From 4c0bd0db489b818c8d3756eaf0dfaf7c672ba84b Mon Sep 17 00:00:00 2001 From: M1nd3r <63123927+M1nd3r@users.noreply.github.com> Date: Fri, 8 Jan 2021 23:12:58 +0100 Subject: [PATCH] Kompletace MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Dodělání a úprava poslední přednášky. Přidán operátor res pro residua. Úprava číslování rovnic (reset na začátku kapitol - provedeno manuálně u každé kapitoly). --- zdroj/main.tex | 3 +- zdroj/prednasky/02_prednaska.tex | 4 +- zdroj/prednasky/03_prednaska.tex | 1 + zdroj/prednasky/04_prednaska.tex | 2 +- zdroj/prednasky/05_prednaska.tex | 4 +- zdroj/prednasky/06_prednaska.tex | 4 +- zdroj/prednasky/08_prednaska.tex | 6 +- zdroj/prednasky/10_prednaska.tex | 24 +- zdroj/prednasky/11_prednaska.tex | 4 +- zdroj/prednasky/14_prednaska.tex | 208 ++++++++++++++---- ...od do komplexn\303\255 anal\303\275zy.pdf" | Bin 715745 -> 733562 bytes ...od do komplexn\303\255 anal\303\275zy.zip" | Bin 83413 -> 87193 bytes ...o komplexn\303\255 anal\303\275zy_UKA.zip" | Bin 763751 -> 0 bytes 13 files changed, 193 insertions(+), 67 deletions(-) rename main_UKA.pdf => "\303\232vod do komplexn\303\255 anal\303\275zy.pdf" (60%) delete mode 100644 "\303\232vod do komplexn\303\255 anal\303\275zy_UKA.zip" diff --git a/zdroj/main.tex b/zdroj/main.tex index 937845b..02c0906 100644 --- a/zdroj/main.tex +++ b/zdroj/main.tex @@ -109,6 +109,7 @@ \DeclareMathOperator{\Img}{Im} \DeclareMathOperator{\Rel}{Re} \DeclareMathOperator{\ind}{ind} +\DeclareMathOperator{\res}{res} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Int}{Int} @@ -139,7 +140,7 @@ \title{\textbf{\Huge Úvod do komplexní analýzy}} \subtitle{doc. RNDr. Roman Lávička, Ph.D.} -\date{6. ledna 2020} +\date{8. ledna 2020} \author{\vspace{-10mm}} %\date{} % Toggle commenting to test diff --git a/zdroj/prednasky/02_prednaska.tex b/zdroj/prednasky/02_prednaska.tex index bf84220..5e03ca5 100644 --- a/zdroj/prednasky/02_prednaska.tex +++ b/zdroj/prednasky/02_prednaska.tex @@ -30,7 +30,7 @@ \subsection{Exponenciála} \item $\exp(z+w)=\exp(z)\exp(w)$ \item $\exp'(z)=\exp(z)$, $ z \in\Comp $ - \begin{equation} + \begin{equation*} \begin{aligned} f(z) = \exp(z), \hspace{3mm} f_1(x,y) &=e^x \cos y, \hspace{3mm} @@ -38,7 +38,7 @@ \subsection{Exponenciála} \frac{\partial f_1}{\partial x} = e^x\cos y = \frac{\partial f_2}{\partial y}&, \hspace{3mm} \frac{\partial f_2}{\partial x} = e^x \sin y= - \frac{\partial f_1}{\partial y} \end{aligned} -\end{equation} +\end{equation*} diff --git a/zdroj/prednasky/03_prednaska.tex b/zdroj/prednasky/03_prednaska.tex index 9e91df9..b5ab2b1 100644 --- a/zdroj/prednasky/03_prednaska.tex +++ b/zdroj/prednasky/03_prednaska.tex @@ -1,5 +1,6 @@ \section{\texorpdfstring{Křivkový integrál}{Krivkový integrál}} %Chceme integrovat komplexní funkci jedné komplexní proměnné podél křivky v komplexní rovině tj. chceme zavést integrál funkce podél křivky. +\setcounter{equation}{0} \begin{definition} Nechť $\varphi:[\alpha,\beta] \rightarrow \Comp $. Potom \begin{enumerate} diff --git a/zdroj/prednasky/04_prednaska.tex b/zdroj/prednasky/04_prednaska.tex index 83e53c4..d94fb5a 100644 --- a/zdroj/prednasky/04_prednaska.tex +++ b/zdroj/prednasky/04_prednaska.tex @@ -41,7 +41,7 @@ \label{eqn:b} \end{equation} -kde první integrand vpravo má PF na $\Comp $ a první integrál je roven $0$. Pro každé $j \in \N_0$ z \cref{eqn:a},\cref{eqn:b} dostaneme +kde první integrand vpravo má PF na $\Comp $ a první integrál je roven $0$. Pro každé $j \in \N_0$ z \cref{eqn:a}, \cref{eqn:b} dostaneme $$ 0<\frac{K}{4^j} \le \left| \int_{\varphi_j}f\,\right|\stackrel{\cref{eqn:b}}{=} \left|\int_{\varphi_j}\varepsilon(z)(z-z_0)\right| \le V^2(\varphi_j) \cdot \max_{\langle\varphi_j\rangle}|\varepsilon|\overset{\cref{eqn:a}}{=}\frac{V^2(\varphi)}{4^j}\cdot \max_{\langle\varphi_j\rangle}|\varepsilon| \text{,} $$ diff --git a/zdroj/prednasky/05_prednaska.tex b/zdroj/prednasky/05_prednaska.tex index 13db181..a777007 100644 --- a/zdroj/prednasky/05_prednaska.tex +++ b/zdroj/prednasky/05_prednaska.tex @@ -149,7 +149,7 @@ \end{proof} % osmá strana -\begin{theorem}[Weierstrass] +\begin{theorem}[Weierstrass]\label{thm:weierstrass} Nechť $G\subset{\Comp }$ je otevřená, $f_n\in\Holo (G)$ pro $n\in\N$ a $f_n\overset{loc}{\rightrightarrows}f$ na $G$. Potom $f\in\Holo (G)$ a $f_n^{(k)}\overset{loc}{\rightrightarrows}f^{(k)}$ na $G$ pro každé $k\in\N$. \end{theorem} @@ -169,7 +169,7 @@ % devátá strana \section{\texorpdfstring{Mocninné řady}{Mocninné rady}} - +\setcounter{equation}{0} \begin{definition} Nechť $\{a_n\}_{n=0}^\infty\subset\Comp $ a $z_0\in\Comp $. Potom \begin{equation} diff --git a/zdroj/prednasky/06_prednaska.tex b/zdroj/prednasky/06_prednaska.tex index 96ad1ea..4744d0e 100644 --- a/zdroj/prednasky/06_prednaska.tex +++ b/zdroj/prednasky/06_prednaska.tex @@ -70,7 +70,7 @@ Protože $g(z_0)=a_p \neq 0$, existuje $r>0$, že $g \neq 0$ na $U(z_0,\,r)$ a $f(z)=(z-z_0)^pg(z) \neq 0$ na $P(z_0,r)$. Obrácené tvrzení z poznámky je snadné. \end{proof} -\begin{theorem}[O jednoznačnosti pro holomorfní funkce] +\begin{theorem}[O jednoznačnosti pro holomorfní funkce]\label{thm:jednoznacnostHolo} Nechť $\emptyset \neq G \subset \Comp $ je oblast a $f,g \in \Holo (G)$. Pak jsou následující tvrzení ekvivalentní: \begin{enumerate} \item $f=g$ na $G$; @@ -122,7 +122,7 @@ Nebo-li $|a_0|^2\geq|a_0|^2+|a_1|^2\rho^2+\cdots$, tudíž $0=a_1=a_2=\cdots$. Dostáváme, že $f=a_0$ na $U(z_0,r)$ a z věty o jednoznačnosti $f=a_0$ na $G$. \end{proof} -\section{\texorpdfstring{Riemannova sféra}{Riemannova sféra}} +\section{\texorpdfstring{Riemannova sféra}{Riemannova sféra}} \setcounter{equation}{0} Rozšíříme $\Comp $ o \emph{nekonečno}. Položíme $\Sp=\Comp \cup\{\infty\}$, kde $\infty\notin\Comp $, a pro $\varepsilon>0$ zavedeme \emph{okolí} kolem $\infty$, následovně $P(\infty,\,\varepsilon):=\{z\in\Comp : |z|>\frac{1}{\varepsilon}\}$, $U(\infty,\,\varepsilon):=P(\infty,\,\varepsilon)\cup \{\infty\}$. diff --git a/zdroj/prednasky/08_prednaska.tex b/zdroj/prednasky/08_prednaska.tex index 48b2412..39f4cdb 100644 --- a/zdroj/prednasky/08_prednaska.tex +++ b/zdroj/prednasky/08_prednaska.tex @@ -140,13 +140,13 @@ \subsection{Izolované singularity 2} \subsection{Reziduum} \begin{definition} -Nechť $f \in \Holo (P(z_0))$ a nechť $f(z)=\sum\limits_{n=-\infty}^{+\infty}a_n(z-z_0)^n$, $z\in P(z_0)$. Potom reziduem $f$ v $z_0$ nazveme číslo res$_{z_0}f:=a_{-1}$. +Nechť $f \in \Holo (P(z_0))$ a nechť $f(z)=\sum\limits_{n=-\infty}^{+\infty}a_n(z-z_0)^n$, $z\in P(z_0)$. Potom reziduem $f$ v $z_0$ nazveme číslo $\res_{z_0}f:=a_{-1}$. \end{definition} \begin{theorem}[Reziduová na hvězdovitých oblastech] Nechť $G \subset \Comp $ je hvězdovitá oblast, $M \subset G$ je konečná a $f \in \Holo (G \setminus M)$. Nechť $\varphi$ je uzavřená křivka v $G \setminus M$. Potom máme \begin{equation} - \int_\varphi f=2\pi i \sum\limits_{s\in M}\text{res}_sf \cdot \ind_\varphi s. + \int_\varphi f=2\pi i \sum\limits_{s\in M}\res_sf \cdot \ind_\varphi s. \tag{RV} \label{eqn:7.31.RV} \end{equation} @@ -159,7 +159,7 @@ \subsection{Reziduum} \begin{proof} Podle předchozí věty existuje $h \in \Holo (G)$ tak, že $f=\sum\limits_{s\in M}H_s+h$ na $G \setminus M$. Potom máme $\int_\varphi f=\sum\limits_{s\in M}\int_\varphi H_s$, protože $\int_\varphi h =0$ z Cauchyho věty pro hvězdovité oblasti. Pro každé $s \in M$: $$ -\int_\varphi H_s(z) \diff z=\int_\varphi \sum\limits_{n=1}^{+\infty}a_{-n}^s\frac{1}{(z-s)^n} \diff z = \sum\limits_{n=1}^{+\infty}a_{-n}^s \int_\varphi \frac{\diff z}{(z-s)^n}=2\pi i \cdot \text{res}_sf\cdot \ind_\varphi s\text{,} +\int_\varphi H_s(z) \diff z=\int_\varphi \sum\limits_{n=1}^{+\infty}a_{-n}^s\frac{1}{(z-s)^n} \diff z = \sum\limits_{n=1}^{+\infty}a_{-n}^s \int_\varphi \frac{\diff z}{(z-s)^n}=2\pi i \cdot \res_sf\cdot \ind_\varphi s\text{,} $$ jelikož suma konverguje stejnoměrně na $\langle\varphi \rangle$ a poslední integrál je roven $0$ pro $n \neq 1$ (neboť jinak má integrand PF, a tudíž je integrál přes uzavřenou křivku nulový) a $2 \pi i \cdot \ind_\varphi s$, je-li $n=1$. \end{proof} diff --git a/zdroj/prednasky/10_prednaska.tex b/zdroj/prednasky/10_prednaska.tex index 025b031..921c975 100644 --- a/zdroj/prednasky/10_prednaska.tex +++ b/zdroj/prednasky/10_prednaska.tex @@ -12,7 +12,7 @@ $$1=\ind_{\varphi\dotplus\tilde\varphi}{s}=\ind_\varphi{s}+\ind_{\tilde{\varphi}}{s}.$$ \end{example} -\section{Speciální typy integrálů} +\section{Speciální typy integrálů} \setcounter{equation}{0} \begin{theorem} Nechť $R=P/Q$, kde $P,Q$ jsou polynomy, které nemají společné kořeny a platí \begin{enumerate} @@ -21,10 +21,10 @@ \section{Speciální typy integrálů} \end{enumerate} Potom \begin{equation} - \int_{-\infty}^\infty{R(x)dx}=2i\pi.\sum_{\begin{array}{cc} + \int_{-\infty}^\infty{R(x)dx}=2i\pi\cdot\sum_{\begin{array}{cc} Q(s)=0 \\ \Img(s)>0 - \end{array}}{res_s{R}}. + \end{array}}{\res_s{R}}. \label{eqn:print} \end{equation} \end{theorem} @@ -34,7 +34,7 @@ \section{Speciální typy integrálů} 2i\pi.\sum_{\begin{array}{cc} Q(s)=0 \\ \Img(s)>0 - \end{array}}{res_s{R}} + \end{array}}{\res_s{R}} \overset{(RV)}{=} \int_{\varphi_r}{R}= \int_{\varphi_r^1}{R} +\int_{\varphi_r^2}{R}. @@ -54,12 +54,12 @@ \section{Speciální typy integrálů} \begin{example} $$I=\int_0^\infty{\underset{\text{sudá}}{\underbrace{\frac{x^2+1}{x^4+1}}}dx} =\frac{1}{2}{\int_{-\infty}^\infty\underset{:=R(x)}{\underbrace{\frac{x^2+1} -{x^4+1}}}dx}=i\pi\cdot(res_{z_0}R+res_{z_1}R)$$ +{x^4+1}}}dx}=i\pi\cdot(\res_{z_0}R+\res_{z_1}R)$$ $$=-\frac{i\pi}{4\sqrt{2}}\left[(1+i)^2-(1-i)^2\right]=-\frac{i\pi}{4\sqrt{2}}2\cdot 2i=\frac{\pi}{\sqrt{2}},$$ protože -$${res}_{z_k}R=\frac{z_k^2+1}{4z^3_k}\cdot\frac{z_k}{z_k}= +$$\res_{z_k}R=\frac{z_k^2+1}{4z^3_k}\cdot\frac{z_k}{z_k}= -\frac{1}{4}(z_k^2+1)z_k,$$ -$${res}_{z_0}R=-\frac{1}{4}(i+1)(1+i)\frac{1}{\sqrt{2}}=-\frac{1}{4\sqrt{2}}(1+i)^2,$$ -$${res}_{z_1}R=-\frac{1}{4}(-i+1)(-1+i)\frac{1}{\sqrt{2}}=\frac{1}{4\sqrt{2}}(1-i)^2.$$ +$$\res_{z_0}R=-\frac{1}{4}(i+1)(1+i)\frac{1}{\sqrt{2}}=-\frac{1}{4\sqrt{2}}(1+i)^2,$$ +$$\res_{z_1}R=-\frac{1}{4}(-i+1)(-1+i)\frac{1}{\sqrt{2}}=\frac{1}{4\sqrt{2}}(1-i)^2.$$ \end{example} \begin{theorem} @@ -74,7 +74,7 @@ \section{Speciální typy integrálů} \int_{-\infty}^\infty{R(x)e^{iax}}dx = 2i\pi\cdot\sum_{\begin{array}{cc} Q(s)=0 \\ \Img(s)>0 - \end{array}}{res_s\left({R(z)e^{iaz}}\right)}. + \end{array}}{\res_s\left({R(z)e^{iaz}}\right)}. \end{equation} \end{theorem} \begin{proof} @@ -99,8 +99,8 @@ \section{Speciální typy integrálů} Spočteme Fourierovu transformaci $\F $ funkce $f(x):=\frac{1}{x^2+1}$, kde $$(\F f)(t):=\frac{1}{2\pi}\int_{-\infty}^\infty{f(x)e^{itx}}dx, \ t\in\Real. $$ \begin{itemize} - \item Nechť $t>0$. Potom $(\F f)(t)=i\cdot{res}_i{(f(z)e^{itz})}=i\cdot\frac{e^{-t}}{2i}=\frac{e^{-t}}{2}$. - \item Nechť $t<0$. Potom $(\F f)(t)=i\cdot{res}_{(-i)}{(f(z)e^{itz})}\cdot(-1)=-i\cdot\frac{e^{-t}}{2\cdot r(-i)}=\frac{e^{t}}{2}$. + \item Nechť $t>0$. Potom $(\F f)(t)=i\cdot\res_i{(f(z)e^{itz})}=i\cdot\frac{e^{-t}}{2i}=\frac{e^{-t}}{2}$. + \item Nechť $t<0$. Potom $(\F f)(t)=i\cdot\res_{(-i)}{(f(z)e^{itz})}\cdot(-1)=-i\cdot\frac{e^{-t}}{2\cdot r(-i)}=\frac{e^{t}}{2}$. \end{itemize} \textbf{Lépe:} $$(\F f)(t)=\frac{1}{2\pi}\int_{-\infty}^\infty\frac{\cos{(tx)}+i\sin{(tx)}}{1+x^2}dx=\frac{e^{-|t|}}{2}, \ t\in\Real .$$ @@ -109,5 +109,5 @@ \section{Speciální typy integrálů} \begin{example} $$\int_0^\infty\frac{x\sin{x}}{x^2+1}dx=\frac{1}{2}\Img \underset{:=I}{\underbrace{\left(\int_{-\infty}^\infty\frac{x.e^{ix}}{x^2+1}dx\right)}}=\frac{\pi}{2e},$$ protože -$$I=2\pi i\,{res}_{i}{f}=2\pi i \frac{ie^{-1}}{2i}=\frac{\pi i}{e}.$$ +$$I=2\pi i\,\res_{i}{f}=2\pi i \frac{ie^{-1}}{2i}=\frac{\pi i}{e}.$$ \end{example} \ No newline at end of file diff --git a/zdroj/prednasky/11_prednaska.tex b/zdroj/prednasky/11_prednaska.tex index 5006778..72f20ce 100644 --- a/zdroj/prednasky/11_prednaska.tex +++ b/zdroj/prednasky/11_prednaska.tex @@ -65,7 +65,7 @@ \section{Index} \begin{aligned} &\ind_\varphi s\stackrel{z=\varphi(t)}{=}\frac{1}{2 \pi i}\int_\alpha^\beta \frac{\varphi'(t)}{\varphi(t)-s} \diff t \stackrel{z=\varphi(t)-s}{=} \frac{1}{2 \pi i} \int_{\widetilde{\varphi}} \frac{\diff z}{z}= -\ind_{\widetilde{\varphi}}0=\stackrel{\text{Věta}}{=}\\ &= \frac{1}{2 \pi i}(\phi(\beta)-\phi(\alpha))=\frac{1}{2 \pi i}(i \Img \phi(\beta)-i \Img \phi(\alpha))=\frac{\theta(\beta)-\theta(\alpha)}{2 \pi}=K +\ind_{\widetilde{\varphi}}0\stackrel{\text{Věta}}{=}\\ &= \frac{1}{2 \pi i}(\phi(\beta)-\phi(\alpha))=\frac{1}{2 \pi i}(i \Img \phi(\beta)-i \Img \phi(\alpha))=\frac{\theta(\beta)-\theta(\alpha)}{2 \pi}=K \end{aligned} \end{equation*} pro nějaké $K \in \Z$, kde $\phi$ je jednoznačná větev logaritmu $\widetilde{\varphi}$, $\Rel(\phi(\beta)) =\log |\widetilde{\varphi}(\beta)|=\log |\widetilde{\varphi}(\alpha)|=\Rel(\phi(\alpha)) $ a $\Img(\phi)$ je jednoznačná větev argumentu $\widetilde{\varphi}$. @@ -121,7 +121,7 @@ \section{Obecná Cauchyho věta a reziduová věta pro cykly} \begin{theorem}[Reziduová pro cykly] Nechť $G\subset\Comp$ je otevřená, $\Gamma$ je cyklus v $G$ a $\Int\Gamma\subset G$. Nechť $K\subset G\setminus\langle\Gamma\rangle$ je konečná a $f\in\Holo(G\setminus K)$. Potom platí \begin{equation} - \int_\Gamma f = 2\pi i \sum_{s\in K}\text{res}_s(f)\cdot\ind_\Gamma(s). + \int_\Gamma f = 2\pi i \sum_{s\in K}\res_s(f)\cdot\ind_\Gamma(s). \tag{RVC} \label{eqn:9.8.rvc} \end{equation} diff --git a/zdroj/prednasky/14_prednaska.tex b/zdroj/prednasky/14_prednaska.tex index 8bc3d8e..0c2cb85 100644 --- a/zdroj/prednasky/14_prednaska.tex +++ b/zdroj/prednasky/14_prednaska.tex @@ -1,48 +1,172 @@ % lednová přednáška \section{Zajímavé funkce} +\setcounter{equation}{0} \subsection{Funkce Gama} -Položme $$\Gamma (z) := \int_0^{+\infty}t^{z-1}e^{-t} \diff t,\ \Rel(z)>0.$$ -Potom platí - -\begin{enumerate} - \item $\Gamma$ je holomorfní funkce na $\Rel(z)>0$. - \begin{proof} - Nechť $z=x+iy$. Potom integrál konverguje jako Lebesgueův, právě když $\Rel(z)>0$. Skutečně, $t^{z-1}=e^{(x+iy-1)\cdot \log t}$, $|t^{z-1}|=t^{x-1}$. Dále z vět o spojitosti a derivování integrálu podle reálných parametrů $x, y$ máme - $$ - \frac{\partial \Gamma}{\partial x}(x,y)=\int_0^{+\infty}t^{z-1}\log (t) e^{-t} \diff t=-i \frac{\partial \Gamma}{\partial y}\text{, protože} - $$ - $$ - \frac{\partial t^{z-1}}{\partial x}=t^{z-1}\log (t)=-i \frac{\partial t^{z-1}}{\partial y}\text{.} - $$ - Z věty \ref{CR} (\nameref{CR}) plyne, že $\Gamma$ je holomorfní na $\Rel(z)>0$. - \end{proof} - \item $\Gamma (z+1)=z \cdot \Gamma (z)$, $\Rel(z)>0$. Speciálně, $\Gamma(n+1)=n!$, $n=0, 1, 2, \dots$ - \begin{proof} - $$\Gamma (1)=\int_0^{+\infty}e^{-t} \diff t=1,$$ - $$\Gamma(z+1)=\int_0^{+\infty}t^{z}e^{-t} \diff t=\underset{=\;0}{\underbrace{[-t^ze^{-t}]_0^{+\infty}}}+z\cdot \Gamma (z).$$ - \end{proof} - \begin{note*} - Z věty o jednoznačnosti stačí rovnost dokázat např. pro reálná $z>0$. - \end{note*} - \item $\Gamma$ lze jednoznačně rozšířit na funkci holomorfní na $\Comp \setminus \{0, -1, -2, -3, \dots \}$, přičemž v bodech $0, -1, -2, -3, \dots$ má $\Gamma$ jednoduché póly a $\text{res}_{-n} \Gamma=\frac{(-1)^n}{n!}$, $n=0, 1, 2, 3, \dots$ - - Víme z 2., že - \begin{equation}\tag{Podíl Gamma}\label{PodilGamma} - \Gamma(z)=\frac{\Gamma(z+1)}{z}\text{ , } \Rel(z)>0 \text{.} - \end{equation} - Protože pravá strana má smysl i pro $0 \geq \Rel(z)>-1$, $z \neq 0$, definujeme pro taková $z$ funkci $\Gamma (z)$ jako $\Gamma(z)=\frac{\Gamma(z+1)}{z}$. - Nyní má ale pravá strana smysl i pro $-1 \geq \Rel(z)>-2$, $z \neq -1, 0$ a pro taková $z$ rozšíříme definici $\Gamma$ opět pomocí \cref{PodilGamma} atd. Navíc máme - $$ - \text{res}_{0} \Gamma (z)=\text{res}_{0}\left(\frac{\Gamma (z+1)}{z} \right)=\Gamma(1)=1; - $$ - $$ - \text{res}_{-1} \Gamma (z)=\text{res}_{-1}\left(\frac{\Gamma (z+2)}{z(z+1)} \right)=-\Gamma(1)=-1; - $$ - $$ - \text{res}_{-n} \Gamma (z)=\text{res}_{-n}\left(\frac{\Gamma (z+n+1)}{z(z+1)\dots (z+n)} \right)=\frac{\Gamma(1)}{n!}\cdot (-1)^n =\frac{(-1)^n}{n!}\text{, } n \in \N_0\text{,} - $$ - protože $\Gamma(z+n+1)=(z+n)(z+n-1) \cdot \dots \cdot z \cdot \Gamma(z)$. -\end{enumerate} + +\begin{definition}[Funkce $\Gamma$] +Položme +\begin{equation}\label{eqn:gamma} + \Gamma (z) := \int_0^{+\infty}t^{z-1}e^{-t} \diff t,\ \Rel(z)>0. +\end{equation} +\end{definition} + +\begin{lemma} +Funkce $\Gamma$ je holomorfní funkce na $\Rel(z)>0$. +\end{lemma} + +\begin{proof} + Nechť $z=x+iy$. Potom integrál konverguje jako Lebesgueův, právě když $\Rel(z)>0$. Skutečně, $t^{z-1}=e^{(x+iy-1)\cdot \log t}$, $|t^{z-1}|=t^{x-1}$. Dále z vět o spojitosti a derivování integrálu podle reálných parametrů $x,\ y$ máme +\begin{align*} + \frac{\partial \Gamma}{\partial x}(x\,,y)&=\int_0^{+\infty}t^{z-1}\log (t) e^{-t} \diff t=-i \frac{\partial \Gamma}{\partial y}\text{, protože}\\ + \frac{\partial t^{z-1}}{\partial x}&=t^{z-1}\log (t)=-i \frac{\partial t^{z-1}}{\partial y}\text{.} +\end{align*} + Z věty \ref{CR} (\nameref{CR}) plyne, že $\Gamma$ je holomorfní na $\Rel(z)>0$. + \end{proof} + +\begin{lemma}\label{lemma:gammaIter} +$\Gamma(z+1)=z\cdot\Gamma(z),\ \Rel(z)>0$. Speciálně, $\Gamma(n+1)=n!,\ n=0,\,1,\,2,\,\ldots$ +\end{lemma} + +\begin{proof} +\begin{align*} + \Gamma(1)&=\int_0^{+\infty}e^{-t}\diff t=1,\\ + \Gamma(z+1)&=\int_0^{+\infty}\underbrace{t^z}_u \underbrace{e^{-t}}_{\frac{dv}{dt}}\diff t \underset{partes}{\underset{per}{=}} \underbrace{\left[-t^z e^{-t}\right]_0^{+\infty}}_{=0}+z\cdot\Gamma(z),\\ + \frac{du}{dt}&=zt^{z-1},\ v:=-e^{-t}. +\end{align*} +\end{proof} + +\begin{note*} +Z věty \cref{thm:jednoznacnostHolo} (\nameref{thm:jednoznacnostHolo}) stačí rovnost dokázat např. pro reálné $z>0$. +\end{note*} + +\begin{lemma} +Funkci $\Gamma$ lze jednoznačně rozšířit na funkci holomorfní na $\Comp\setminus\{0,\,-1,\,-2,\,-3,\,\ldots\}$, přičemž v bodech $0,\,-1,\,-2,\,-3,\,\ldots$ +má $\Gamma$ jednoduché póly a $\res_{-n}\Gamma=(-1)^n/n!,\ n=0,\,1,\,2,\,3,\,\ldots$ +\end{lemma} + +\begin{proof} +Víme z \cref{lemma:gammaIter}, že +\begin{equation}\tag{*}\label{eqn:gammaIterFraction} + \Gamma(z)=\frac{\Gamma(z+1)}{z},\ \Rel(z)>0, +\end{equation} +protože pravá strana má smysl i pro $0\geq\Rel(z)>-1,\ z\neq 0$, definujeme pro taková $z$ funkci $\Gamma(z)$ jako $\frac{\Gamma(z+1)}{z}$. Nyní má ale pravá strana smysl pro $-1\geq\Rel(z)>-2, z\neq-1,\,0$ a pro taková $z$ rozšíříme definici $\Gamma$ opět pomocí \cref{eqn:gammaIterFraction}, atd. Navíc máme +\begin{align*} + \res_0\Gamma(z)&=\res_0\left(\frac{\Gamma(z+1)}{z}\right)=\Gamma(1)=1, \\ + \res_{-1}\Gamma(z)&=\res_{-1}\left(\frac{\Gamma(z+2)}{z(z+1)}\right)=-\Gamma(1)=-1, \\ + \res_{-n}\Gamma(z)&=\res_{-n}\left(\frac{\Gamma(z+n+1)}{z(z+1)\cdots(z+n)}\right)=\frac{\Gamma(1)}{n!}\cdot(-1)^n=\frac{(-1)^n}{n!},\ n \in \N_0, +\end{align*} +protože $\Gamma(z+n+1)=(z+n)(z+n-1) \cdots z \cdot \Gamma(z)$. +\end{proof} \subsection{Riemannova zeta funkce} + +\begin{definition}[Riemannova $\zeta$ funkce] +Pro $\Rel(z)>1$ položme +\begin{equation}\label{eqn:zeta} + \zeta(z):=\sum_{n=1}^\infty \frac{1}{n^z}. +\end{equation} +\end{definition} + +\begin{lemma} +Funkce $\zeta$ je holomorfní funkce na $\Rel z>1$. +\end{lemma} + +\begin{proof} +Skutečně, řada v \cref{eqn:zeta} konverguje absolutně a lokálně stejnoměrně na $\Rel z>1$, protože $$\left|\frac{1}{n^z}\right|=\left|e^{-z\cdot \log(n)}\right|=e^{-\Rel(z)\cdot\log n}\leq\frac{1}{n^{x_0}},$$ +je-li $\Rel (z)\geq x_0>1$ a $\sum_{n=1}^\infty\frac{1}{n^{x_0}}<+\infty$. Z Věty \cref{thm:weierstrass} (\nameref{thm:weierstrass}) plyne\\ $\zeta\in\Holo\big(\{\Rel(z)>1\}\big)$. +\end{proof} + +\begin{definition}[Dirichletova eta funkce] +\begin{equation}\label{eqn:eta} + \eta(z):=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^z},\ \Rel(z)>0. +\end{equation} +\end{definition} + +\begin{lemma} +Funkce $\eta$ je holomorfní na $\Rel(z)>0$. +\end{lemma} + +\begin{proof} +Na $\Rel(z)>1$ je důkaz stejný jako pro $\zeta$. Na $0<\Rel(z)<1$ řada v \cref{eqn:eta} konverguje neabsolutně - pro reálná $z$ je důkaz snadný. +\end{proof} + +\begin{lemma}\label{lemma:etaZetaRelation} Platí +\begin{equation*} + \eta(z)=(1-2^{1-z})\zeta(z),\ \Rel(z)>1. +\end{equation*} +\end{lemma} + +\begin{proof} +Nechť $\Rel(z)>1$. Potom \begin{align*} + \zeta(z)+\eta(z&)\,=\,2\cdot\sum_{n=1}^\infty \frac{1}{(2n-1)^z} \text{ a}\\ + \eta(z)&\underset{konv.}{\underset{abs.}{=}}\sum_{n=1}^\infty \frac{1}{(2n-1)^z}-\sum_{n=1}^\infty\frac{1}{(2n)^z}=\frac{\zeta(z)+\eta(z)}{2}-\frac{1}{2^z}\zeta(z),\ \text{tudíž}\\ + \eta(z&)\,=\,\zeta(z)-2^{1-z}\zeta(z). +\end{align*} +\end{proof} + +\begin{lemma} +$\varphi(z):=1-2^{1-z}$ je celá funkce, která má \emph{nulové body} právě v bodech $$z_k:=1+k\cdot\frac{2\pi i}{\log 2},\ k\in \Z,$$ a to \emph{jednoduché}. +\end{lemma} + +\begin{proof}\mbox{} +\begin{itemize} + \item $2^{1-z}=e^{(1-z\log 2)}=1\iff(1-z)\cdot\log2=2k\pi i,\ k\in \Z$, + \item $\varphi'(z)=2^{1-z}\cdot\log2,\ \varphi'(z_k)=\log2\neq0$. +\end{itemize} +\end{proof} + +\begin{lemma} +Funkci $\zeta$ lze jednoznačně rozšířit na holomorfní funkci na $\Comp\setminus\{1\}$ s jednoduchým pólem v 1 a $\res_1\zeta=1$. +\end{lemma} + +\begin{proof} +Z \cref{lemma:etaZetaRelation} víme, že pro $\Rel(z)>1$ je +\begin{equation}\tag{*}\label{eqn:zetaExpansion} + \zeta(z)=\frac{\eta(z)}{\left(1-2^{1-z}\right)}. +\end{equation} +Protože pravá strana \cref{eqn:zetaExpansion} má smysl i pro $1\geq\Rel(z)>0,\ z\neq z_k$ pro všechna $k\in\Z$, rozšíříme $\zeta$ holomorfně pro taková $z$ pomocí \cref{eqn:zetaExpansion}. Dále z \cref{eqn:zetaExpansion} máme $\res_1\zeta=\frac{\eta(1)}{\varphi'(1)}=\frac{\log2}{\log2}=1$. +Pro další vlastnosti $zeta$ viz např. [CONWAY, J. B.; Functions of one complex variable, Springer, 1978]. +\end{proof} + +\begin{lemma} +Mimo \emph{kritický pás} $\{z\in\Comp|0<\Rel(z)<1\}$ má $\zeta$ +nulové body právě v $-2,\,-4,\,-6,\,$\\$-8,\,\ldots$ \emph{triviální nulové body}. +\end{lemma} +\fbox{Reimannova hypotéza} Je-li $z$ nulový bod funkce $\zeta$ v kritickém pásu, potom $\Rel(z)=\frac{1}{2}$. Jeden z nejslavnějších otevřených problémů v matematice (odměna j milion \$). Má úzkou souvislost s rozložením prvočísel, viz [RIEMANN, B.; On the Number of Prime Numbers less than a Given Quantity, 1859] + +\begin{theorem} [Eulerova]\label{thm:EulerovaV} +Je-li $\Rel(z)>1$, potom +\begin{equation}\label{eqn:eulerovaV1} + \zeta(z)=\prod_{n=1}^\infty\left(\frac{1}{1-p_n^{-z}}\right), +\end{equation} +kde $\{p_n\}_{n=1}^\infty$ je posloupnost všech prvočísel. +\end{theorem} + +\begin{note*} +Zde $\prod_{n=1}^\infty a_n:=\lim_{N\rightarrow+\infty}\prod_{n=1}^N a_n$, má-li pravá strana smysl. +\end{note*} + +\begin{proof} +Pro každé $n\in\N$ platí +$$\frac{1}{1-\underset{\phantom{.}|\cdots|<1}{p_n^{-z}}}=\sum_{m=0}^{+\infty}p_n^{-mz}.$$ +Nechť $n\in\N$. Potom +\begin{equation}\tag{*}\label{eqn:eulerV} + \prod_{k=1}^n\left(\frac{1}{1-p_k^{-z}}\right)=\sum_{j=1}^{+\infty}n_j^{-z} +\end{equation} +jsou-li $n_1,\,n_2,\,\ldots$ všechna přirozená čísla, která se dají rozložit na součin mocnin jen prvočísel $p_1,\,p_2,\,\ldots,\,p_n$. +Zde se využívá jednoznačnosti faktorizace $n_j$ jako součinu provčísel. Pokud v \cref{eqn:eulerV} pošleme $n\rightarrow\infty$, dostaneme \cref{eqn:eulerovaV1}. +\end{proof} + +\subsection{Prvočíselná věta} + +\begin{theorem}[Prvočíselná] \emph{[HADAWARD, J.; POUSSIN, Ch. J. de Pa Valleé; 1896]} +Nechť $\pi(x)$ je počet prvočísel $p\leq x$ pro každé $x>0$. Potom +$$\pi(x)\sim \frac{x}{\log x}\text{, pro }x\rightarrow\infty $$ +$$\text{tzn. }\lim_{x\rightarrow+\infty}\frac{\pi(x)}{\left(\frac{x}{\log x}\right)}=1 +$$ +\end{theorem} + +\begin{note*} +První velký úspěch KA. 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