From d118ee1b3890f2b18b1c68795f0299b23241e9c4 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Tue, 20 Nov 2007 12:25:38 +0100
Subject: [PATCH] Pridany slidy o komplexnich cislech.

---
 slides/Makefile     | 10 +++++
 slides/complex.tex  | 85 +++++++++++++++++++++++++++++++++++++++
 slides/slidemac.tex | 98 +++++++++++++++++++++++++++++++++++++++++++++
 3 files changed, 193 insertions(+)
 create mode 100644 slides/Makefile
 create mode 100644 slides/complex.tex
 create mode 100644 slides/slidemac.tex

diff --git a/slides/Makefile b/slides/Makefile
new file mode 100644
index 0000000..a11f216
--- /dev/null
+++ b/slides/Makefile
@@ -0,0 +1,10 @@
+all: complex.ps
+
+%.dvi: %.tex slidemac.tex
+	csplain $<
+
+%.ps: %.dvi
+	dvips -o $@ -D600 -T250mm,187mm -O-1in,-1in $<
+
+clean:
+	rm -f *~ *.log *.tfm *.*pk *.*gf *.ps *.dvi *.pdf
diff --git a/slides/complex.tex b/slides/complex.tex
new file mode 100644
index 0000000..99d1e5b
--- /dev/null
+++ b/slides/complex.tex
@@ -0,0 +1,85 @@
+\input slidemac.tex
+
+\language=\czech
+\chyph
+
+\def\i{{\rm i}}
+\def\bb{\fam\bbfam}
+\advance\parskip by 10pt
+
+\slide{Komplexní èísla: Slo¾kový tvar}
+
+Definice: ${\bb C} = \{ a+b\i \mid a,b\in {\bb R} \}$.
+
+Sèítání: $(a+b\i)\pm(p+q\i) = (a\pm p) + (b\pm q)\i$.
+
+Násobení: $(a+b\i)\cdot(p+q\i) = ap + (aq+bp)\i + bq\i^2 = (ap-bq)+(aq+bp)\i$.
+
+\noindent\qquad Pro $\alpha\in{\bb R}$ je $\alpha(a+b\i) = \alpha a + \alpha b\i$.
+
+Komplexní sdru¾ení: $\overline{a+b\i} = a-b\i$.
+
+\noindent\qquad $\overline{\overline x} = x$, $\overline{x\pm y} = \overline{x} \pm \overline{y}$, $\overline{x\cdot y} = \overline x \cdot \overline y$, $x\cdot \overline x \in {\bb R}$.
+
+Absolutní hodnota: $\vert z \vert = \sqrt{z\cdot\overline{z}}$, tak¾e $\vert a+b\i \vert = \sqrt{a^2+b^2}$.
+
+\noindent\qquad Také $\vert \alpha x \vert = \alpha \cdot \vert x \vert$.
+
+Dìlení: $x/y = (x\cdot \overline{y}) / (y \cdot \overline{y})$.
+
+\endslide
+
+\slide{Komplexní èísla: Gau{\ss}ova rovina a goniometrický tvar}
+
+Komplexním èíslùm pøiøadíme body v~${\bb R}^2$: $a+b\i \leftrightarrow (a,b)$.
+
+$\vert x\vert$ je vzdálenost od~bodu $(0,0)$.
+
+$\vert x\vert = 1$ pro èísla le¾ící na~jednotkové kru¾nici ({\sit komplexní jednotky\/}).
+
+\noindent\qquad Pak platí $x=\cos\varphi + \i\sin\varphi$ pro nìjaké $\varphi\in\left[ 0,2\pi \right)$.
+
+Pro libovolné $x\in{\bb C}$: $x=\vert x \vert \cdot (\cos\varphi(x) + \i\sin\varphi(x))$.
+
+\noindent\qquad Èíslu $\varphi(x)\in\left[ 0,2\pi \right)$ øíkáme {\sit argument\/} èísla~$x$, nìkdy znaèíme $\mathop{\rm arg} x$.
+
+Navíc $\varphi({\overline{x}}) = -\varphi(x)$.
+
+\endslide
+
+\slide{Komplexní èísla: Exponenciální tvar}
+
+Eulerova formule: $e^{i\varphi} = \cos\varphi + \i\sin\varphi$.
+
+Ka¾dé $x\in{\bb C}$ lze tedy zapsat jako $\vert x\vert \cdot e^{\i\cdot\varphi(x)}$.
+
+Násobení: $xy = \left(\vert x\vert\cdot e^{\i\cdot\varphi(x)}\right) \cdot \left(\vert y\vert\cdot e^{\i\cdot\varphi(y)}\right) =
+\vert x\vert \cdot \vert y\vert \cdot e^{\i\cdot(\varphi(x) + \varphi(y))}$.
+
+\noindent\qquad (absolutní hodnoty se násobí, argumenty sèítají)
+
+Umocòování: $x^\alpha = \left(\vert x\vert\cdot e^{\i\cdot\varphi(x)}\right)^n = {\vert x\vert}^n\cdot e^{\i n \varphi(x)}$.
+
+Odmocòování: $\root n\of x = {\vert x\vert}^{1/n} \cdot e^{\i\cdot \varphi(x)/n}$.
+
+\noindent\qquad \dots\ pozor, odmocnina není jednoznaèná: $1^4=(-1)^4=\i^4=(-\i)^4=1$.
+
+\endslide
+
+\slide{Komplexní èísla: Odmocniny z~jednièky}
+
+Je-li nìjaké $x\in{\bb C}$ $n$-tou odmocninou z~jednièky, musí platit:
+
+\noindent\qquad $\vert x \vert = 1$, tak¾e $x=e^{\i\varphi}$ pro nìjaké~$\varphi$,
+
+\noindent\qquad $e^{\i\varphi n} = \cos{\varphi n} + \i\sin\varphi n = 1$, proèe¾ $\varphi n = 2k\pi$ pro nìjaké $k\in{\bb Z}$.
+
+Z~toho plyne: $\varphi = 2k\pi/n$
+
+\noindent\qquad (pro $k=0,\ldots,n-1$ dostáváme rùzné $n$-té odmocniny).
+
+Obecné odmocòování: $\root n \of x = {\vert x\vert}^{1/n} \cdot e^{\i\varphi(x)/n} \cdot u$, kde $u=\root n\of 1$.
+
+\endslide
+
+\end
diff --git a/slides/slidemac.tex b/slides/slidemac.tex
new file mode 100644
index 0000000..6bda976
--- /dev/null
+++ b/slides/slidemac.tex
@@ -0,0 +1,98 @@
+\hsize=230mm
+\vsize=157mm
+\voffset=10mm
+\hoffset=10mm
+\nopagenumbers
+
+\font\srm=csss12 scaled \magstep3
+\font\stit=csb12 scaled \magstep3
+\font\sem=csssbx12 scaled \magstep3
+\font\sit=csssi12 scaled \magstep3
+\font\stt=cstt12 scaled \magstep3
+\font\stitle=cscsc12 scaled \magstep4
+
+\baselineskip=25pt
+\lineskip=2.1pt
+\parindent=0pt
+\parskip=4pt
+\abovedisplayskip=24pt plus 6pt minus 18pt
+\abovedisplayshortskip=0pt plus 6pt
+\belowdisplayskip=24pt plus 6pt minus 18pt
+\belowdisplayshortskip=14pt plus 6pt minus 8pt
+
+\def\em#1{{\sem #1}}
+\srm
+
+\newfam\bbfam
+\font\rmfont=cmr10 scaled \magstep4
+\font\ttfont=cmtt10 scaled \magstep4
+\font\ifont=cmmi10 scaled \magstep4
+\font\symfont=cmsy10 scaled \magstep4
+\font\exfont=cmex10 scaled \magstep4
+\font\rmfonts=cmr7 scaled \magstep4
+\font\ifonts=cmmi7 scaled \magstep4
+\font\symfonts=cmsy7 scaled \magstep4
+\font\exfonts=cmex7 scaled \magstep4
+\font\rmfontss=cmr5 scaled \magstep4
+\font\ifontss=cmmi5 scaled \magstep4
+\font\symfontss=cmsy5 scaled \magstep4
+\font\exfontss=cmex5 scaled \magstep4
+\font\bbfont=bbold10 scaled \magstep4
+\textfont0=\rmfont
+\textfont1=\ifont
+\textfont2=\symfont
+\textfont3=\exfont
+\textfont\bbfam=\bbfont
+\scriptfont0=\rmfonts
+\scriptfont1=\ifonts
+\scriptfont2=\symfonts
+\scriptfont3=\exfonts
+\scriptscriptfont0=\rmfontss
+\scriptscriptfont1=\ifontss
+\scriptscriptfont2=\symfontss
+\scriptscriptfont3=\exfontss
+
+\def\slide#1{\begingroup
+\ifx:#1:\else
+\line{\vrule width 0pt height 25pt depth 4pt \stit #1\hfill}
+\medskip
+\hrule height 2pt
+\bigskip
+\bigskip
+\fi
+}
+\def\endslide{\vfill\eject\endgroup}
+
+\def\\{\hfil\break}
+\def\itemize#1{\par{\advance\leftskip by 35pt{\parskip=5pt #1}\par}}
+\def\:{\par\leavevmode\llap{$\bullet$\hskip 7pt}}
+\def\>{\par\leavevmode\llap{$\circ$\hskip 7pt}}
+\def\<#1>{\hbox{\sit #1\/}}
+\def\bbold{\bbfont\fam\bbfam}
+\def\O{{\cal O}}
+
+\newcount\itemcount
+\def\interlistskip{\medskip}
+\def\algo{
+\begingroup
+\let\:=\algoitem
+\let\*=\algohang
+\parskip=1pt plus 1pt minus 0.3pt
+\rightskip=2em
+\itemcount=0
+\interlistskip
+}
+\def\endalgo{\interlistskip\endgroup}
+\def\algoitem{\par
+\parindent=2em
+\hangindent=4em
+\hangafter=1
+\advance\itemcount by 1
+\leavevmode\hbox to 2em{\hss \the\itemcount. }%
+\futurelet\next\algoitemh}
+\def\algoitemh{\ifx\next:\let\next=\algohang\else\let\next=\relax\fi\next}
+\def\algohang:{\advance\hangindent by 2em \hskip 2em\futurelet\next\algoitemh}
+
+\def\popcolor{\special{color pop}}
+\def\pushcolor#1{\special{color push #1}}
+\def\color#1{\pushcolor{#1}\aftergroup\popcolor}
-- 
2.39.2