofcontents + +{\small Следующий список задач для семинаров основан на листочках, подготовленных А.В. Забродиным. +Любая из перечисленных ниже задач, за исключением задач со звездочкой, + может встретиться на промежуточной контрольной или итоговом экзамене + (а в качестве бонусной задачи может встретиться и задача со звездочкой). + +Если задача помечена как \textbf{(ДЗ)}, то она может встретиться в домашнем задании. +Эти задачи можно обсуждать на семинаре, но только по инициативе студентов, пытавшихся их + решить, но испытавших какие-либо сложности. +Целью обсуждения в этом случае является преодоление возникших сложностей. + +Если задача помечена звездочкой, то разбор ее решения на семинаре следует отложить + на некоторое время --- чтобы студенты могли самостоятельно над ней подумать. +Задачи со звездочкой можно сдавать лично преподавателю вне семинарских часов. +Преподаватель может устанавливать дедлайны по отдельным задачам со звездочкой так, что до дедлайна + можно эту задачу сдавать в индивидуальном порядке, а после дедлайна задача + может быть разобрана на семинаре, если возникнет такой запрос со стороны студентов. + +Задачи, помеченные контурным кружочком ($\circ$), рекомендуется разобрать на семинаре в каждой группе. +Остальные задачи можно опускать при нехватке времени. +Задачи, помеченные заполненным кружочком ($\bullet$), возможно, будут обсуждаться на лекции.} + +\section{Комплексные числа} + +\begin{p} +Найдите действительные и мнимые части следующих комплексных чисел: + +\medskip + +(а)$\circ$ $\displaystyle{\frac{5}{1+2i}+\frac{5}{2-i}}$, + +(б) $\displaystyle{\left (\frac{1-i}{1+i}\right )^3}$, + +(в) $\displaystyle{\frac{(1-i\sqrt{3})^6}{(1+i)^4}}$, + +(г) $\displaystyle{\left (\frac{i^5+2}{i^{19}+1}\right )^2}$ +\end{p} + +\begin{p} +Найдите модули и аргументы следующих комплексных чисел: + +\medskip + +(а)$\circ$ $\displaystyle{(\sqrt{3}-i)^{2026}}$, + +(б) $\displaystyle{\left (\frac{1+i}{1-i}\right )^2}$, + +(в) $\displaystyle{\frac{1+\cos \alpha +i\sin \alpha}{1+\cos \alpha -i\sin \alpha}, +\quad 0<\alpha <\frac{\pi}{2}}$, + +(г) \textbf{(ДЗ)} + $\displaystyle{\frac{e^{i\alpha}+1}{e^{i\alpha}-1}, \quad 0<\alpha <2\pi}$. +\end{p} + +\begin{p}\textbf{(ДЗ)} +Докажите, что многочлен +$$ +f(x)=(\cos \alpha +x\sin \alpha )^n -\cos n\alpha - x\sin n \alpha$$ +делится на $x^2+1$. +\emph{Указание:} воспользуйтесь тем, что $f(i)=0$. +\end{p} + +\begin{p} +Выразите $\sin 4x$ как многочлен от $\sin x$, $\cos x$. +\end{p} + +\begin{p}$\circ$ +Решите уравнение $\bar z =z^{n-1}$, $n\in \mathbb{N}$. +\end{p} + +\begin{p} +Решите уравнения $z^2=5-12i$, $z^4+4=0$. +\end{p} + +\begin{p}$\circ$ +Пусть $\zeta_1, \ldots , \zeta_n$ --- все корни уравнения $z^n=1$, где $n$ --- +натуральное число. Найдите сумму $\zeta_1^k+\ldots +\zeta_n^k$ для каждого +натурального $k$. +\end{p} + +\begin{p}\textbf{(ДЗ)} +Точки $z_1$ и $z_2$ --- смежные вершины правильного $n$-угольника. +Найдите вершину $z_3$, смежную с $z_2$ ($z_3\neq z_1$, нумерация против +часовой стрелки). +\end{p} + +\begin{p} +Докажите, что комплексные числа $a,b,c$ представляют вершины равностороннего +треугольника тогда и только тогда, когда +$$ +a^2+b^2+c^2=ab+bc+ac. +$$ +\end{p} + +\begin{p}\textbf{*} +Найдите произведение длин отрезков, соединяющих вершину правильного $n$-угольника +со всеми остальными вершинами, если длина стороны равна 1. +\end{p} + +\begin{p} +Дайте геометрическое описание множеств + +(а) $\{ z \colon |z-z_1|=|z-z_2|\}$, + +(б) $\{ z \colon |z-1|+|z+1|=2a \}$ ($a>1$), + +(в) \textbf{(ДЗ)} $\{ z \colon \mbox{Re}(1/z)=\frac{1}{2} \}$. +\end{p} + +\begin{p} +Пусть $a,b,c,d$ --- четыре различные точки на единичной окружности. +Найдите точку перечесения прямых $ab$ и $cd$. +\end{p} + +\begin{p}\textbf{*} +Докажите равенство +$$ +|\sqrt{z^2-1}+z|+|\sqrt{z^2-1}-z|=|z-1|+|z+1|. +$$ +В левой части в оба раза берется одно и то же значение квадратного корня. +\end{p} + +\begin{p}\textbf{*} +Пусть $P(z)$ --- многочлен степени $n$, имеющий $n$ вещественных корней +(с учетом кратности), $a$ --- произвольное вещественное число. Докажите, +что многочлен $Q(z)=P(z+ia)+P(z-ia)$ тоже имеет $n$ вещественных корней. +\end{p} + +\begin{p} +Найдите радиус сходимости степенного ряда $\sum_{n=1}^{\infty} n! z^{n!}$. +\end{p} + +\begin{p} +Верно ли, что степенной ряд для $e^z$ сходится равномерно на $\C$? +Обоснуйте ответ. +\end{p} + + +\section{Дробно-линейные преобразования} +\label{s:fraclin} + +{\small +\emph{Дробно-линейной функцией}, или \emph{дробно-линейным преобразованием}, + называется функция вида +$$ +f(z)=\frac{az+b}{cz+d}\,, \qquad a,b,c,d \in \C, \qquad +ad-bc \neq 0. +$$ +} + +\begin{p} +Докажите, что дробно-линейное преобразование осуществляет взаимно-однозначное +отображение $\bar{\C}\to \bar{\C}$. +\end{p} + +\begin{p}$\bullet$ +Докажите, что дробно-линейные преобразования образуют группу. +Какой группе она изоморфна? +\end{p} + +\begin{p}$\bullet$ +Докажите, что дробно-линейная функция переводит окружности +и прямые в окружности (или прямые). +\end{p} + +\begin{p} +Напишите дробно-линейную функцию, переводящую три произвольные +(различные) точки $z_1$, $z_2$, $z_3$ в три другие (различные) точки $w_1$, $w_2$, $w_3$. +Единственна ли она? +\end{p} + +\begin{p} +Во что преобразуется квадрант $x>0, y>0$ при отображении $\displaystyle{ +w=\frac{z-i}{z+i}}\, $? +\end{p} + +\begin{p} +Найдите все дробно-линейные функции $f$ такие, что $f(f(z))=z$ для всех $z\in\C$. +\end{p} + + +\begin{p}\textbf{*} +Докажите, что бидифференциал $\displaystyle{\frac{dz_1 dz_2}{(z_1-z_2)^2}}$ +инвариантен относительно дробно-линейных преобразований. +Как именно дробно линейное преобразование должно действовать на + бидифференциал, предлагается восстановить из здравого смысла; + решение задачи должно начинаться с четкого определения этого действия. +\end{p} + +\vspace{5mm} + +{\small +\emph{Двойным отношением} четырех различных точек $z_1, z_2, z_3, z_4$ называется +$$ +(z_1, z_2, z_3, z_4)=\frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}. +$$ +} + +\begin{p} +Докажите, что дробно-линейная функция сохраняет двойное отношение: +$$ +(f(z_1), f(z_2), f(z_3), f(z_4))=(z_1, z_2, z_3, z_4). +$$ +\end{p} + +\begin{p} +Докажите, что четыре различные точки лежат на одной окружности +(или на одной прямой), если и только если их двойное отношение +вещественно. +\end{p} + +\begin{p} +Найдите общий вид дробно-линейных изоморфизмов + + (а)$\bullet$ $\mathbb{H} \to \mathbb{H}$, + + (б)$\bullet$ $\mathbb{D} \to \mathbb{D}$, + + (в) \textbf{(ДЗ)} $\mathbb{D} \to \mathbb{H}$ + +($\mathbb{H}$ --- верхняя полуплоскость, $\mathbb{D}$ --- единичный круг). +\end{p} + +\begin{p} +Отобразите верхнюю полуплоскость на единичный круг так, чтобы +$w(2i)=0$, $\mbox{arg}\, w'(2i)=0$. +\end{p} + +\begin{p} +Сколько неподвижных точек может иметь дробно-линейное преобразование? +Докажите, что если дробно-линейное преобразование имеет две неподвижные точки в $\C$, +то произведение производных в этих точках равно 1. +\end{p} + +\begin{p} +Дробно-линейное преобразование с одной неподвижной точкой $z_0$ называется \emph{параболическим}. +Докажите, что параболическое преобразование можно записать в канонической форме +$$ +\frac{1}{w-z_0}=\frac{1}{z-z_0}+h +$$ +если $z_0\neq \infty$ и $w=z+h$ если $z_0=\infty$. +\end{p} + +\begin{p} +Докажите, что дробно-линейное преобразование с двумя различными неподвижными точками $z_1$, $z_2$ +можно представить в канонической форме +$$ +\frac{w-z_1}{w-z_2}=k\frac{z-z_1}{z-z_2} +$$ +если $z_{1,2}\neq \infty$ и $w-z_1=k(z-z_1)$ если $z_2=\infty$. Преобразование +называется гиперболическим, если $k>0$, эллиптическим если $|k|=1$ ($k\neq 1$) и +локсодромическим в общем случае комплексного $k$. +\end{p} + +\section{Комплексная дифференцируемость} + +\begin{p}$\circ$ +Найдите все точки, в которых $\C$-дифференцируемы функции\\ +а) $|z|^2$, б) $|z|z$, в) \textbf{(ДЗ)} $({\rm Re}\, z)^2$. +\end{p} + +\begin{p} +а) При каких комплексных $a,b,c$ функция $f(z)=az+b\bar z +z^2 +c$ имеет +комплексную производную в точке $z=0$? б) При каких вещественных $a,b$ +функция $f(x,y)=ax^2+by^2+ixy$ голоморфна на всем $\C$? +\end{p} + +\begin{p}$\circ$ +Пусть $f$ --- дифференцируемая в точке $a$ функция комплексного переменного. +Докажите, что функция $z\mapsto\overline{f(\bar z)}$ дифференцируема в точке $\bar a$. +\end{p} + +\begin{p} $\circ$ +Покажите, что функция $f: re^{i\varphi}\mapsto re^{2i\varphi}$ не +голоморфна на $\C \setminus \{0\}$. +\end{p} + +\begin{p} +а) +Докажите, что если голоморфная в некоторой области (связном открытом множестве) функция +$f(z)$ вещественна (т.е. принимает только вещественные значения), то она +постоянна. + +б) \textbf{(ДЗ)} +Пусть функция $f(z)$ голоморфна в некоторой области, + и $|f(z)|=1$ всюду в этой области. Докажите, что $f(z)\equiv {\rm const}$. +\end{p} + +\begin{p} +Восстановите аналитическую функцию $f(z)$ по условию: + +а) ${\rm Re}\, f(z)=\sin x \cosh y$, $f(0)=0$, + +б) ${\rm Re}\, f(z)= +x\sin x \cosh y -y\sinh y \cos x$, $f(0)=0$, + +в) ${\rm Im}\, f(z)= +y\cos x \cosh y-x\sin x \sinh y$, $f(0)=2$, + +г) \textbf{(ДЗ)} $|f(z)|=(x^2+y^2)e^x$, + +д) $\mbox{arg}\, f(z)=xy$. +\end{p} + +\begin{p} +Пусть $f$ --- многочлен от $z$ с комплексными коэффициентами. +Запишем $f(x+iy)$ как $u(x,y)+iv(x,y)$, где $u$ и $v$ --- многочлены от $x$, $y$ с вещественными коэффициентами. +Докажите, что %аналитическую в $\C$ функцию +%$f(z)=u(x,y)+iv(x,y)$ +функцию $f$ можно восстановить по ее вещественной части $u$ +или мнимой части $v$ с помощью формул +$$ +f(z)=2u\Bigl (\frac{z+z_0}{2}, \, \frac{z-z_0}{2i}\Bigr )- +\overline{f(z_0)}, +$$ +$$ +f(z)=2iv\Bigl (\frac{z+z_0}{2}, \, \frac{z-z_0}{2i}\Bigr )+ +\overline{f(z_0)}, +$$ +где $z_0 \in \C$ --- некоторая точка. +\end{p} + +\vspace{0.5cm} +{\small +Вещественнозначная функция двух вещественных +переменных $F(x,y)$ называется +{\it гармонической}, если она дважды дифференцируема и +$$ +\frac{\partial^2 F}{\partial x^2}+\frac{\partial^2 F}{\partial y^2}=0. +$$ +} + +\begin{p} +Запишите оператор Лапласа $\Delta =\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}$ +%в координатах $z$, $\bar z$. +через $\frac{\partial}{\partial z}$, $\frac{\partial}{\partial \overline{z}}$. +\end{p} + +\begin{p} +Пусть $f(z)$ --- голоморфная функция. +Докажите, что функции ${\rm Re} f$ и ${\rm Im} f$ --- гармонические +(т.е. $\Delta {\rm Re} f=\Delta {\rm Im} f=0$). +Что можно сказать о функциях $|f|$ и $\mbox{arg}\, f$? +В этой задаче можно предполагать, что вещественная и мнимая части функции $f$ + имеют непрерывные частные производные порядка $\le 2$. +\end{p} + +\begin{p} +Пусть $F:U\to\R$ --- гармоническая функция на связном открытом множестве $U\subset \C$. +Для каких функций $G:\R\to \R$ функция $G\circ F$ тоже будет гармонической на $U$? +\end{p} + +\begin{p} +Найдите все гармонические функции вида + +а) $F=\varphi (x^2+y^2)$, + +б) $F=\varphi (x^2-y^2)$, + +в) $F=\varphi (y/x)$. +\end{p} + +\begin{p} +Пусть $u(x,y)$ -- гармоническая функция. Являются ли гармоническими +функции $u(x, -y)$, $u(x^2-y^2, 2xy)$? +\end{p} + +\begin{p} +Найдите гармоническую функцию $u(x,y)$ на $\C$ такую, что\\ +$u(\cos t, \sin t)=\cos^2 t$. +\end{p} + +\section{Производная Шварца} +\emph{Задачи из этого раздела не войдут в итоговый контроль.} + +\medskip + +{\small +\emph{Производной Шварца} (шварцианом) голоморфной +функции $w(z)$ (во всех задачах про производную Шварца можно считать эту функцию +многочленом, а кратные производные воспринимать алгебраически --- как производные многочлена) +называется выражение +$$ +\displaystyle{S(w;z):=\frac{w'''(z)}{w'(z)}-\frac{3}{2} +\left (\frac{w''(z)}{w'(z)}\right )^2}. +$$ +} + +\begin{p} +Докажите следующее тождество, связывающее двойное отношение со шварцианом: +$$ +\frac{(w(z+ta), w(z+tb), w(z+tc), w(z+td))}{(a,b,c,d)}= +$$ +$$ +=1+\frac{t^2}{6}\, (a-b)(c-d)S(w;z) +O(t^3)\quad +(t\to 0). +$$ +\end{p} + +\begin{p} +Докажите, что +$S(w;z)=0$ тогда и только тогда, когда $w(z)$ --- дробно-линейная функция. +\end{p} + +\begin{p} +Докажите, что +$$ +S(w;z)=6 \lim_{\zeta \to z}\, \frac{\partial }{\partial z}\frac{\partial }{\partial \zeta}\, +\log \frac{w(z)-w(\zeta )}{z-\zeta}. +$$ +\end{p} + +\begin{p} +Пусть $w(z)$, $f(w)$ --- две функции, $F$ --- их композиция: +$F(z)=f(w(z))$. Докажите, что +$$ +S(F;z)=S(f;w)\Bigl (\frac{dw}{dz}\Bigr )^2 +S(w;z). +$$ +\end{p} + +\begin{p}\textbf{*} +Пусть $f_1, f_2$ --- два линейно независимых решения дифференциального уравнения $f''+Q(z)f=0$. +Покажите, что $Q(z)=\frac{1}{2}\, S(f_1/f_2; z)$. +\end{p} + + +\section{Геометрия комплексных функций} + +\begin{p} +Найдите, во что переводят координатную сетку следующие отображения из +$\C$ в $\C$: +(а) $z\mapsto z^2$; +(б) $z\mapsto e^z$; +(в) $z\mapsto 1/z$. +\end{p} + +\begin{p} +Отображение $z\mapsto z+(1/z)$ определено на множестве всех ненулевых +комплексных чисел. + +(а) Во что это отображение переводит множество $\{z\colon |z|>1\}$? + +(б) Является ли это отображение взаимно-однозначным на данном множестве? +\end{p} + +\begin{p} +Пусть $A>0, C$ действительные, а $B$ -- комплексная постоянные и пусть $AC<|B|^2$. +Докажите, что уравнение +$$ +A|z|^2 +\bar B z +B\bar z +C=0 +$$ +является уравнением окружности и найдите центр этой окружности и ее радиус. +\end{p} + +\begin{p} +Напишите уравнение окружности, проходящей через три различные +точки $z_1$, $z_2$, $z_3$, не лежащие на одной прямой. +\end{p} + +\begin{p}\textbf{*} +Пусть отображение задано формулой $z\mapsto Az+B\bar z$, + в которой $A$, $B$ --- заданные комплексные числа, причем $|A|>|B|$. +Докажите, что образ окружности с центром в нуле при таком преобразовании --- эллипс. +Выразите длины главных полуосей этого эллипса, а также отношение площади внутри + эллипса к площади внутри окружности в терминах $A$ и $B$. +%Во что оно переводит окружность с центром в нуле? +\end{p} + +\begin{p}$\circ$ +Найдите конформное отображение полукруга $|z|<1$, ${\rm Im}\, z>0$ +на верхнюю полуплоскость. +\end{p} + +\begin{p}$\circ$ +Найдите конформное отображение полуполосы ${\rm Re}\, z>0$, +$0< {\rm Im}\, z <1$ на верхнюю полуплоскость. +\end{p} + +\begin{p} +Найдите конформное отображение верхней полуплоскости с выброшенным отрезком +от $0$ до $i$ на верхнюю полуплоскость. +\end{p} + +\begin{p}\textbf{*} +Найдите конформное отображение внешности эллипса с полуосями $a$, $b$ ($a>b$) +на внешность единичного круга. +\end{p} + +\begin{p} +Найдите конформное отображение области +$$ +\{z:\,\mathrm{Im}(z)>0\}\setminus \{z:\, |z|=1,\ 0\leq {\rm arg}\, z \leq \alpha\}, +$$ +%полуплоскостью ${\rm Im}\, z>0$ с разрезом по дуге окружности +%$|z|=1$, $0\leq {\rm arg}\, z \leq \alpha$, +где $0<\alpha <\pi$, +на верхнюю полуплоскость. +\end{p} + +\begin{p} +Найдите конформное отображение области, являющейся полуполосой +${\rm Re}\, z>0$, $0<{\rm Im}\, z < \pi$ с разрезом по отрезку +$[\frac{1}{2}\pi i, 1+\frac{1}{2}\pi i]$, на верхнюю полуплоскость. +\end{p} + +\begin{p}\textbf{*} +Найдите конформное отображение $z(w)$ внешности единичного круга +$\mathbb{D}$ на расширенной комплексной плоскости +$\hat \C$ (т.е. некомпактной области $\hat \C \setminus \overline{\mathbb{D}}$) +на расширенную комплексную плоскость с разрезом по дуге +$\gamma$ +единичной окружности с углом раствора +$2\alpha$, +$$\gamma = \bigl \{z\in \C\mid \, |z|=1, + -\alpha < \mbox{arg}\, z < \alpha \bigr \} +\quad (0 < \alpha <\pi ), +$$ +(т.е. на область $\hat \C \setminus \gamma$), +такое, что $z(\infty )=\infty$ и $z'(\infty )\in \R_{>0}$. +Последнее условие означает, что $z(w)=\lambda w+o(w)$ при $w\to\infty$ + для некоторого положительного действительного числа $\lambda$. +\end{p} + +%\begin{p} +%Для каждого из перечисленных ниже открытых множеств постройте его конформное отображение на верхнюю полуплоскость. +% +%(а) $\{z:\, 1<\Re(z)+\Im(z)<2\}$, +% +%(б) $\{z:\, \Re(z)^2<\Im(z/2)^2-1\}$, +% +%(в) $\{z:\, \left|z-\sqrt{3}\right|>2,\ \left|z-\frac{\sqrt{3}}{3}\right|<1\}$, +% +%(г) $\mathbb{D}\setminus[0;1]$, где $\mathbb{D}$ --- единичный круг $\{z:\, |z|<1\}$, а $[0;1]$ --- отрезок +% вещественной оси, соединяющий точки 0 и 1, +% +%(д) $\mathbb{D}\setminus [1/2;1]$. +%\end{p} + + +\begin{p}\textbf{*} +Пусть $\Gamma$ --- гладкая кривая $\Gamma : [0, +\infty )\to \mathbb{H}$ в верхней полуплоскости +$\mathbb{H}$, начинающаяся в точке $\Gamma (0)\in \R$ вещественной оси, и пусть $\Gamma_t$ -- +часть кривой $\Gamma_t:=\Gamma ([0,t])$. Через $g(z,t)$ обозначим функцию, осуществляющую +конформное отображение области $\mathbb{H} \setminus \Gamma_t$ на $\mathbb{H}$, нормированную условием +$g(z, t)=z+u(t)z^{-1}+O(z^{-2})$, $z\to \infty$, $t>0$. Докажите, что существует +непрерывная вещественнозначная функция $\xi (t)$, такая, что функция $g(z, t)$ удовлетворяет +дифференциальному уравнению +$$ +\frac{\partial g(z, t)}{\partial t} = \frac{1}{g(z, t) -\xi (t)} \, +\frac{\partial u}{\partial t}, \qquad g(z, 0)=z. +$$ +Это уравнение называется (хордовым) уравнением Левнера и играет важную роль в теории +конформных отображений. +\end{p} + +\begin{p} \textbf{(ДЗ)} + Укажите хотя бы один конформный изоморфизм между верхней полуплоскостью и внешностью параболы. +Обоснуйте ответ. +\end{p} + +\begin{p} \textbf{(ДЗ)} +Укажите хотя бы один конформный изоморфизм между $\mathbb{D}$ и $\mathbb{D}\sm [1/2;1]$, + где $\mathbb{D}$ --- это открытый единичный диск, а $[1/2;1]$ --- отрезок вещественной оси + между точками $1/2$ и $1$. +Обоснуйте ответ. +\end{p} + +\section{Контурные интегралы} +Направление обхода контуров в контурных интегралах предполагается положительным +(против часовой стрелки). + +\begin{p} +Вычислите интеграл $\displaystyle{\int_{\gamma}(z-a)^n dz}$ +по контуру $\gamma (t)=a+re^{it}$, \\$0\leq t<2\pi$, $r>0$, $n\in \mathbb{Z}$. +\end{p} + +\begin{p} +Вычислите интеграл $\displaystyle{\int_{\gamma} +z^n dz}$, где $n\in \mathbb{Z}$, а $\gamma$ --- некоторый путь из $a$ в $b$. +(Убедитесь, что при $n\neq -1$ интеграл +не зависит от выбора пути, а при $n=-1$ зависит, и опишите как.) +\end{p} + +\begin{p} +Вычислите интеграл $\displaystyle{\oint_{|z|=1} +\frac{dz}{z^2 +\frac{1}{4}}}$ +по единичной окружности. +\end{p} + +\begin{p} +Вычислите интеграл +$\displaystyle{\oint_{\gamma}\frac{2z^2+5z+8}{z(z+1)^2}}\, dz$, где контур +$\gamma$ --- окружность $|z|=3$. +\end{p} + +\begin{p} +Найдите все возможные значения интеграла $\displaystyle{ +\int_{C} \frac{dz}{z(z^2-1)}}$ при различных выборах контура $C$ +(здесь $C$ --- замкнутый контур без самопересечений, не проходящий ни через +одну из точек $0,1,-1$). +\end{p} + +\begin{p} \textbf{(ДЗ)} +Покажите, что если путь $\gamma$, соединяющий точки 0 и 1, не проходит через +точки $\pm i$, то +$$ +\int_{\gamma}\frac{dz}{1+z^2} =\frac{\pi}{4}+k\pi, +$$ +где $k$ --- целое число. +\end{p} + +\begin{p} +Вычислите интеграл $\displaystyle{\oint_{|z|=1}\cos (z^{-1})dz}$. +\end{p} + +\begin{p} +Приведите пример рациональной функции $f$ и замкнутого контура $\gamma$ +таких, что $\displaystyle{\oint_{\gamma}f(z)dz=1}$ и +$\displaystyle{\oint_{\gamma}zf(z)dz =-1}$. +\end{p} + +\begin{p} +Вычислите интеграл $\displaystyle{\int_{\gamma}\bar z dz}$, +где $\gamma$ -- окружность радиуса $R$ с центром в точке $a\in \C$, +проходимая против часовой стрелки. +\end{p} + +\begin{p} \textbf{(ДЗ)} +Вычислите интеграл $\displaystyle{\int_{\gamma} +\frac{\bar z}{z}\, dz}$, +где $\gamma$ --- окружность радиуса $R$ с центром в точке $a$ +($|a|\neq R$). +\end{p} + +\begin{p} +Пусть $P(z)$ --- многочлен. Докажите, что +$$\displaystyle{ +\frac{1}{2\pi i}\oint_{|z-a|=R}P(z)d\bar z =-R^2 P'(a)}. +$$ +\end{p} + +\begin{p} +Пусть $\gamma$ --- гладкий замкнутый контур, а функция $f$ голоморфна в +окрестности кривой $\gamma$. Докажите, что +$\displaystyle{\oint_{\gamma}\overline{f(z)}f'(z)dz}$ --- чисто мнимое +число. +\end{p} + +\begin{p} +Вычислите интегралы +$$ +\hbox{(а)}\ \int \!\!\! \int_{\C} +e^{-\alpha |z|^2}\, dx \, dy,\qquad +\hbox{(б)}\ \int \!\!\! \int_{\C} +e^{-\alpha |z|^2 +az+\bar a \bar z}\, dx \, dy. +$$ +\end{p} + +\begin{p}\textbf{*} +Вычислите интеграл +$$ +\int_{0}^{2\pi} e^{\cos\varphi}\cos\left(\sin\varphi - 3\varphi\right)\, d\varphi. +$$ +\end{p} + +\section{Формула Коши, вычеты} + +\begin{p} +С помощью интегральной формулы Коши вычислите интегралы + +(а) $\displaystyle{\int_{|z|=2}\frac{e^z dz}{z^2-1}},\quad$ +(б) \textbf{(ДЗ)} $\displaystyle{\int_{|z-a|=a}\frac{zdz}{z^4-1}},\quad a>1$, + +\medskip + +(в) $\displaystyle{\int_{|z-1|=1}\frac{ze^zdz}{(z-1)^3}}$. +\end{p} + +\begin{p} +Пусть $f:\overline{\mathbb{H}}\cup\{\infty\}\to\C$ --- непрерывная функция, голоморфная в $\mathbb{H}$ +% в верхней полуплоскости, непрерывно продолжаемая +%на вещественную ось, причем $\lim\limits_{|z|\to \infty} f(z)=0$. + и такая, что $f(\infty)=0$. +Докажите \emph{интегральную формулу Шварца} +$$ +f(z)=\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{{\rm Im} f(x)}{x-z}\, dx,\quad \forall z\in\mathbb{H}. +$$ +Напомним, что $\mathbb{H}$ обозначает открытую верхнюю полуплоскость. +Интеграл понимается в смысле \emph{главного значения}, то есть как предел $\int_{-R}^{R}$ при $R\to +\infty$. +\end{p} + +\begin{p}\textbf{*} +Пусть $f(z)$ --- голоморфная функция в единичном круге, непрерывно продолжаемая +на его границу. Докажите \emph{интегральную формулу Пуанкаре} +$$ +f(z)=\frac{1}{2\pi}\int_{0}^{2\pi} \!{\rm Re} f(e^{i\theta}) +\frac{e^{i\theta}+z}{e^{i\theta}-z}\, d\theta +i{\rm Im} f(0). +$$ +\end{p} + +\begin{p} +Найдите вычеты дифференциальных форм в их особых точках:\\ + +а) $\displaystyle{\frac{z dz}{z^2+1}},\quad$ +б) $\displaystyle{\frac{e^z dz}{(z-1)^2}},\quad$ +в) $\displaystyle{\frac{z^{}dz}{\sin ^2 z}},\quad$ +г) $\displaystyle{\frac{z^{2n}dz}{(z-1)^n}},\quad$ +д) $\displaystyle{\frac{e^{1/z}dz}{1-z}}$, + +\medskip + +е) $\sin z \, \sin (1/z) \, dz,\quad$ +ж) \textbf{(ДЗ)} $\displaystyle{\frac{\sin 3z - 3\sin z}{\sin z \, (\sin z -z)}\, dz}$ в точке +$z=0,\quad$ + +\medskip + +з) $\displaystyle{z^2 \sin \frac{1}{z+1} \, dz},\quad$ +и) $\displaystyle{\frac{dz}{z(e^{2z}-1)}}$ в точке $z=0,\quad$ +к) $\displaystyle{e^{z/(1-z)}dz}$. +\end{p} + + +\begin{p} \textbf{(ДЗ)} +Докажите, что для четной функции $f(z)$ имеет место равенство +$$\mbox{res}_{z=a}f(z)dz =-\mbox{res}_{z=-a}f(z)dz,$$ +а для нечетной +$$\mbox{res}_{z=a}f(z)dz =\mbox{res}_{z=-a}f(z)dz.$$ +\end{p} + +\begin{p} +Вычислите интегралы: + +а) $\displaystyle{\oint_{|z|=4}\frac{z^4 dz}{e^z +1}},\quad$ +б) $\displaystyle{\oint_{|z|=1}\frac{e^z dz}{(2z+1)^2 (z+2)}},\quad$ +в) $\displaystyle{\oint_{|z|=1}(z+1)e^{1/z}dz},$ + +\medskip + +г) $\displaystyle{\oint_{|z|=1}\frac{z^3 dz}{2z^4+1}},\quad$ +д) $\displaystyle{\oint_{|z|=3}\frac{z^3-1}{z-1}\left (e^{\frac{1}{z}}+ +e^{\frac{1}{z-1}}+e^{\frac{1}{z-2}}\right )dz}$, + +\medskip + +е) $\displaystyle{\oint_{|z|=1}\frac{dz}{e^{2/z}-e^{1/z}}},\quad$ +ж) \textbf{(ДЗ)} $\displaystyle{\oint_{|z|=3}\frac{dz}{(z-1)^2(1-\cos z)}}$, + +\medskip + +з) $\displaystyle{\oint_{|z|=1}\frac{z-(\log 2)^{-1}}{e^{1/z}-2}\, dz}$. +\end{p} + +\begin{p} +Пусть функция $f(z)$ регулярна в замыкании конечной области $D$, +а точки $a_1, \ldots , a_n$ лежат в $D$ и попарно различны. Обозначим +$P(z)=(z-a_1)\ldots (z-a_n)$ и +$$ +\Phi (z)=-\frac{P(z)}{2\pi i}\oint_{\partial D} \frac{f(\xi )}{P(\xi )} +\, \frac{d\xi}{\xi -z}, \quad z\notin D. +$$ +Докажите, что функция $\Phi (z)$ аналитически продолжается на всю +плоскость и представляет собой многочлен степени $n-1$, причем +$\Phi (a_k)=f(a_k)$, $k=1, \ldots , n$. (Многочлен $\Phi (z)$ +называется \emph{интерполяционным многочленом Лагранжа}.) +\end{p} + +\begin{p} +Пусть $p_1, \ldots , p_N$ --- $N$ различных комплексных чисел. +Докажите тождество +$$ +\sum_{i=1}^N \prod_{j\neq i}^N \frac{p_i+p_j}{p_i-p_j}= +\frac{1}{2}\Bigl (1-(-1)^N\Bigr ). +$$ +{\it Указание: воспользуйтесь тем, что +сумма вычетов рациональной функции, включая вычет +в $\infty$, равна 0.} +\end{p} + +\begin{p}\textbf{*} +Пусть $w_1, \ldots , w_N$ --- $N$ различных комплексных чисел. +Найдите сумму +$$ +S=\sum_{i=1}^N \prod_{j\neq i}^N \frac{1-w_iw_j}{w_i-w_j}. +$$ +\end{p} + +\section{Изолированные особенности и ряды Лорана} + +\begin{p} +Для каждой из следующих функций найдите все их изолированные особенности и укажите тип этих + особенностей (устранимая, полюс, существенная). +В случае полюсов, укажите их порядок. + +$$ +\hbox{(a)}\ f(z)=\frac{1}{z(z^2-4)^2},\quad \hbox{(б)}\ f(z)=\frac{z}{\sin(z^2)},\quad \hbox{(в)}\ f(z)=e^{\tg z}. +$$ +\end{p} + +\begin{p} +Существует ли функция $f$, голоморфная в единичном круге $\mathbb{D}=\{z: |z|<1\}$ и удовлетворяющая + неравенствам $|z|^{5/3}\le |f(z)|\le |z|^{4/3}$ для всех $z\in\mathbb{D}$? +\end{p} + + +\begin{p} +Разложите функцию $\displaystyle f(z)=\frac{1}{(z-1)(z-2)}$ в ряд Лорана: + +a) в круге $\{z:|z|<1\}$; + +б) в кольце $\{z:1<|z|<2\}$; + +в) в кольце $\{z:|z|>2\}$. + +\end{p} + +\begin{p} +Пусть $a$ и $b$ --- два различных ненулевых комплексных числа. + +(а) Докажите, что функция $f(z)=\sqrt{(z-a)(z-b)}$ + имеет однозначную ветвь в бесконечном кольце $\{z: |z|>\max(|a|,|b|)\}$; + найдите коэффициенты при $z^2$ и $z^{-2}$ в ее ряде Лорана в этом кольце. + +(б) Те же вопросы про функцию $f(z)=\log\frac{z-a}{z-b}$ + (через $\log$ обозначается натуральный логарифм). +\end{p} + +\begin{p} +Пусть $f$ голоморфна в проколотом круге $\mathbb{D}^*=\{z:0<|z|<1\}$ + и удовлетворяет условию $|f(z)|\le |z|^{-\sqrt{7}}$ для всех $z\in\mathbb{D}^*$. +Что можно сказать об изолированной особенности функции $f$ в нуле? +\end{p} + +\begin{p}\textbf{(ДЗ)} +Пусть $f$ голоморфна в проколотом круге $\mathbb{D}^*=\{z:0<|z|<1\}$ + и удовлетворяет условию $|f(z)|\le 1-\log |z|$ для всех $z\in\mathbb{D}^*$. +Что можно сказать об изолированной особенности функции $f$ в нуле? +\end{p} + +\begin{p} +Пусть $f$ голоморфна в проколотом круге $\mathbb{D}^*=\{z:0<|z|<1\}$ +и при этом $\mathrm{Re}\,f(z)>0$ для всех $z\in\mathbb{D}^*$. + +(а) Может ли $f$ иметь в точке 0 существенную особенность? + +(б) Может ли $f$ иметь в точке 0 полюс? +\end{p} + +\begin{p}\textbf{*} +Существует ли голоморфная функция $f:\C\setminus\{0\}\to\C$, такая, что $f(z)+e^{f(z)}=1/z$? +\end{p} + +\begin{p} +Пусть $P$ --- многочлен степени $n$ без кратных корней; обозначим его корни через $a_1$, $\dots$, $a_n$. +Докажите, что +$$ +\frac{a_1^k}{P'(a_1)} +\dots +\frac{a_n^k}{P'(a_n)}=0 +$$ +для всякого целого $k$, такого, что $0\le k\le n-2$. +\end{p} + +\section{Аналитическое продолжение} + +\begin{p} +Рассмотрим аналитическое продолжение ростка аналитической функции +$f(z)=\sqrt{z^2+9}$, заданного в окрестности точки +$z=4$ условием $f(4)=5$, вдоль +а) полуокружности $|z|=4$, +${\rm Im} \, z\geq 0$, б) полуокружности $|z|=4$, +${\rm Im} \, z< 0$, с) отрезка $[-4, 4]$. +Чему равно $f(-4)$ во всех этих случаях? +\end{p} + +\begin{p} \textbf{(ДЗ)} +Фиксируем какую-либо ветвь функции +$f(z)=\sqrt{z^2-9}$ в окрестности точки $z=5$ и +рассмотрим два +способа ее аналитического продолжения +из точки $z=5$ в точку $x\in [-3,3]$: вдоль +дуги полуокружности в верхней полуплоскости и +вдоль +дуги полуокружности в нижней полуплоскости. +Результаты обозначим $f(x+i0)$ и $f(x-i0)$ соответственно. +Как связаны между собой $f(x+i0)$ и $f(x-i0)$? +\end{p} + +\begin{p} +Пусть $D$ --- вся комплексная плоскость с разрезами по отрезкам +$[-1, i]$ и $[-i, 1]$, а $h(z)$ --- регулярная ветвь функции $\sqrt{1-z^4}$, +положительная на интервале $(-1, 1)$. Найдите значения $h(\sqrt{5/3})$, +$h(i\sqrt{5/3})$. +\end{p} + +\begin{p} +Пусть $h(z)$ --- регулярная ветвь функции $\sqrt[3]{1-z^2}$ в области $D$, +удовлетворяющая условию $h(0)=1$. Найдите значение $h(-3)$ в случаях, когда +область $D$ а) вся комплексная плоскость с разрезами по лучам $[1, +\infty ]$ и +$[-1, -1+i\infty ]$, б) вся комплексная плоскость с разрезами по лучам +$[1, 1-i\infty ]$ и $[-1, -1-i\infty ]$. +\end{p} + +\begin{p} +Пусть $h(z)$ --- регулярная ветвь многозначной функции +$\displaystyle{\log \frac{1-z}{1+z}}$ в области $\C \setminus [-1, 1]$, такая, +что $h(0+i0)=0$. Найдите значения $h(0-i0)$, $h(i)$, $h(\infty )$. Разложите +функцию $h(z)$ в ряд Лорана по степеням $z$ в окрестности $\infty$. +\end{p} + +\begin{p} \textbf{(ДЗ)} +Пусть $D$ --- вся плоскость с разрезами по лучам $(-\infty , -1]$ и +$[1, +\infty)$, а $h(z)$ --- регулярная ветвь многозначной функции $\log (1-z^2)$ +в области $D$, удовлетворяющая условию $h(0)=0$. Найдите $h(i)$, $h(-i)$, +$h(2+i0)$, $h(2-i0)$. +\end{p} + +\begin{p} +Исследуйте поведение каждой из однозначных ветвей заданных аналитических +функций в окрестности указанных точек и определите тип особенностей +(если они есть): \\ +а) $\displaystyle{\frac{z}{1+\sqrt{z-3}}\, , \,\, z=4}$ , б) $\displaystyle{ +z+\sqrt{z^2 -1}\, , \, z=\infty} $, +в) $\displaystyle{\frac{2z+3}{1+z -2\sqrt{z}}\, , \, z=1} $, \\ +г) $\displaystyle{\cos \frac{1}{1+\sqrt{z}}}\,, z=1$, +д) \textbf{(ДЗ)} $\displaystyle{\sin \frac{1}{1+\sqrt{z/(z-1)}}}\,, z=\infty$. +\end{p} + + +\begin{p} +Найдите вычеты $\mbox{res}_{z=1}(f(z)dz)$ для каждой из однозначных ветвей функции +а) $f(z)=\displaystyle{\frac{1}{\sqrt{2-z}+1}}$, б) $f(z)= +\exp \left ( \displaystyle{\frac{1}{1+\sqrt{z}}}\right )$. +\end{p} + + + + +\section{Вычисление интегралов} +В этом разделе не следует разбирать все задачи. +Можно разобрать только только некоторые (примерно одну треть, на выбор аудитории или семинариста). + +\begin{p} +Вычислите интегралы Френеля +$\displaystyle{\int_{0}^{\infty}\! \cos x^2 dx, \;\; \int_{0}^{\infty}\! \sin x^2 dx}.$ +\end{p} + +\begin{p} +Вычислите интеграл +$\displaystyle{\int_{0}^{\infty}\frac{\cos ax -\cos bx}{x^2}\, dx}$. +\end{p} + +\begin{p} +Вычислите интегралы + +\smallskip + +а) $\bullet$ $\displaystyle{\int_{0}^{\infty}\frac{\sin x}{x}\, dx}$,\quad +б) \textbf{(ДЗ)} $\displaystyle{\int_{0}^{\infty}\frac{\sin x}{x}\, +\frac{\sin (x/3)}{x/3}\, dx}$. +\end{p} + +\begin{p} +Вычислите интеграл +$\displaystyle{\int_{0}^{\infty}\cos x\, \frac{\sin x}{x}\, dx}$. +\end{p} + + + +\begin{p} +Вычислите определенные интегралы + (параметры $a$, $b$, $k$ предполагаются вещественными): + +\medskip + +а) $\displaystyle{\int_{0}^{2\pi}\!\!\frac{d\varphi}{a+b\cos \varphi}}$ ($a>b>0$),\quad +%б) $\displaystyle{\int_{0}^{2\pi}\!\! +%e^{\cos \varphi}\cos (n\varphi -\sin \varphi )d\varphi}$, +б) $\displaystyle{\int_{0}^{\infty}\frac{x^2+1}{x^4+1}\, dx}$,\quad +в) $\displaystyle{\int_{0}^{\infty}\frac{dx}{x^n+1}}$,\\ + +\medskip + +г) $\displaystyle{\int_{0}^{\infty}\frac{\cos ax}{x^2+b^2}\, dx}$,\quad +д) $\bullet$ $\displaystyle{\int_{0}^{\infty}\frac{x\sin ax}{x^2+b^2}\, dx}$,\quad +е) \textbf{(ДЗ)} $\displaystyle{\int_{0}^{\infty}\frac{\sin ax}{x(x^2+b^2)}\, dx}$,\\ + +\medskip + +ж) $\displaystyle{\int_{-\infty}^{\infty}\frac{e^{ax}dx}{1+e^x}}\;\;$ ($00)$,\quad +(б) \textbf{*} $\displaystyle{\int_{0}^{\infty}e^{-\pi x}\, \frac{\sin ax}{\sinh \pi x}\, dx}$,\\ + +\medskip + +(в) $\displaystyle{\int_{0}^{\infty}\frac{x-\sin x}{x^3}\, dx}$,\quad +г) $\displaystyle{\int_{-\infty}^{\infty}\frac{e^{ax}-e^{bx}}{1-e^x}\, dx}\;\; +(00)}$; + +в) $\displaystyle{\int\limits_{0}^{\infty}\frac{\log x \, dx}{(x+1)(x+2)^2}\,}$;\qquad +г) $\displaystyle{\int\limits_{0}^{\infty}\frac{\log x \, dx}{x^2-1}}$. +\end{p} + +\begin{p} +Вычислите интегралы с помощью вычетов: + +а) \textbf{*} $\displaystyle{\int\limits_{0}^{\infty}\left (\frac{\log x}{x-1}\right )^2 dx}$;\qquad +б) $\displaystyle{\int\limits_{0}^{\infty}\frac{\sqrt{x}\, \log x dx}{x^2+1}}$; + +в) \textbf{*} $\displaystyle{\int\limits_{-\infty}^{\infty}\frac{\log |x^2-1|}{x^2+1}\, dx}$;\qquad +г) \textbf{*} $\displaystyle{\int\limits_{0}^{\infty} +\frac{\log (1+x^2)}{x^2}\, \frac{dx}{1+x^2}}$. +\end{p} + + + +\section{Принцип аргумента} + +\begin{p} +Сколько корней многочлена $z^7 +3z+1$ находится в круге +$|z|<1$? +\end{p} + +\begin{p} +Сколько корней многочлена $z^5 +z^2 +3z+1$ находится в круге +$|z|<1$? +\end{p} + +\begin{p} +Докажите, что уравнение $e^z = z+2$ имеет в +левой полуплоскости (${\rm Re} \, z <0$) единственный (и при том +действительный) корень. +\end{p} + +\begin{p} +Докажите, что при $\lambda >1$ уравнение $ze^{\lambda -z}=1$ имеет +в единичном круге ровно один корень (и при том действительный). +\end{p} + +\begin{p} \textbf{(ДЗ)} +Найдите количество корней многочлена +$$ +z^6 +z^5 +6z^4+5z^3 +8z^2 +4z +1 +$$ +в правой полуплоскости $\{\mathrm{Re}(z)>0\}$. +\end{p} + +\begin{p} +Найдите количество корней уравнения +$$ +z^4 +2z^3 +3z^2 +z +2=0 +$$ +в правой полуплоскости и в правом верхнем квадранте. +\end{p} + +\section{Принцип максимума} %конформные инварианты} + +\begin{p} +Пусть $f$ --- целая непостоянная функция. Положим $$M= +\mbox{max}_{|z|=r} |f(z)|.$$ Докажите, что +функция $r\mapsto M(r)$ является строго возрастающей. +Чему может быть равен ее предел при $r\to +\infty$? +\end{p} + +\begin{p} +Пусть $\Delta$ --- равносторонний треугольник со стороной длины $a$, + рассматриваемый как множество своих внутренних и граничных точек. +Найдите максимум произведения расстояний от точки из $\Delta$ до трех вершин треугольника. +\end{p} + +\begin{p} \textbf{(ДЗ)} +Рассмотрим функцию $f$, непрерывную в единичном диске $\{|z|\le 1\}$ + и голоморфную в его внутренности $\{|z|<1\}$. +Если $|f(z)|=1$ для всех $z$ на единичной окружности $\{|z|=1\}$, то либо $f=const$ на $\{|z|\le 1\}$, + либо уравнение $f(z)=0$ имеет хотя бы одно решение при $|z|<1$. +Докажите. +\end{p} + +\begin{p} +Докажите, что всякий конформный автоморфизм комплексной плоскости +имеет вид $z\mapsto az+b$, $a\neq 0$. +\end{p} + +\begin{p} +Докажите, что не существует конформного автоморфизма между множествами +$\C \setminus \{0,1,2\}$ и $\C \setminus \{0,1,1000\}$. +\end{p} + +\begin{p} $\bullet$ +Докажите, что всякий конформный автоморфизм единичного круга --- +дробно-линейный автоморфизм. +\end{p} + +\begin{p} \textbf{(ДЗ)} +Пусть $D$ --- единичный круг (без границы). Существует ли голоморфное +отображение $f:D\to D$, для которого $f(0)=\frac{1}{2}$, +$f(\frac{1}{2})=\frac{7}{8}$? +\end{p} + +\begin{p}\textbf{*} +Предположим, что функция $f$ определена и непрерывна на замкнутой полосе $S$ между двумя + параллельными прямыми, что она ограниченна на этой полосе и голоморфна в ее внутренности. +Докажите, что $\sup_{z\in S} |f(z)|=\sup_{z\in\partial S} |f(z)|$. +\end{p} + +\section{Гиперболическая метрика} + +\begin{p} +Даны две точки $\alpha$, $\beta$ внутри единичного круга $D=\{|z|<1\}$. +Найдите максимум $|f'(\alpha)|$ по всем голоморфным функциям $f:D\to D$, + таким, что $f(\alpha)=\beta$. +\end{p} + +\begin{p} +Найдите гиперболическую метрику в полосе $|\Im(z)|<\frac\pi 2$. +\end{p} + +\begin{p} +В каждом непустом кольце $A=\{r<|z|0$ + с метрикой Пуанкаре $\frac{|dz|}{\mathrm{Im}(z)}$ выражается формулой +$$ +\log|(a,b,\alpha,\beta)|, +$$ + где $\alpha$, $\beta$ --- точки, в которых геодезическая через $a$, $b$ подходит к абсолюту, + причем порядок точек $\alpha$, $a$, $b$, $\beta$ соответствует + порядку точек на замыкании геодезической (для прямой $\alpha=\infty$ или $\beta=\infty$). +\end{p} + +\begin{p} \textbf{(ДЗ)} +Докажите, что все конформные автоморфизмы верхней полуплоскости переводят + геодезические (метрики Пуанкаре) в геодезические. +\end{p} + +\begin{p} \textbf{*} +%Обоснуйте следующие утверждения. +Зафиксируем непустое кольцо $A=\{r<|z|\frac 25\right\} +$$ +и каким-нибудь кольцом вида $\{r<|z|0$; определим \emph{эллиптическую кривую} +$E_{\tau}=\C /(\Z +\tau \Z )$ и \emph{тэта-функцию} с характеристиками $a$, $b$ +рядом +$$ +\theta_{a,b}(u)=\theta_{a,b}(u,\tau )=\sum_{k\in \Z}q^{(k+a)^2}e^{2\pi i (k+a)(u+b)}, \quad +q=e^{i\pi \tau}, +$$ +а также введем \emph{тэта-функции Якоби} +$$ +\theta_1(u)=-\theta_{\frac{1}{2}, \frac{1}{2}}(u), \quad +\theta_2(u)=\theta_{\frac{1}{2},0}(u), \quad \theta_3(u)=\theta_{0, 0}(u), \quad +\theta_4(u)=\theta_{0,\frac{1}{2}}(u). +$$ +%\end{p} + +\begin{p} +Докажите, что ряд абсолютно сходится на $\C$, и тэта-функции являются целыми функциями, +причем $\theta_1$ --- нечетная, а $\theta_2$, $\theta_3$, $\theta_4$ --- четные функции. +\end{p} + +\begin{p} +Докажите, что $\theta_2(u)=\theta_1(u+\frac{1}{2})$, $\theta_3(u)= +e^{\pi i(u+\tau /4)}\theta_1(u+\frac{\tau +1}{2})$, +$\theta_4(u)=-ie^{\pi i(u+\tau /4)}\theta_1(u+\frac{\tau}{2})$. +\end{p} + +\begin{p} +Докажите следующие свойства тэта-функции при сдвигах на векторы решетки +$1$, $\tau$: +$$ +\left \{ \begin{array}{l} +\theta_1(u+1)=-\theta_1(u)\\ +\theta_1(u+\tau)=-q^{-1}e^{-2\pi iu}\theta_1(u). +\end{array} \right. +$$ +\end{p} + +\begin{p} +Найдите нули функций +$\theta_1 (u)$, $\theta_2(u)$, $\theta_3(u)$, $\theta_4(u)$ и покажите, что они простые. +\end{p} + +\begin{p} +Докажите, что тэта-функции удовлетворяют <<уравнению теплопроводности>> +$4\pi i \partial_{\tau}\theta = \partial_u^2 \theta$. +\end{p} + +\begin{p} +Покажите, что если для набора $\alpha_i\in \C$ выполняется +$\displaystyle{\sum_{i=1}^n \alpha _i=0}$, то при любых $a_i\in E_{\tau}$ +$$ +f(u)=\sum_{i=1}^n\alpha_i \, \frac{\partial_u \theta_1(u-a_i)}{\theta_1(u-a_i)} +$$ +является эллиптической (т.е. двоякопериодической) функцией на $\C$ с периодами $1$, $\tau$. +\end{p} + +\begin{p} +Покажите, что +$ +P(u)=\partial_u^2 \log \theta_1(u) +$ +является эллиптической функцией на $\C$ с периодами $1$, $\tau$ и найдите ее полюса. +\end{p} + +\begin{p} +Докажите тождество +$$ +\theta_1(a+b)\theta_1(a-b)\theta_1(u+c)\theta_1(u-c)+ +\theta_1(b+c)\theta_1(b-c)\theta_1(u+a)\theta_1(u-a) +$$ +$$ ++\, \theta_1(c+a)\theta_1(c-a)\theta_1(u+b)\theta_1(u-b)=0. +$$ +\end{p} + +\begin{p} +Докажите тождество +$$ +P(v)-P(u)=(\theta_1'(0))^2 \, \frac{\theta_1(u-v)\theta_1(v+u)}{\theta_1^2(u)\, \theta_1^2(v)}. +$$ +\end{p} + +\begin{p} +Докажите следующие представления тэта-функций в виде бесконечных произведений: +$$ +\theta_1(u)=2q^{\frac{1}{4}}\sin (\pi u) \prod_{n=1}^{\infty} +(1-q^{2n}) (1-q^{2n}e^{2\pi i u})(1-q^{2n}e^{-2\pi i u}), +$$ +$$ +\theta_2(u)=2q^{\frac{1}{4}}\cos (\pi u) \prod_{n=1}^{\infty} +(1-q^{2n}) (1+q^{2n}e^{2\pi i u})(1+q^{2n}e^{-2\pi i u}), +$$ +$$ +\theta_3(u)= \prod_{n=1}^{\infty} +(1-q^{2n}) (1+q^{2n-1}e^{2\pi i u})(1+q^{2n-1}e^{-2\pi i u}), +$$ +$$ +\theta_4(u)=\prod_{n=1}^{\infty} +(1-q^{2n}) (1-q^{2n-1}e^{2\pi i u})(1-q^{2n-1}e^{-2\pi i u}). +$$ +\end{p} + +\begin{p} +Докажите тождество $\theta '_1(0)=\pi \theta_2(0)\theta_3(0)\theta_4(0)$. +\end{p} + +\begin{p} +Докажите формулы модулярного преобразования $\tau \to -1/\tau$: +$$ +\theta_1(u/\tau , -1/\tau )=-i\sqrt{-i\tau} e^{\pi i u^2/\tau} +\theta_1(u, \tau ), +$$ +$$ +\theta_2(u/\tau , -1/\tau )=\sqrt{-i\tau} e^{\pi i u^2/\tau} +\theta_4(u, \tau ), +$$ +$$ +\theta_3(u/\tau , -1/\tau )=\sqrt{-i\tau} e^{\pi i u^2/\tau} +\theta_3(u, \tau ), +$$ +$$ +\theta_4(u/\tau , -1/\tau )=\sqrt{-i\tau} e^{\pi i u^2/\tau} +\theta_2(u, \tau ). +$$ +\end{p} + + +\section{Эллиптические функции} + +\emph{Задачи из этого раздела не войдут в итоговый контроль.} + +\medskip + +\noindent +$\wp$-функция Вейерштрасса с периодами $2\omega_1, +2\omega_2$ (${\rm Im}\, \omega_2/ \omega_1 >0$): +$$ +\wp (z)=\frac{1}{z^2}+\sum_{s\neq 0} \Bigl (\frac{1}{(z-s)^2}- +\frac{1}{s^2}\Bigr ), +\quad s=2\omega_1 n_1 +2 \omega_2 n_2, \quad n_{1,2}\in \Z . +$$ +Разложение в окрестности $z=0$: +$$ +\wp (z)=\frac{1}{z^2}+\frac{g_2}{20}\, z^2 +\frac{g_3}{28}\, z^4+O(z^6), +$$ +$$ +g_2=60\sum_{s\neq 0} s^{-4}, \quad g_3=140\sum_{s\neq 0} s^{-6}. +$$ +$\zeta$- и $\sigma$-функции Вейерштрасса: $\zeta '(z)=-\wp (z)$, $\sigma '(z)/\sigma (z)= +\zeta (z)$, +$$ +\zeta (z)=\frac{1}{z}-\int_{0}^{z}\Bigl (\wp (x)-\frac{1}{x^2}\Bigr )dx, +\quad +\log (\sigma (z)/z)=\int_{0}^{z}\Bigl (\zeta (x)-\frac{1}{x}\Bigr )dx. +$$ +$$ +\zeta (z)=\frac{1}{z}+\sum_{s\neq 0} \Bigl (\frac{1}{z-s}+\frac{1}{s}+\frac{z}{s^2}\Bigr ), +\quad +\sigma (z)=z\prod_{s\neq 0} \Bigl (1-\frac{z}{s}\Bigr )e^{\frac{z}{s}+\frac{z^2}{2s^2}}. +$$ +$\zeta (z+2\omega_{\alpha} )=\zeta (z)+2\eta_{\alpha}$, +$\sigma (z+2\omega_{\alpha})=-e^{2\eta_{\alpha}(z+\omega_{\alpha})} +\sigma (z)$, +$\eta_{\alpha}=\zeta (\omega_{\alpha})$. + + +\noindent +Выражение $\sigma$-функции через тэта-функцию: +$$ +\sigma (z)=2\omega_1 e^{\eta_1 z^2/(2\omega_1 )}\frac{\theta_1(z/(2\omega_1 ))}{\theta_1'(0)}. +$$ + +\noindent +Функции ${\rm sn}$, ${\rm cn}$ и ${\rm dn}$: +$$ +{\rm sn}\, u = \frac{\theta_3(0)}{\theta_2(0)}\, +\frac{\theta_1 \Bigl (\frac{u}{\pi \theta_3^2(0)} +\Bigr )}{\theta_4 \Bigl (\frac{u}{\pi \theta_3^2(0)}\Bigr )}, +\; +{\rm cn}\, u = \frac{\theta_4(0)}{\theta_2(0)}\, +\frac{\theta_2 \Bigl (\frac{u}{\pi \theta_3^2(0)} +\Bigr )}{\theta_4 \Bigl (\frac{u}{\pi \theta_3^2(0)}\Bigr )}, +\; +{\rm dn}\, u = \frac{\theta_4(0)}{\theta_3(0)}\, +\frac{\theta_3 \Bigl (\frac{u}{\pi \theta_3^2(0)} +\Bigr )}{\theta_4 \Bigl (\frac{u}{\pi \theta_3^2(0)}\Bigr )}. +$$ + + +\begin{p} +Докажите, что функция $\wp (z)$ удовлетворяет дифференциальному уравнению +$$ +(\wp '(z))^2 =4\wp ^3(z)-g_2\wp (z)-g_3. +$$ +\end{p} + +\begin{p} +Докажите тождество +$ +\wp '''(z)=12\wp (z)\wp '(z). +$ +\end{p} + +\begin{p} +Докажите соотношение $2\eta_1 \omega_2 -2\eta_2 \omega_1 =\pi i$. +\end{p} + +%\nz +%Докажите тождество +%$$ +%\wp (z+y)-\wp (z-y)=-\frac{\wp '(z)\wp '(y)}{(\wp (z)-\wp (y))^2}. +%$$ + +\begin{p} +Докажите, что при $x+y+z=0$ справедливо тождество +$$ +\left | \begin{array}{lll} +1 & \wp (x) & \wp '(x) +\\ +1 & \wp (y) & \wp '(y) +\\ +1 & \wp (z) & \wp '(z) +\end{array} +\right |=0. +$$ +\end{p} + +\begin{p} +Выразите +$\displaystyle{ +\left | \begin{array}{lll} +1 & \wp (x) & \wp '(x) +\\ +1 & \wp (y) & \wp '(y) +\\ +1 & \wp (z) & \wp '(z) +\end{array} +\right |} +$ +при произвольных $x,y,z \in \C$ через $\sigma$-функции. +\end{p} + +\begin{p} +Докажите, что при $x+y+z=0$ справедливо тождество +$$ +\Bigl (\zeta (x)+\zeta (y) +\zeta (z)\Bigr )^2= +\wp (x)+\wp (y)+\wp (z). +$$ +\end{p} + +\begin{p} +Докажите тождество +$$ +\zeta (x-a_1)-\zeta (x-a_2)+\zeta (a_1-a_2)=-\frac{1}{2}\, +\frac{\wp '(x-a_1)+\wp '(x-a_2)}{\wp (x-a_1)-\wp (x-a_2)}. +$$ +\end{p} + +\begin{p} +Докажите тождества +$$ +{\rm sn}^2\, u +{\rm cn}^2\, u =1, \quad +k^2{\rm sn}^2\, u +{\rm dn}^2\, u =1, +$$ +где $k^2=\theta_2^4(0)/\theta_3^4(0)$. +\end{p} + +\section{Теоретические вопросы к промежуточной контрольной + и итоговому экзамену} + +\begin{p} +Как выглядит (в базисе $1$, $i$) матрица линейного над $\mathbb{R}$ отображения $f:\mathbb{C}\to\mathbb{C}$, + такого, что $z\mapsto f(\overline{z})$ голоморфно? +\end{p} + +\begin{p} +В чем состоит геометрический смысл числа $\arg f'(a)$ для функции $f$, + определенной и голоморфной в окрестности точки $a\in\C$, при условии, что $f'(a)\ne 0$? +\end{p} + +\begin{p} +Пусть функция $f$ определена и непрерывна в замыкании области $D$ с кусочно гладкой границей, + и пусть $f$ голоморфна в области $D$. +Рассмотрим функцию $g:\mathbb{C}\setminus\partial D\to\mathbb{C}$, заданную уравнением +$$ +g(z)=\frac{1}{2\pi i}\int_{\partial D}\frac{f(\zeta)d\zeta}{\zeta-z}. +$$ +Что можно сказать про значения функции $g$ внутри и вне области $D$? +В каких случаях эту функцию можно продолжить по непрерывности на всю плоскость? +\end{p} + +\begin{p} +Выразите $\mathrm{Res}_0 f(5z)dz$ через $\mathrm{Res}_0 f(z)dz$. +Здесь $f$ --- функция, определенная и голоморфная в проколотой окрестности точки $z=0$. +\end{p} + +\begin{p} +Как может выглядеть область сходимости ряда Лорана (т.е. максимальное \emph{открытое} + множество, во всех точках которого ряд Лорана сходится)? +Укажите все возможные области и строго обоснуйте ответ. +\end{p} + +\begin{p} +Функция $f$ определена и непрерывна в замкнутом кольце $\overline{A}=\{z:1\le |z|\le 2\}$, + а также голоморфна в открытом кольце $A=\{z:1<|z|<2\}$. +Известны значения $f(z)$ при $|z|=1$ и $|z|=2$. +Как найти коэффициенты разложения функции $f$ в ряд Лорана в кольце $A$ + (выведите соответствующие интегральные формулы)? +\end{p} + +\begin{p} +Опишите множество всех комплексных чисел, которые можно представить + как $\mathrm{Res}_0\frac{df}{f}$ для + +а) некоторой аналитической функции $f$, имеющей в точке $z=0$ изолированную особенность; + +б) некоторой аналитической функции $f$, имеющей в точке $z=0$ полюс? +\end{p} + +\begin{p} +Пусть функция $f(z)$ определена и голоморфна в проколотой окрестности точки $z=0$. +Верно ли следующее утверждение (строго обоснуйте ответ)? +Точка $z=0$ является существенной особенностью функции $f(z)$ тогда и только тогда, + когда она является существенной особенностью функции $f(z)(1+e^{f(z)})$. +\end{p} + +\begin{p} +Докажите, что следующие ростки голоморфных функций не +допускают аналитического продолжения вдоль отрезка $[-1;1]$ с направлением от $1$ к $-1$: + +(а) росток функции $1/z$ в точке $1$; + +(б) росток функции $\sqrt{z}$ в точке $1$, такой, что $\sqrt 1 = 1$. +\end{p} + +\begin{p} +Приведите пример многозначной аналитической функции $f$ со следующими свойствами: + $f$ определена и голоморфна всюду на $\mathbb{C}\setminus \{0,1\}$, + причем множество значений функции $f$ в каждой точке $z\in \mathbb{C}\setminus \{0,1\}$ + всюду плотно в $\mathbb{C}$. +\end{p} + +\begin{p} +Пользуясь свойствами голоморфных функций, обоснуйте следующее утверждение про + вещественные гармонические функции. +Если вещественнозначная функция $u$ непрерывна на компактном множестве $K$ + и гармонична во внутренности множества $K$, то ее минимальное значение + достигается в некоторой граничной точке множества $K$. +\end{p} + +\begin{p} +Сформулируйте и докажите теорему о среднем для гармонических функций. +\end{p} + +\begin{p} +Докажите, что существует не более одной, с точностью до умножения на положительную константу, + конформной метрики на диске $\{|z|<1\}$, + инвариантной относительно всех конформных автоморфизмов диска. +\end{p} + +\begin{p} +Найдите гиперболическую метрику на проколотом диске $\{0<|z|<1\}$. +Покажите, что площадь проколотой окрестности нуля относительно этой метрики конечна. +\end{p} + +\begin{p} +Последовательность голоморфных функций $f_n:U\to\C$ на области $U\subset \C$ + сходится равномерно на компактах к функции $f:U\to\C$. +Докажите, что производные $f'_n$ сходятся (равномерно на компактах) к производной $f'$. +\end{p} + +\begin{p} +Последовательность многочленов степени 2026 сходится равномерно на компактах на всем $\C$. +Может ли предел этой последовательности быть многочленом степени 2025? +А может ли он быть многочленом степени 2027? +А может ли он вообще не быть многочленом? +Строго обоснуйте ответы. +\end{p} + +\begin{p} +Существует ли конформный изоморфизм между $\mathbb{H}\setminus \{i,2i\}$ и $\mathbb{H}\setminus\{i,3i\}$, + где $\mathbb{H}=\{z : \Im(z)>0\}$ --- верхняя полуплоскость? +Строго обоснуйте ответы. +\end{p} + +\begin{p} +Пользуясь теоремой Римана об отображении и принципом соответствия границ, + постройте голоморфное накрытие $p:\{|z|<1\}\to \C\setminus\{-1,0,1\}$. +Выведите, что на $\C\setminus\{-1,0,1\}$ имеется гиперболическая метрика. +\end{p} + +%\begin{p} +%Выпишите (в виде интеграла от явно заданной функции) конформный изоморфизм +% между внутренностью единичного диска и внутренностью правильного шестиугольника. +%\end{p} + + + +\end{document} diff --git a/ForStudents/bd.png b/ForStudents/bd.png new file mode 100644 index 0000000000000000000000000000000000000000..1b1d11af919ef9b65f7bbf40375b16c690988dab GIT binary patch literal 56682 zcmeEv_dnMC`}Z3~TSF72oK50tNtrDpMP!vwsY~{#I5To~wUyb>RiZ*xkrgrvMIVrt0)5i>5FPGhJP6 zrw+qR3}P7G^hx~qjos>~wfKKLCv{Y}GcoxhE%;%A`L?~=7$(kV>d29a_<6Fu+JTb{ zBYK(skH_a*_i=`)8CT!AO_y!bUhB-({WkQoTW<2?3-eqATGRwH3~ov74V$!Tnb^@? z$}e9ePn?D7;E8t%;MSb(=?@6B-O(>Z+jflKd?BdQF;T77mM{*#%&xv=r<2s_E5 zSjAsAo$X*#8J}aqi!ThMKYRAq#S0f!ty!~sY;0`n>DHU;|NeMFnvW4vKb*6HvvafK zSGDrkWV`y99hYb1+@&q|&Aq-bBqCy!y3VOniD!peR=Q4R7wVwY^2Q2+f*x zM0LufNuS3?hi}O_+sg4WV%l+y3zZ)^986lOt`Q(r`u)>An`tzlz**udHP27o4#kY~ z)p!_|It@PO@!orh)|Kmp^BQCBaEGxCi*ae%!lO-wHFJxx^7A~+nXAjB!hT6-jXc5= 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zyJi~^0?{MmMt*zcz<~oS_=r+-Rw4wVM}BriP0i(!L$Kf@_{G_X5Qx>nVavhywuc}H z7JP*E2M$7nBqCx<0v$PP78SH6>vinkkDctCga|3bdU1d|Mx+j$%E5xK1y3a}Awn84 zE>5M( zjF1uY@Iw(5@CgylpAuj&EbMspq|FwBTzs7k{eAt=;uyC8Zzmr|cP~W9&0?PnXOO6{ z;LpXkLWpoKk{g`z`vF{0lO<;WWaF#7bN1PTZid>u5~)(IO{UBu|qu=*PP{umhq3qGOPb049{5bGLtlw$LP7BUDH zd_vV5F+u`j_~tzkP2Uwp2El?)NWCLLNI)!%@xj3&+!kaIEck?ZUMWHXVqF|v_<39Z zH3Y$ePf!n%CM0Yqi@(E*gN}(S3IweL*kFO>Zo^frEVD*{kNVcK2x*%fR^}-d3WoC# z4G&daAAZsu6-aXU`<0k0_rX0M1Hpf8Eug+W9OLNZg)+eed!f)s_m`ZIlY}2Z#ewMB zx{j-?P6`OEg1$), + +(в) $\{ z \colon \mbox{Re}(1/z)=\frac{1}{2} \}$. + +\nz +Пусть $a,b,c,d$ -- четыре различные точки на единичной окружности. +Найдите точку перечесения прямых $ab$ и $cd$. + +\nz +Докажите равенство +$$ +|\sqrt{z^2-1}+z|+|\sqrt{z^2-1}-z|=|z-1|+|z+1|. +$$ + +\nz +Пусть $P(z)$ -- многочлен степени $n$, имеющий $n$ вещественных корней +(с учетом кратности), $a$ -- произвольное вещественное число. Докажите, +что многочлен $Q(z)=P(z+ia)+P(z-ia)$ тоже имеет $n$ вещественных корней. + +\nz +Найдите, во что переводят координатную сетку следующие отображения из +\C в~\C: +(а) $z\mapsto z^2$; +(б) $z\mapsto e^z$; +(в) $z\mapsto 1/z$. + +\nz +Отображение $z\mapsto z+(1/z)$ определено на множестве всех ненулевых +комплексных чисел. + +(а) Во что это отображение переводит множество $\{z\colon |z|>1\}$? + +(б) Является ли это отображение взаимно-однозначным на данном множестве? + + +\nz +Пусть $A>0, C$ действительные, а $B$ -- комплексная постоянные и пусть $AC<|B|^2$. +Докажите, что уравнение +$$ +A|z|^2 +\bar B z +B\bar z +C=0 +$$ +является уравнением окружности и найдите центр этой окружности и ее радиус. + +\nz +Напишите уравнение окружности, проходящей через три различные +точки $z_1, z_2, z_3$, не лежащие на одной прямой. + +\nz +Пусть отображение задано формулой $z\mapsto Az+B\bar z$. Во что оно +переводит окружность с центром в +нуле? + + + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok10.tex b/ForStudents/листки Забродина/listok10.tex new file mode 100644 index 0000000..07a0592 --- /dev/null +++ b/ForStudents/листки Забродина/listok10.tex @@ -0,0 +1,109 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} +\newcommand*{\HH}{\ensuremath{\mathbb H}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc + Теория функций комплексного переменного 2025\\ листок 10 +\end{center} + + + + + + + + +\nz +Сколько корней многочлена $z^7 +3z+1$ находится в круге +$|z|<1$? + +\nz +Сколько корней многочлена $z^5 +z^2 +3z+1$ находится в круге +$|z|<1$? + +\nz +Докажите, что уравнение $e^z = z+2$ имеет в +левой полуплоскости (${\rm Re} \, z <0$) единственный (и при том +действительный) корень. + +\nz +Докажите, что при $\lambda >1$ уравнение $ze^{\lambda -z}=1$ имеет +в единичном круге ровно один корень (и при том действительный). + +\nz +Найдите количество корней многочлена +$$ +z^6 +z^5 +6z^4+5z^3 +8z^2 +4z +1 +$$ +в правой полуплоскости. + +\nz +Найдите количество корней уравнения +$$ +z^4 +2z^3 +3z^2 +z +2=0 +$$ +в правой полуплоскости и в правом верхнем квадранте. + +\nz +Пусть $f$ -- целая не постоянная функция. Положим $$M= +\mbox{max}_{|z|=r} |f(z)|.$$ Докажите, что +функция $r\mapsto M(r)$ является строго возрастающей. +Чему может быть равен ее предел при $r\to +\infty$? + +\nz +Докажите, что всякий конформный автоморфизм комплексной плоскости +имеет вид $z\mapsto az+b$, $a\neq 0$. + +\nz +Докажите, что не существует конформного автоморфизма между множествами +$\C \setminus \{0,1,2\}$ и $\C \setminus \{0,1,1000\}$. + +\nz +Докажите, что всякий конформный автоморфизм единичного круга -- +дробно-линейный автоморфизм. + +\nz +Пусть $U$ -- единичный круг (без границы). Существует ли голоморфное +отображение $f:U\to U$, для которого $f(0)=\frac{1}{2}$, +$f(\frac{1}{2})=\frac{7}{8}$? + + + + + + + + + + + + + + + + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok11.tex b/ForStudents/листки Забродина/listok11.tex new file mode 100644 index 0000000..46e6d7a --- /dev/null +++ b/ForStudents/листки Забродина/listok11.tex @@ -0,0 +1,111 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc + Теория функций комплексного переменного -- 2025\\ листок 11 +\end{center} + + +\nz +Рассмотрим аналитическое продолжение функции +$f(z)=\sqrt{z^2+9}$, заданной в окрестности точки +$z=4$ условием $f(4)=5$, вдоль +а) полуокружности $|z|=4$, +${\rm Im} \, z\geq 0$, б) полуокружности $|z|=4$, +${\rm Im} \, z< 0$, с) отрезка $[-4, 4]$. +Чему равно $f(-4)$ во всех этих случаях? + + +\nz +Фиксируем какую-либо ветвь функции +$f(z)=\sqrt{z^2-9}$ в окрестности точки $z=5$ и +рассмотрим два +способа ее аналитического продолжения +из точки $z=5$ в точку $x\in [-3,3]$: вдоль +дуги полуокружности в верхней полуплоскости и +вдоль +дуги полуокружности в нижней полуплоскости. +Результаты обозначим $f(x+i0)$ и $f(x-i0)$ соответственно. +Как связаны между собой $f(x+i0)$ и $f(x-i0)$? + +\nz +Пусть $D$ -- вся комплексная плоскость с разрезами по отрезкам +$[-1, i]$ и $[-i, 1]$, а $h(z)$ -- регулярная ветвь функции $\sqrt{1-z^4}$, +положительная на интервале $(-1, 1)$. Найдите значения $h(\sqrt{5/3})$, +$h(i\sqrt{5/3})$. + +\nz +Пусть $h(z)$ -- регулярная ветвь функции $\sqrt[3]{1-z^2}$ в области $D$, +удовлетворяющая условию $h(0)=1$. Найдите значение $h(-3)$ в случаях, когда +область $D$ а) вся комплексная плоскость с разрезами по лучам $[1, +\infty ]$ и +$[-1, -1+i\infty ]$, б) вся комплексная плоскость с разрезами по лучам +$[1, 1-i\infty ]$ и $[-1, -1-i\infty ]$. + + +\nz +Пусть $h(z)$ -- регулярная ветвь многозначной функции +$\displaystyle{\log \frac{1-z}{1+z}}$ в области $\C \setminus [-1, 1]$, такая, +что $h(0+i0)=0$. Найдите значения $h(0-i0)$, $h(i)$, $h(\infty )$. Разложите +функцию $h(z)$ в ряд Лорана по степеням $z$ в окрестности $\infty$. + + +\nz +Пусть $D$ -- вся комплексная плоскость с разрезами по лучам $[-\infty , -1]$ и +$[1, +\infty ]$, а $h(z)$ -- регулярная ветвь многозначной функции $\log (1-z^2)$ +в области $D$, удовлетворяющая условию $h(0)=0$. Найдите $h(i)$, $h(-i)$, +$h(2+i0)$, $h(2-i0)$. + + + + +\nz +Исследуйте поведение каждой из однозначных ветвей заданных аналитических +функций в окрестности указанных точек и определите тип особенностей +(если они есть): \\ +а) $\displaystyle{\frac{z}{1+\sqrt{z-3}}\, , \,\, z=4}$ , б) $\displaystyle{ +z+\sqrt{z^2 -1}\, , \, z=\infty} $, +в) $\displaystyle{\frac{2z+3}{1+z -2\sqrt{z}}\, , \, z=1} $, \\ +г) $\displaystyle{\cos \frac{1}{1+\sqrt{z}}}\,, z=1$, +д) $\displaystyle{\sin \frac{1}{1+\sqrt{z/(z-1)}}}\,, z=\infty$. + + + +\nz +Найдите вычеты $\mbox{res}_{z=1}(f(z)dz)$ для каждой из однозначных ветвей функции +а) $f(z)=\displaystyle{\frac{1}{\sqrt{2-z}+1}}$, б) $f(z)= +\exp \left ( \displaystyle{\frac{1}{1+\sqrt{z}}}\right )$. + + +\end{document} + + + + + + + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok12.tex b/ForStudents/листки Забродина/listok12.tex new file mode 100644 index 0000000..35c917e --- /dev/null +++ b/ForStudents/листки Забродина/listok12.tex @@ -0,0 +1,106 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc + -- 2025\\ 12 +\end{center} + + + + +\nz + $g(z)$ $\sqrt{z^2-4}$ $D$, + +$|z|=2$, ${\rm Im} z\geq 0$, $g(z)$ + $\infty$ $z$. +$$ +I=\oint _{|z|=1}\frac{dz}{g(z)-3z}. +$$ + +\nz + +$$ +\oint_{|z|=\frac{1}{2}}\frac{dz}{(2+g(z))\sin z}, +$$ + $g(z)$ -- $\sqrt{z-1}$ +$|z|<\frac{1}{2}$, , $g(0)=i$. + +\nz + $f(z)$ -- $\sqrt[3]{2z-8}$ + $[4, 4-i\infty ]$ , $f(8)=-1-i\sqrt{3}$. +$$ +I=\oint _{|z-2|=3/2}\frac{dz}{f(z)-z+2}. +$$ + + +\nz + : + +) $\displaystyle{\int\limits_{-1}^{1} +\frac{dx}{\sqrt{1-x^2}}}\,\,$; +) $\displaystyle{\int\limits_{0}^{1} +\sqrt{\frac{1-x}{x}}\, dx} \,\,$. + +\nz + : +$\displaystyle{\int\limits_{0}^{\infty} +\frac{x^{\alpha -1}}{x+1}\,dx} \,\,$ ($0<\alpha <1$). + +\nz + : + + +) $\displaystyle{\int\limits_{0}^{1}\left (\frac{x}{1-x}\right )^{\alpha} +\frac{dx}{x+1}\,\,}$ ($-1<\alpha <1$); +) $\displaystyle{\int\limits_{1}^{2}\sqrt[5]{\frac{(2-x)^3}{(x-1)^3}}\, dx}$;\\ +) $\displaystyle{\int\limits_{-2}^{2}\frac{dx}{\sqrt[4]{(2+x)^2 (4-x^2)}}}\,$. + +\nz + : + +) $\displaystyle{\int\limits_{0}^{\infty}\frac{dx}{x^3+1}}\,$; +) $\displaystyle{\int\limits_{0}^{\infty}\frac{\log x \, dx}{x^2+a^2} \,\,\, (a>0)}$; +) $\displaystyle{\int\limits_{0}^{\infty}\frac{\log x \, dx}{(x+1)(x+2)^2}\,}$;\\ +) $\displaystyle{\int\limits_{0}^{\infty}\frac{\log x \, dx}{x^2-1}}\,$. + +\nz + : + +) $\displaystyle{\int\limits_{0}^{\infty}\left (\frac{\log x}{x-1}\right )^2 dx}$; +) $\displaystyle{\int\limits_{0}^{\infty}\frac{\sqrt{x}\, \log x dx}{x^2+1}}$; +) $\displaystyle{\int\limits_{-\infty}^{\infty}\frac{\log |x^2-1|}{x^2+1}\, dx}$;\\ +) $\displaystyle{\int\limits_{0}^{\infty} +\frac{\log (1+x^2)}{x^2}\, \frac{dx}{1+x^2}}$. + + + + + + + + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok13.tex b/ForStudents/листки Забродина/listok13.tex new file mode 100644 index 0000000..b297de4 --- /dev/null +++ b/ForStudents/листки Забродина/listok13.tex @@ -0,0 +1,148 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} +\newcommand*{\Z}{\ensuremath{\mathbb Z}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc + Теория функций комплексного переменного 2025\\ Листок 13 +\end{center} + +\noindent +Пусть задан параметр $\tau \in \C$, ${\rm Im} \, \tau >0$; определим эллиптическую кривую +$E_{\tau}=\C /(\Z +\tau \Z )$ и тэта-функцию с характеристиками $a,b$ +рядом +$$ +\theta_{a,b}(u)=\theta_{a,b}(u,\tau )=\sum_{k\in \Z}q^{(k+a)^2}e^{2\pi i (k+a)(u+b)}, \quad +q=e^{i\pi \tau}, +$$ +а также введем тэта-функции Якоби $$\theta_1(u)=-\theta_{\frac{1}{2}, \frac{1}{2}}(u), \quad +\theta_2(u)=\theta_{\frac{1}{2},0}(u), \quad \theta_3(u)=\theta_{0, 0}(u), \quad +\theta_4(u)=\theta_{0,\frac{1}{2}}(u).$$ + +\nz +Докажите, что ряд абсолютно сходится на $\C$, и тэта-функции являются целыми функциями, +причем $\theta_1$ -- нечетная, а $\theta_2$, $\theta_3$, $\theta_4$ -- четные функции. + +\nz +Докажите, что $\theta_2(u)=\theta_1(u+\frac{1}{2})$, $\theta_3(u)= +e^{\pi i(u+\tau /4)}\theta_1(u+\frac{\tau +1}{2})$, +$\theta_4(u)=-ie^{\pi i(u+\tau /4)}\theta_1(u+\frac{\tau}{2})$. + +\nz +Докажите следующие свойства тэта-функции при сдвигах на векторы решетки +$1$, $\tau$: +$$ +\left \{ \begin{array}{l} +\theta_1(u+1)=-\theta_1(u)\\ +\theta_1(u+\tau)=-q^{-1}e^{-2\pi iu}\theta_1(u). +\end{array} \right. +$$ + +\nz +Найдите нули функций +$\theta_1 (u)$, $\theta_2(u)$, $\theta_3(u)$, $\theta_4(u)$ и покажите, что они простые. + +\nz +Докажите, что тэта-функции удовлетворяют ``уравнению теплопроводности'' +$4\pi i \partial_{\tau}\theta = \partial_u^2 \theta$. + + +\nz +Покажите, что если для набора $\alpha_i\in \C$ выполняется +$\displaystyle{\sum_{i=1}^n \alpha _i=0}$, то при любых $a_i\in E_{\tau}$ +$$ +f(u)=\sum_{i=1}^n\alpha_i \, \frac{\partial_u \theta_1(u-a_i)}{\theta_1(u-a_i)} +$$ +является эллиптической (т.е. двоякопериодической) функцией на $\C$ с периодами $1$, $\tau$. + +\nz +Покажите, что +$ +P(u)=\partial_u^2 \log \theta_1(u) +$ +является эллиптической функцией на $\C$ с периодами $1$, $\tau$ и найдите ее полюса. + +\nz +Докажите тождество +$$ +\theta_1(a+b)\theta_1(a-b)\theta_1(u+c)\theta_1(u-c)+ +\theta_1(b+c)\theta_1(b-c)\theta_1(u+a)\theta_1(u-a) +$$ +$$ ++\, \theta_1(c+a)\theta_1(c-a)\theta_1(u+b)\theta_1(u-b)=0. +$$ + + +\nz +Докажите тождество +$$ +P(v)-P(u)=(\theta_1'(0))^2 \, \frac{\theta_1(u-v)\theta_1(v+u)}{\theta_1^2(u)\, \theta_1^2(v)}. +$$ + +\nz +Докажите следующие представления тэта-функций в виде бесконечных произведений: +$$ +\theta_1(u)=2q^{\frac{1}{4}}\sin (\pi u) \prod_{n=1}^{\infty} +(1-q^{2n}) (1-q^{2n}e^{2\pi i u})(1-q^{2n}e^{-2\pi i u}), +$$ +$$ +\theta_2(u)=2q^{\frac{1}{4}}\cos (\pi u) \prod_{n=1}^{\infty} +(1-q^{2n}) (1+q^{2n}e^{2\pi i u})(1+q^{2n}e^{-2\pi i u}), +$$ +$$ +\theta_3(u)= \prod_{n=1}^{\infty} +(1-q^{2n}) (1+q^{2n-1}e^{2\pi i u})(1+q^{2n-1}e^{-2\pi i u}), +$$ +$$ +\theta_4(u)=\prod_{n=1}^{\infty} +(1-q^{2n}) (1-q^{2n-1}e^{2\pi i u})(1-q^{2n-1}e^{-2\pi i u}). +$$ + +\nz +Докажите тождество $\theta '_1(0)=\pi \theta_2(0)\theta_3(0)\theta_4(0)$. + +\nz +Докажите формулы модулярного преобразования $\tau \to -1/\tau$: +$$ +\theta_1(u/\tau , -1/\tau )=-i\sqrt{-i\tau} e^{\pi i u^2/\tau} +\theta_1(u, \tau ), +$$ +$$ +\theta_2(u/\tau , -1/\tau )=\sqrt{-i\tau} e^{\pi i u^2/\tau} +\theta_4(u, \tau ), +$$ +$$ +\theta_3(u/\tau , -1/\tau )=\sqrt{-i\tau} e^{\pi i u^2/\tau} +\theta_3(u, \tau ), +$$ +$$ +\theta_4(u/\tau , -1/\tau )=\sqrt{-i\tau} e^{\pi i u^2/\tau} +\theta_2(u, \tau ). +$$ + + + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok14.tex b/ForStudents/листки Забродина/listok14.tex new file mode 100644 index 0000000..4a7e29e --- /dev/null +++ b/ForStudents/листки Забродина/listok14.tex @@ -0,0 +1,162 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} +\newcommand*{\Z}{\ensuremath{\mathbb Z}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc + 2025\\ 14 +\end{center} + + +\noindent +$\wp$- $2\omega_1, +2\omega_2$ (${\rm Im}\, \omega_2/ \omega_1 >0$): +$$ +\wp (z)=\frac{1}{z^2}+\sum_{s\neq 0} \Bigl (\frac{1}{(z-s)^2}- +\frac{1}{s^2}\Bigr ), +\quad s=2\omega_1 n_1 +2 \omega_2 n_2, \quad n_{1,2}\in \Z . +$$ + $z=0$: +$$ +\wp (z)=\frac{1}{z^2}+\frac{g_2}{20}\, z^2 +\frac{g_3}{28}\, z^4+O(z^6), +$$ +$$ +g_2=60\sum_{s\neq 0} s^{-4}, \quad g_3=140\sum_{s\neq 0} s^{-6}. +$$ +$\zeta$- $\sigma$- : $\zeta '(z)=-\wp (z)$, $\sigma '(z)/\sigma (z)= +\zeta (z)$, +$$ +\zeta (z)=\frac{1}{z}-\int_{0}^{z}\Bigl (\wp (x)-\frac{1}{x^2}\Bigr )dx, +\quad +\log (\sigma (z)/z)=\int_{0}^{z}\Bigl (\zeta (x)-\frac{1}{x}\Bigr )dx. +$$ +$$ +\zeta (z)=\frac{1}{z}+\sum_{s\neq 0} \Bigl (\frac{1}{z-s}+\frac{1}{s}+\frac{z}{s^2}\Bigr ), +\quad +\sigma (z)=z\prod_{s\neq 0} \Bigl (1-\frac{z}{s}\Bigr )e^{\frac{z}{s}+\frac{z^2}{2s^2}}. +$$ +$\zeta (z+2\omega_{\alpha} )=\zeta (z)+2\eta_{\alpha}$, +$\sigma (z+2\omega_{\alpha})=-e^{2\eta_{\alpha}(z+\omega_{\alpha})} +\sigma (z)$, +$\eta_{\alpha}=\zeta (\omega_{\alpha})$. + + +\noindent + $\sigma$- -: +$$ +\sigma (z)=2\omega_1 e^{\eta_1 z^2/(2\omega_1 )}\frac{\theta_1(z/(2\omega_1 ))}{\theta_1'(0)}. +$$ + +\noindent + ${\rm sn}$, ${\rm cn}$ ${\rm dn}$: +$$ +{\rm sn}\, u = \frac{\theta_3(0)}{\theta_2(0)}\, +\frac{\theta_1 \Bigl (\frac{u}{\pi \theta_3^2(0)} +\Bigr )}{\theta_4 \Bigl (\frac{u}{\pi \theta_3^2(0)}\Bigr )}, +\; +{\rm cn}\, u = \frac{\theta_4(0)}{\theta_2(0)}\, +\frac{\theta_2 \Bigl (\frac{u}{\pi \theta_3^2(0)} +\Bigr )}{\theta_4 \Bigl (\frac{u}{\pi \theta_3^2(0)}\Bigr )}, +\; +{\rm dn}\, u = \frac{\theta_4(0)}{\theta_3(0)}\, +\frac{\theta_3 \Bigl (\frac{u}{\pi \theta_3^2(0)} +\Bigr )}{\theta_4 \Bigl (\frac{u}{\pi \theta_3^2(0)}\Bigr )}. +$$ + + +\nz +, $\wp (z)$ +$$ +(\wp '(z))^2 =4\wp ^3(z)-g_2\wp (z)-g_3. +$$ + +\nz + +$ +\wp '''(z)=12\wp (z)\wp '(z). +$ + +\nz + $2\eta_1 \omega_2 -2\eta_2 \omega_1 =\pi i$. + + +%\nz +% +%$$ +%\wp (z+y)-\wp (z-y)=-\frac{\wp '(z)\wp '(y)}{(\wp (z)-\wp (y))^2}. +%$$ + +\nz +, $x+y+z=0$ +$$ +\left | \begin{array}{lll} +1 & \wp (x) & \wp '(x) +\\ +1 & \wp (y) & \wp '(y) +\\ +1 & \wp (z) & \wp '(z) +\end{array} +\right |=0. +$$ + +\nz + +$\displaystyle{ +\left | \begin{array}{lll} +1 & \wp (x) & \wp '(x) +\\ +1 & \wp (y) & \wp '(y) +\\ +1 & \wp (z) & \wp '(z) +\end{array} +\right |} +$ + $x,y,z \in \C$ $\sigma$-. + +\nz +, $x+y+z=0$ +$$ +\Bigl (\zeta (x)+\zeta (y) +\zeta (z)\Bigr )^2= +\wp (x)+\wp (y)+\wp (z). +$$ + +\nz + +$$ +\zeta (x-a_1)-\zeta (x-a_2)+\zeta (a_1-a_2)=-\frac{1}{2}\, +\frac{\wp '(x-a_1)+\wp '(x-a_2)}{\wp (x-a_1)-\wp (x-a_2)}. +$$ + +\nz + +$$ +{\rm sn}^2\, u +{\rm cn}^2\, u =1, \quad +k^2{\rm sn}^2\, u +{\rm dn}^2\, u =1, +$$ + $k^2=\theta_2^4(0)/\theta_3^4(0)$. + + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok2.tex b/ForStudents/листки Забродина/listok2.tex new file mode 100644 index 0000000..73b6a48 --- /dev/null +++ b/ForStudents/листки Забродина/listok2.tex @@ -0,0 +1,171 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc + Теория функций комплексного переменного -- 2025\\ Листок 2 +\end{center} + +{\small +Дробно-линейной функцией (д.-л. функцией) называется функция вида +$$f(z)=\frac{az+b}{cz+d}\,, \qquad a,b,c,d \in \C, \qquad +ad-bc \neq 0. +$$ +} + +\nz +Докажите, что д.-л. функция осуществляет взаимно-однозначное +отображение $\bar{\C}\to \bar{\C}$. + +\nz +Докажите, что д.-л. +отображения образуют группу. +Какой группе она изоморфна? + +\nz +Докажите, что д.-л. функция переводит окружности +и прямые в окружности (или прямые). + + + +\nz +Напишите д.-л. функцию, переводящую три произвольные +(различные) точки $z_1, z_2, z_3$ в три другие (различные) +точки $w_1, w_2, w_3$. Единственна ли она? + +\nz +Во что преобразуется квадрант $x>0, y>0$ при отображении $\displaystyle{ +w=\frac{z-i}{z+i}}\, $? + +\nz +Найдите все дробно-линейные функции $f$ такие, что $f(f(z))=z$. + + + +\nz +Докажите, что бидифференциал $\displaystyle{\frac{dz_1 dz_2}{(z_1-z_2)^2}}$ +инвариантен относительно д.-л. преобразований. + + +\vspace{5mm} + +{\small +Двойным отношением четырех различных точек $z_1, z_2, z_3, z_4$ называется +$$ +(z_1, z_2, z_3, z_4)=\frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}. +$$ +Производной Шварца (шварцианом) голоморфной +функции $w(z)$ называется +выражение +$$\displaystyle{S(w;z):=\frac{w'''(z)}{w'(z)}-\frac{3}{2} +\left (\frac{w''(z)}{w'(z)}\right )^2}.$$ +} + +\nz +Докажите следующее тождество, связывающее двойное отношение со шварцианом: +$$ +\frac{(w(z+ta), w(z+tb), w(z+tc), w(z+td))}{(a,b,c,d)}= +1+\frac{t^2}{6}\, (a-b)(c-d)S(w;z) +O(t^3) +$$ +($t\to 0$). + +\nz +Докажите, что д.-л. функция сохраняет двойное отношение: +$$ +(f(z_1), f(z_2), f(z_3), f(z_4))=(z_1, z_2, z_3, z_4). +$$ + + + + +\nz +\ +Докажите, что +$S(w;z)=0$ тогда и только тогда, когда $w(z)$ -- д.-л. функция. + +\nz +Докажите, что четыре различные точки лежат на одной окружности +(или на одной прямой), если и только если их двойное отношение +вещественно. + +\nz +Докажите, что +$$ +S(w;z)=6 \lim_{\zeta \to z}\, \frac{\partial }{\partial z}\frac{\partial }{\partial \zeta}\, +\log \frac{w(z)-w(\zeta )}{z-\zeta}. +$$ + + +\nz +Пусть $w(z)$, $f(w)$ -- две функции, $F$ -- их композиция: +$F(z)=f(w(z))$. Докажите, что +$$ +S(F;z)=S(f;w)\Bigl (\frac{dw}{dz}\Bigr )^2 +S(w;z). +$$ + +\nz +Пусть $f_1, f_2$ -- два линейно независимых решения дифференциального уравнения +$f''+Q(z)f=0$. Покажите, что $Q(z)=\frac{1}{2}\, S(f_1/f_2; z)$. + + + + +\nz +Найдите общий вид д.-л. изоморфизмов +(а) $\mathbb{H} \to \mathbb{H}$, (б) $\mathbb{U} \to \mathbb{U}$, +(в) $\mathbb{U} \to \mathbb{H}$ +($\mathbb{H}$ -- верхняя полуплоскость, $\mathbb{U}$ -- единичный круг). + +\nz +Отобразите верхнюю полуплоскость на единичный круг так, чтобы +$w(2i)=0$, $\mbox{arg}\, w'(2i)=0$. + + + +\nz +Сколько неподвижных точек может иметь д.-л. преобразование? +Докажите, что если д.-л. преобразование имеет две неподвижные точки, +то произведение производных в этих точках равно 1. + +\nz +Д.-л. преобразование с одной неподвижной точкой $z_0$ называется параболическим. +Докажите, что параболическое преобразование можно записать в канонической форме +$$ +\frac{1}{w-z_0}=\frac{1}{z-z_0}+h +$$ +если $z_0\neq \infty$ и $w=z+h$ если $z_0=\infty$. + +\nz +Докажите, что д.-л. преобразование с двумя различными неподвижными точками $z_1, z_2$ +можно представить в канонической форме +$$ +\frac{w-z_1}{w-z_2}=k\frac{z-z_1}{z-z_2} +$$ +если $z_{1,2}\neq \infty$ и $w-z_1=k(z-z_1)$ если $z_2=\infty$. Преобразование +называется гиперболическим, если $k>0$, эллиптическим если $|k|=1$ ($k\neq 1$) и +локсодромическим в общем случае комплексного $k$. + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok3.tex b/ForStudents/листки Забродина/listok3.tex new file mode 100644 index 0000000..d6ec048 --- /dev/null +++ b/ForStudents/листки Забродина/listok3.tex @@ -0,0 +1,134 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc + Теория функций комплексного переменного -- 2025\\ + Листок 3 +\end{center} + + +\nz +Найдите все точки, в которых $\C$-дифференцируемы функции\\ +а) $|z|^2$, б) $|z|z$, в) $({\rm Re}\, z)^2$. + +\nz +а) При каких комплексных $a,b,c$ функция $f(z)=az+b\bar z +z^2 +c$ имеет +комплексную производную в точке $z=0$? б) При каких вещественных $a,b$ +функция $f(x,y)=ax^2+by^2+ixy$ голоморфна на всем $\C$? + +\nz +Пусть $f(z)$ -- дифференцируемая функция комплексного +переменного в точке $a$. Докажите, что функция $\overline{f(\bar z)}$ +дифференцируема в точке $\bar a$. + +\nz +Покажите, что функция $f: re^{i\varphi}\mapsto re^{2i\varphi}$ не +голоморфна на $\C \setminus \{0\}$. + +\nz +а) +Докажите, что если голоморфная в некоторой области функция +$f(z)$ вешественна (т.е. принимает только вещественные значения), то она +постоянна. +б) +Пусть функция $f(z)$ голоморфна в некоторой области $D$, и $|f(z)|=1$ +всюду в этой области. Докажите, что $f(z)\equiv {\rm const}$. + + +\nz +Восстановите аналитическую функцию $f(z)$ по условию: + +а) ${\rm Re}\, f(z)=\sin x \cosh y$, $f(0)=0$, + +б) ${\rm Re}\, f(z)= +x\sin x \cosh y -y\sinh y \cos x$, $f(0)=0$, + +в) ${\rm Im}\, f(z)= +y\cos x \cosh y-x\sin x \sinh y$, $f(0)=2$, + +г) $|f(z)|=(x^2+y^2)e^x$, + +д) $\mbox{arg}\, f(z)=xy$. + +\nz +Докажите, что аналитическую в $\C$ функцию +$f(z)=u(x,y)+iv(x,y)$ можно восстановить по ее вещественной части $u$ +или мнимой части $v$ с помощью формул +$$ +f(z)=2u\Bigl (\frac{z+z_0}{2}, \, \frac{z-z_0}{2i}\Bigr )- +\overline{f(\bar z_0)}, +$$ +$$ +f(z)=2iv\Bigl (\frac{z+z_0}{2}, \, \frac{z-z_0}{2i}\Bigr )+ +\overline{f(\bar z_0)}, +$$ +где $z_0 \in \C$ -- некоторая точка. + +\vspace{0.5cm} +\small{ +Вещественнозначная функция двух вещественных +переменных $F(x,y)$ называется +{\it гармонической}, если $\displaystyle{ +\frac{\partial^2 F}{\partial x^2}+\frac{\partial^2 F}{\partial y^2}=0}$. +} + +\nz +Запишите оператор Лапласа $\Delta =\displaystyle{ +\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}}$ в координатах +$z$, $\bar z$. + +\nz +Пусть $f(z)$ -- голоморфная функция. Докажите, что функции +${\rm Re} f$ и ${\rm Im} f$ -- гармонические +(т.е. $\Delta {\rm Re} f=\Delta {\rm Im} f=0$). Что можно сказать +о функциях $|f|$ и $\mbox{arg}\, f$? + +\nz +Пусть $F(x,y)$ -- гармоническая функция. Для каких функций +$G$ функция $G(F(x,y))$ тоже будет гармонической? + + +\nz +Найдите все гармонические функции вида а) $F=\varphi (x^2+y^2)$, \\ б) +$F=\varphi (x^2-y^2)$, в) $F=\varphi (y/x)$. + +\nz +Пусть $u(x,y)$ -- гармоническая функция. Являются ли гармоническими +функции $u(x, -y)$, $u(x^2-y^2, 2xy)$? + +\nz +Найдите гармоническую функцию $u(x,y)$ на $\C$ такую, что\\ +$u(\cos t, \sin t)=\cos^2 t.$ + + + + + + + + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok4.tex b/ForStudents/листки Забродина/listok4.tex new file mode 100644 index 0000000..1025ab3 --- /dev/null +++ b/ForStudents/листки Забродина/listok4.tex @@ -0,0 +1,121 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc +Теория функций комплексного переменного -- 2025\\ листок 4 +\end{center} + + + + +\nz +Вычислите интеграл $\displaystyle{\int_{\gamma}(z-a)^n dz}$ +по контуру $\gamma (t)=a+re^{it}$, \\$0\leq t<2\pi$, $r>0$, $n\in \mathbb{Z}$. + + +\nz +Вычислите интеграл $\displaystyle{\int_{\gamma} +z^n dz}$, где $n\in \mathbb{Z}$, а $\gamma$ -- некоторый путь из $a$ в $b$. +(Убедитесь, что при $n\neq -1$ интеграл +не зависит от выбора пути, а при $n=-1$ зависит, и опишите как.) + +\nz +Вычислите интеграл $\displaystyle{\oint_{|z|=1} +\frac{dz}{z^2 +\frac{1}{4}}}$ +по единичной окружности. + +\nz +Вычислите интеграл +$\displaystyle{\oint_{\gamma}\frac{2z^2+5z+8}{z(z+1)^2}}\, dz$, где контур +$\gamma$ -- окружность $|z|=3$. + +\nz +Найдите все возможные значения интеграла $\displaystyle{ +\int_{C} \frac{dz}{z(z^2-1)}}$ при различных выборах контура $C$ +($C$ -- замкнутый контур без самопересечений, не проходящий ни через +одну из точек $0,1,-1$). + +\nz +Покажите, что если путь $\gamma$, соединяющий точки 0 и 1, не проходит через +точки $\pm i$, то +$$ +\int_{\gamma}\frac{dz}{1+z^2} =\frac{\pi}{4}+k\pi, +$$ +где $k$ -- целое число. + + +\nz +Вычислите интеграл $\displaystyle{\oint_{|z|=1}\cos (z^{-1})dz}$. + +\nz +Приведите пример рациональной функции $f$ и замкнутого контура $\gamma$ +таких, что $\displaystyle{\oint_{\gamma}f(z)dz=1}$ и +$\displaystyle{\oint_{\gamma}zf(z)dz =-1}$. + +\nz +Вычислите интеграл $\displaystyle{\int_{\gamma}\bar z dz}$, +где $\gamma$ -- окружность радиуса $R$ с центром в точке $a\in \C$, +проходимая против часовой стрелки. + +\nz +Вычислите интеграл $\displaystyle{\int_{\gamma} +\frac{\bar z}{z}\, dz}$, +где $\gamma$ -- окружность радиуса $R$ с центром в точке $a$ +($|a|\neq R$). + +\nz +Пусть $P(z)$ -- многочлен. Докажите, что +$$\displaystyle{ +\frac{1}{2\pi i}\oint_{|z-a|=R}P(z)d\bar z =-R^2 P'(a)}. +$$ + +\nz +Пусть $\gamma$ -- гладкий замкнутый контур, а функция $f$ голоморфна в +окрестности кривой $\gamma$. Докажите, что +$\displaystyle{\oint_{\gamma}\overline{f(z)}f'(z)dz}$ -- чисто мнимое +число. + +\nz +Вычислите интегралы + +(а) $\displaystyle{\int \!\!\! \int_{\C} +e^{-\alpha |z|^2}\, dx \, dy}$, + +(б) $\displaystyle{\int \!\!\! \int_{\C} +e^{-\alpha |z|^2 +az+\bar a \bar z}\, dx \, dy}$. + + + +\vspace{1cm} + + + +Направление обхода контуров в контурных интегралах предполагается положительным +(против часовой стрелки). + + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok5.tex b/ForStudents/листки Забродина/listok5.tex new file mode 100644 index 0000000..35affe8 --- /dev/null +++ b/ForStudents/листки Забродина/listok5.tex @@ -0,0 +1,99 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} +\newcommand*{\HH}{\ensuremath{\mathbb H}\xspace} +\newcommand*{\UU}{\ensuremath{\mathbb U}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc + Теория функций комплексного переменного -- 2026\\ + Листок 5 +\end{center} + + +\nz +Найдите конформное отображение полукруга $|z|<1$, ${\rm Im}\, z>0$ +на верхнюю полуплоскость. + +\nz +Найдите конформное отображение полуполосы ${\rm Re}\, z>0$, +$0< {\rm Im}\, z <1$ на верхнюю полуплоскость. + + +\nz +Найдите конформное отображение верхней полуплоскости с выброшенным отрезком +от $0$ до $i$ на верхнюю полуплоскость. + +\nz +Найдите конформное отображение внешности эллипса с полуосями +$a$, $b$ ($a>b$) +на внешность единичного круга. + +\nz +Найдите конформное отображение области $D$, являющейся верхней +полуплоскостью ${\rm Im}\, z>0$ с разрезом по дуге окружности +$|z|=1$, $0\leq {\rm arg}\, z \leq \alpha$, где $0<\alpha <\pi$, +на верхнюю полуплоскость. + +\nz +Найдите конформное отображение области $D$, являющейся полуполосой +${\rm Re}\, z>0$, $0<{\rm Im}\, z < \pi$ с разрезом по отрезку +$[\frac{1}{2}\pi i, 1+\frac{1}{2}\pi i]$, на верхнюю полуплоскость. + +\nz +Найдите конформное отображение $z(w)$ внешности единичного круга +$\UU$ на расширенной комплексной плоскости +$\hat \C$ (т.е. некомпактной области $\hat \C \setminus \UU$) +на расширенную комплексную плоскость с разрезом по дуге +$\gamma$ +единичной окружности с углом раствора +$2\alpha$, +$$\gamma = \bigl \{z\in \C \bigl | \, |z|=1, +2\pi \! -\! \alpha \! < \! \mbox{arg}\, z \! < \! \alpha \bigr \} +\quad (0 < \alpha <\pi ), +$$ +(т.е. на область $\hat \C \setminus \gamma$), +такое, что $z(\infty )=\infty$ и $z'(\infty )\in \R_{+}$. + + +\nz +Пусть $\Gamma$ -- гладкая кривая $\Gamma : [0, +\infty )\to \HH$ в верхней полуплоскости +$\HH$, начинающаяся в точке $\Gamma (0)\in \R$ вещественной оси, и пусть $\Gamma_t$ -- +часть кривой $\Gamma_t:=\Gamma ([0,t])$. Через $g(z,t)$ обозначим функцию, осуществляющую +конформное отображение области $\HH \setminus \Gamma_t$ на $\HH$, нормированную условием +$g(z, t)=z+u(t)z^{-1}+O(z^{-2})$, $z\to \infty$, $t>0$. Докажите, что существует +непрерывная вещественнозначная функция $\xi (t)$, такая, что функция $g(z, t)$ удовлетворяет +дифференциальному уравнению +$$ +\frac{\partial g(z, t)}{\partial t} = \frac{1}{g(z, t) -\xi (t)} \, +\frac{\partial u}{\partial t}, \qquad g(z, 0)=z. +$$ +Это уравнение называется (хордовым) уравнением Левнера и +играет важную роль в теории +конформных отображений. + + + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok6.tex b/ForStudents/листки Забродина/listok6.tex new file mode 100644 index 0000000..1af7d8b --- /dev/null +++ b/ForStudents/листки Забродина/listok6.tex @@ -0,0 +1,134 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc + Теория функций комплексного переменного - 2025\\ листок 6 +\end{center} + + + +\nz +Разложите в ряд Тейлора следующие функции +и найдите области сходимости рядов: + +(а) $f(z)=\displaystyle{\frac{z}{1-z-z^2}}\,\,$ в точке $z=0$, + +(б) $f(z)=\displaystyle{\frac{z}{z^2-2z+5}}\,\,$ в точке $z=1$, + +(в) $f(z)=\displaystyle{\frac{z}{(1-z)^2}}\,\,$ в точке $z=0$ +({\small эта функция называется {\it функцией} $\phantom{aaaaaa}$ {\it Кёбе}. На что она +отображает единичный диск?}), + +(г) $f(z)=\displaystyle{\sin ^2 z}\,\,$ в точке $z=0$, + +(д) $f(z)=\displaystyle{\frac{e^z-z-1}{z^2}}\,\,$ в точке $z=0$, + +(е) $f(z)=\displaystyle{\int_0^z \frac{e^t -1}{t}\, dt}\,\,$ в точке $z=0$. + + +\nz +Пусть голоморфная в некотором круге с центром в нуле функция $f(z)$ удовлетворяет уравнению +$f(z)=z+f(z^2)$. Найдите разложение $f(z)$ в ряд и область сходимости ряда. + +\nz +Найдите $f'''(0)$, где $f(z)=e^{\sin z}$. + + +\nz +Найдите радиусы сходимости рядов: + +(а) $\displaystyle{\sum_{n=0}^{\infty} +\frac{n}{2^n}\, z^n},\,\, $ +(б) $ \displaystyle{\sum_{n=0}^{\infty}\frac{n!}{n^n}\, z^n},\,\, $ +(в) $ \displaystyle{\sum_{n=0}^{\infty}z^{n!},}\,\,$ +(г) $\displaystyle{\sum_{n=0}^{\infty}\cos (in) \, z^{n}, \,\, }$\\ +$\phantom{aaa}$ (д) $\displaystyle{\sum_{n=0}^{\infty} (3+(-1)^n)^n z^n }$. + +\nz +Найдите суммы рядов $\displaystyle{\sum_{n=1}^{\infty}z^n/n}$, +$\displaystyle{\sum_{n=1}^{\infty}n^2z^n}$. + + + + + +\nz +Докажите, что в разложении +$\displaystyle{ +\frac{z}{e^z-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}\, z^n} +$ +все коэффициенты $B_n$ (числа Бернулли) с нечетными номерами равны $0$ кроме $B_1$ +и найдите радиус сходимости ряда. + + + + + + +\nz +Для любого $a\in \C$ положим $(a)_n =a(a+1)(a+2)\ldots (a+n-1)$ (символ +Похгаммера). + +(a) Определите радиус сходимости ряда +$$ +F(a,b;c;z)= 1+\sum_{n=1}^{\infty}\frac{(a)_n (b)_n}{n! (c)_n} \, z^n. +$$ + +(б) Докажите, что функция $F=F(a,b;c;z)$, задаваемая этим рядом, +удовлетворяет дифференциалному уравнению +$$ +z(1-z)\frac{d^2 F}{d z^2}+[c-(a+b+1)z]\frac{d F}{d z}-ab F=0. +$$ +{\small Функция $F(a,b;c;z)$ называется {\it гипергеометрической +функцией}}. + +(в) Найдите второе решение гипергеометрического дифференциального уравнения в +предположении, что оно имеет вид $z^{\alpha}f(z)$, где $\alpha$ -- константа, +а $f(z)$ голоморфна в 0. + + + + +\nz +Найдите все особые точки и определите их характер +у следующих функций: +(а) $\displaystyle{\frac{1}{1-\sin z}}\,\,$, +(б) $\displaystyle{\frac{z}{\sin (z^3)}}\,\,$, +(в) $\displaystyle{\ctg (1/z)}\,\,$, +(г) $\displaystyle{\frac{1}{e^{z^2}+1}}\,\,$,\\ +(д) $\displaystyle{\frac{1}{e^{z}-1}-\frac{1}{z}}\,\,$, +(е) $\displaystyle{\frac{z^3}{\sin ^2 \frac{\pi}{z+1}}}\,$. + + + + +%\nz +%Пусть функция $f$ голоморфна в проколотой +%окрестности точки $a$ и ${\rm Im}\, f(z)>0$ всюду в этой окрестности. +%Какого типа особенность может быть в точке $a$? + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok7.tex b/ForStudents/листки Забродина/listok7.tex new file mode 100644 index 0000000..c63c739 --- /dev/null +++ b/ForStudents/листки Забродина/listok7.tex @@ -0,0 +1,129 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +%\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +%\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} + +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} +\newcommand*{\HH}{\ensuremath{\mathbb H}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc +Теория функций комплексного переменного - 2026\\ листок 7 +\end{center} + + + + + + + +\nz +С помощью интегральной формулы Коши вычислите интегралы +(а) $\displaystyle{\int_{|z|=2}\frac{e^z dz}{z^2-1}}$, +(б) $\displaystyle{\int_{|z-a|=a}\frac{zdz}{z^4-1}}$, $a>1$, +(в) $\displaystyle{\int_{|z-1|=1}\frac{ze^zdz}{(z-1)^3}}$. + +\nz +Пусть $f(z)$ -- голоморфная функция в верхней полуплоскости, непрерывно продолжаемая +на вещественную ось, причем $\lim\limits_{|z|\to \infty} f(z)=0$. +Докажите интегральную формулу Шварца +$$ +f(z)=\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{{\rm Im} f(x)}{x-z}\, dx. +$$ + +\nz +Пусть $f(z)$ -- голоморфная функция в единичном круге, непрерывно продолжаемая +на его границу. Докажите интегральную формулу Пуанкаре +$$ +f(z)=\frac{1}{2\pi}\int_{0}^{2\pi} \!{\rm Re} f(e^{i\theta}) +\frac{e^{i\theta}+z}{e^{i\theta}-z}\, d\theta +i{\rm Im} f(0). +$$ + +\nz +Найдите вычеты дифференциальных форм в их особых точках:\\ +а) $\displaystyle{\frac{z dz}{z^2+1}}$, +б) $\displaystyle{\frac{e^z dz}{(z-1)^2}}$, +в) $\displaystyle{\frac{z^{}dz}{\sin ^2 z}}$, +г) $\displaystyle{\frac{z^{2n}dz}{(z-1)^n}}$, +д) $\displaystyle{\frac{e^{1/z}dz}{1-z}}$, +е) $\sin z \, \sin (1/z) \, dz$,\\ +ж) $\displaystyle{\frac{\sin 3z - 3\sin z}{\sin z \, (\sin z -z)}\, dz}$ в точке +$z=0$, +з) $\displaystyle{z^2 \sin \frac{1}{z+1} \, dz}$, +и) $\displaystyle{\frac{dz}{z(e^{2z}-1)}}$ в точке +$z=0$, +к) $\displaystyle{e^{z/(1-z)}dz}$. + +\nz +Докажите, что для четной функции $f(z)$ имеет место равенство +$$\mbox{res}_{z=a}f(z)dz =-\mbox{res}_{z=-a}f(z)dz$$ а для нечетной +$$\mbox{res}_{z=a}f(z)dz =\mbox{res}_{z=-a}f(z)dz$$ + +\nz +Вычислите интегралы: +а) $\displaystyle{\oint_{|z|=4}\frac{z^4 dz}{e^z +1}}$, +б) $\displaystyle{\oint_{|z|=1}\frac{e^z dz}{(2z+1)^2 (z+2)}}$, \\ +в) $\displaystyle{\oint_{|z|=1}(z+1)e^{1/z}dz}$, +г) $\displaystyle{\oint_{|z|=1}\frac{z^3 dz}{2z^4+1}}$, +д) $\displaystyle{\oint_{|z|=3}\frac{z^3-1}{z-1}\left (e^{\frac{1}{z}}+ +e^{\frac{1}{z-1}}+e^{\frac{1}{z-2}}\right )dz}$, +е) $\displaystyle{\oint_{|z|=1}\frac{dz}{e^{2/z}-e^{1/z}}}$, +ж) $\displaystyle{\oint_{|z|=3}\frac{dz}{(z-1)^2(1-\cos z)}}$, +з) $\displaystyle{\oint_{|z|=1}\frac{z-(\log 2)^{-1}}{e^{1/z}-2}\, dz}$. + +\nz +Пусть функция $f(z)$ регулярна в замыкании конечной области $D$, +а точки $a_1, \ldots , a_n$ лежат в $D$ и попарно различны. Обозначим +$P(z)=(z-a_1)\ldots (z-a_n)$ и +$$ +\Phi (z)=-\frac{P(z)}{2\pi i}\oint_{\partial D} \frac{f(\xi )}{P(\xi )} +\, \frac{d\xi}{\xi -z}, \quad z\notin D. +$$ +Докажите, что функция $\Phi (z)$ аналитически продолжается на всю +плоскость и представляет собой многочлен степени $n-1$, причем +$\Phi (a_k)=f(a_k)$, $k=1, \ldots , n$. (Многочлен $\Phi (z)$ +называется интерполяционным многочленом Лагранжа.) + +\nz +Пусть $p_1, \ldots , p_N$ -- $N$ различных комплексных чисел. +Докажите тождество +$$ +\sum_{i=1}^N \prod_{j\neq i}^N \frac{p_i+p_j}{p_i-p_j}= +\frac{1}{2}\Bigl (1-(-1)^N\Bigr ). +$$ +{\it Указание: воспользуйтесь тем, что +сумма вычетов рациональной функции, включая вычет +в $\infty$, равна 0.} + +\nz +Пусть $w_1, \ldots , w_N$ -- $N$ различных комплексных чисел. +Найдите сумму +$$ +S=\sum_{i=1}^N \prod_{j\neq i}^N \frac{1-w_iw_j}{w_i-w_j}. +$$ + + + +\end{document} \ No newline at end of file diff --git a/ForStudents/листки Забродина/listok8.tex b/ForStudents/листки Забродина/listok8.tex new file mode 100644 index 0000000..7bfb9ee --- /dev/null +++ b/ForStudents/листки Забродина/listok8.tex @@ -0,0 +1,84 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb,amsfonts,amsmath} +\usepackage{xspace} + +\pagestyle{empty} + +\newcounter{zadacha} +\setcounter{zadacha}0 + +\newcommand*{\nz}{% +\par\medskip\refstepcounter{zadacha} +\textbf{\thezadacha.} } + +\let\C\relax +\newcommand*{\C}{\ensuremath{\mathbb C}\xspace} +\newcommand*{\R}{\ensuremath{\mathbb R}\xspace} + +\let\Re\relax +\let\Im\relax +\DeclareMathOperator{\Re}{Re} +\DeclareMathOperator{\Im}{Im} + + +\begin{document} + +\begin{center}\large\sc + Теория функций комплексного переменного -- 2025\\ листок 8 +\end{center} + + + + + +\nz +Вычислите интегралы Френеля +$\displaystyle{ +\int_{0}^{\infty}\! \cos x^2 dx, \;\; \int_{0}^{\infty}\! \sin x^2 dx} +$. + +\nz +Вычислите интеграл +$\displaystyle{\int_{0}^{\infty}\frac{\cos ax -\cos bx}{x^2}\, dx}$. + +\nz +Вычислите интегралы +а) $\displaystyle{\int_{0}^{\infty}\frac{\sin x}{x}\, dx}$, +б) $\displaystyle{\int_{0}^{\infty}\frac{\sin x}{x}\, +\frac{\sin (x/3)}{x/3}\, dx}$. + +\nz +Вычислите интеграл +$\displaystyle{\int_{0}^{\infty}\cos x\, \frac{\sin x}{x}\, dx}$. + + + + +\nz +Вычислите определенные интегралы: +а) $\displaystyle{\int_{0}^{2\pi}\!\!\frac{d\varphi}{a+b\cos \varphi}}$ ($a>b>0$),\\ +б) $\displaystyle{\int_{0}^{2\pi}\!\! +e^{\cos \varphi}\cos (n\varphi -\sin \varphi )d\varphi}$, +в) $\displaystyle{\int_{0}^{\infty}\frac{x^2+1}{x^4+1}\, dx}$, +г) $\displaystyle{\int_{0}^{\infty}\frac{dx}{x^n+1}}$,\\ +д) $\displaystyle{\int_{0}^{\infty}\frac{\cos ax}{x^2+b^2}\, dx}$, +е) $\displaystyle{\int_{0}^{\infty}\frac{x\sin ax}{x^2+b^2}\, dx}$, +ж) $\displaystyle{\int_{0}^{\infty}\frac{\sin ax}{x(x^2+b^2)}\, dx}$, +з) $\displaystyle{\int_{-\infty}^{\infty}\frac{e^{ax}dx}{1+e^x}}\;\;$ ($00)$, +(б) $\displaystyle{\int_{0}^{\infty}e^{-\pi x}\, \frac{\sin ax}{\sinh \pi x}\, dx}$, +(в) $\displaystyle{\int_{0}^{\infty}\frac{x-\sin x}{x^3}\, dx}$, +г) $\displaystyle{\int_{-\infty}^{\infty}\frac{e^{ax}-e^{bx}}{1-e^x}\, dx}\;\; +(0`z=8_b0VB>b3Bt{?1~mnMCM4%LDFtl`IxpNsnX9{J;oXT~uNbM0Yz zIjoQ{@>y=sGd}k`3eO*5XO)eW?6Ly)DrBLnQEN+icaw8od&woHn+4&Qs_&xb4UCx? 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ze6654mc-`9!|I6Y<)xEt5iC4SV7B5Jj8|Y5EEZ@ehp_PkeEp>b9@im4Ie=n;yo=dI zgklA0$3Qkg?ZN=na2bo)o`Fky(YR!I9*b2LN)=Qu3vqEaQF`h!0 zV>}fg6wtl`%VBkx2-^%)uSiqq8bv~Bi`Et(W6*db30NBuyMRo@YG4wUa|9n87A>MV z8){bgqwhCZ$piu6nzXvXZsav5wQ6N>n7!dBoEjEzE=O4vdjKXr=i@0)nbU?-{3D%IMK ReG^GZmNPUpwV7dS_#f?b$o&8S literal 0 HcmV?d00001 diff --git a/ForStudents/листки Забродина/prob1.txt b/ForStudents/листки Забродина/prob1.txt new file mode 100644 index 0000000..e69de29 diff --git a/README.md b/README.md new file mode 100644 index 0000000..23b3711 --- /dev/null +++ b/README.md @@ -0,0 +1,68 @@ +# ТФКП / Complex Analysis + +Курс теории функций комплексного переменного, ВШЭ — совместный бакалавриат НИУ ВШЭ и ЦПМ, 2 курс 2025/26. +Лектор: **В.А. Тиморин**. Страница курса: +Видео лекций: · Материалы (Яндекс.Диск): + +## Цель репы + +Подготовка к пересдаче. Главная цель — уметь решать **абсолютно всё** из +списка задач (`ForStudents/CA-prob.tex` / `problem-list.pdf`). +Идём по темам снизу вверх по списку, закрываем по очереди. Параллельно читаем +учебник. + +**Основной учебник:** Львовский (фрагмент лежит локально в `ForStudents/Литература/`, +не публикуется по соображениям копирайта). Дополнительно — любые другие из +локальной библиотеки (Ahlfors, Conway, Needham *Visual Complex Analysis*, +Stein–Shakarchi, Rudin, Lang, Gamelin…). + +## Содержимое + +- `problem-list.pdf` — итоговый список задач (25 стр., 182 задачи + теоретические вопросы). +- `ForStudents/CA-prob.tex` — LaTeX-исходник задачника (читаемая кириллица, канон). +- `ForStudents/CA-prob.pdf` — скомпилированный задачник. +- `ForStudents/листки Забродина/` — исходные листочки А.В. Забродина (`listok1.tex`…`listok14.tex`), + на которых основан задачник. +- `ForStudents/лекция 13.03.2026.pdf` / `.pptx` — лекция. +- `ForStudents/bd.png` — рисунок к условию. + +Не в git (локально только): `ForStudents/Литература/` (копирайтные учебники), +`ForStudents/текущая оценка.pdf` (оценки группы, приватно), `ForStudents.zip` +(оригинальный архив). + +## Темы (чеклист) + +Задачи пронумерованы сквозной нумерацией 1–182 и сгруппированы по темам. +Идти по порядку: + +- [ ] 1. Комплексные числа +- [ ] 2. Дробно-линейные преобразования +- [ ] 3. Комплексная дифференцируемость +- [ ] 4. Производная Шварца +- [ ] 5. Геометрия комплексных функций +- [ ] 6. Контурные интегралы +- [ ] 7. Формула Коши, вычеты +- [ ] 8. Изолированные особенности и ряды Лорана +- [ ] 9. Аналитическое продолжение +- [ ] 10. Вычисление интегралов +- [ ] 11. Принцип аргумента +- [ ] 12. Принцип максимума +- [ ] 13. Гиперболическая метрика +- [ ] 14. Принцип симметрии и принцип соответствия границ +- [ ] 15. Тэта-функции +- [ ] 16. Эллиптические функции + +Отдельный отдел (не тема): «Теоретические вопросы к промежуточной контрольной и +итоговому экзамену» — список утверждений для устного/письменного экзамена. + +## Маркеры задач + +- `◦` — разобрать на семинаре в каждой группе +- `•` — возможно обсуждается на лекции +- `*` — звёздочка, сложная; разбор откладывается, сдаётся преподу вне семинара + (есть дедлайны по сдаче) +- `(ДЗ)` — может быть в домашнем задании + +## Оценка + +Итог = 0.5×(тесты + ДЗ + работа на семинарах) + 0.2×промежуточная к/р + 0.3×экзамен. diff --git a/problem-list.pdf b/problem-list.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2d5ea69b4a74cba01167f3ec3a91963218f2d448 GIT binary patch literal 396915 zcmcG#V~}Orwziv9iAvj?Y1_7K+qUhjv~AnAZ96M%J1g<6wZFa2UMJ4J5&PV@cm8V8 zW46&pj9<^w-`0ucghZ(6X_z62XBUT7AsGR509ykKNG>i&S}9{2QztV313f(h;2%Fo zS`l+ACu0Wyt%#Mrld+Jop{xT`_m?caC|s<4V{E)P z^(^uUdrNUkzV{bvDG^P+RTL>ZGzqTw=4)L3zRlOx=4FiT)y>Doam_Pa80+;44*e$2 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