%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Tuesday, October 23, 2007 07:38:39} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za III online - Diferenci\U{e1}lny po\U{10d}et funkci\U{ed} komplexnej premennej - Deriv\U{e1}cia funkcie komplexnej premennej\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Diferenci\'{a}lny po\v{c}et funkci\'{\i} komplexnej premennej} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{mcindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{mcindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{K2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{K2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}c}{}{}{K22.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}c}{}{}{K22.tex}% %EndExpansion %TCIMACRO{\hyperref{a str\'{a}nka}{}{}{K22.tex}}% %BeginExpansion \msihyperref{a str\'{a}nka}{}{}{K22.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{Ot2.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{Ot2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{Cv2.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{Cv2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Ind.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Ind.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \section{Deriv\'{a}cia funkcie komplexnej premennej} \begin{definition} Nech $f:A\left( \subset \mathbf{C}\right) \longrightarrow \mathbf{C}$ je funkcia komplexnej premennej, $\,A$ je otvoren\'{a}, $a\in A$. Ak existuje kone\v{c}n\'{a} limita $\lim_{z\longrightarrow a}\frac{f\left( z\right) -f\left( a\right) }{z-a},$ potom t\'{u}to limitu naz\'{y}vame \emph{deriv% \'{a}cia\label{1}} funkcie $f$ v bode $a,$ ozna\v{c}ujeme $f\,^{\prime }\left( a\right) $ a hovor\'{\i}me, \v{z}e funkcia $f$ je \emph{diferencovate% \v{l}n\'{a}} v bode $a$. Ak je funkcia $f$ diferencovate\v{l}n\'{a} v ka\v{z}% dom bode z $A$, hovor\'{\i}me, \v{z}e $f$ je \emph{diferencovate\v{l}n\'{a} funkcia\label{2}} a funkciu \[ f\,^{\prime }:A\left( \subset \mathbf{C}\right) \longrightarrow \mathbf{C}% ,\,f\,^{\prime }\left( a\right) =\lim_{z\longrightarrow a}\frac{f\left( z\right) -f\left( a\right) }{z-a} \]% naz\'{y}vame \emph{deriv\'{a}ciou} funkcie $f$. \end{definition} Preto\v{z}e defin\'{\i}cia deriv\'{a}cie funkcie komplexnej premennej v bode je tak\'{a} ist\'{a} ako pre funkciu re\'{a}lnej premennej, platia v\v{s}% etky pravidl\'{a}, ktor\'{e} platili pre derivovanie funkci\'{\i} re\'{a}% lnej premennej, ako aj v\v{s}etky vety o diferencovate\v{l}nosti, napr\'{\i}% klad diferencovate\v{l}nos\v{t} funkcie komplexnej premennej $f\left( z\right) $ v nejakom bode z defini\v{c}n\'{e}ho oboru implikuje spojitos\v{t} funkcie $f$ v tomto bode. \begin{example} N\'{a}jdite deriv\'{a}ciu funkcie \[ f:\mathbf{C}\setminus \left\{ \frac{4}{3}i\right\} \longrightarrow \mathbf{C}% ,\,f\left( z\right) =\frac{z-2i}{3iz+4}. \] \end{example} \begin{solution} Preto\v{z}e funkcia $f$ je spojit\'{a} na celom defini\v{c}nom obore, tak m% \'{a}me \[ f\,^{\prime }:\mathbf{C}\setminus \left\{ \frac{4}{3}i\right\} \longrightarrow \mathbf{C},\,f\,^{\prime }\left( z\right) =\frac{-2}{\left( 3iz+4\right) ^{2}}.\square \] \end{solution} \begin{example} N\'{a}jdite deriv\'{a}ciu funkcie \[ f:\mathbf{C}\setminus \left\{ -\frac{2+i}{3}\right\} \longrightarrow \mathbf{% C},\,f\left( z\right) =\ln \left( 2+i+3z\right) . \] \end{example} \begin{solution} Preto\v{z}e funkcia $\ln z$ nie je spojit\'{a} pre re\'{a}lne z\'{a}porn\'{e} \v{c}\'{\i}sla, deriv\'{a}cia $f\,^{\prime }\left( z\right) $ pre tieto hodnoty neexistuje. Ak $z=x+iy,$ potom \[ 2+i+3z=2+3x+i\left( 1+3y\right) \]% a toto \v{c}\'{\i}slo je re\'{a}lne z\'{a}porn\'{e} vtedy a len vtedy ak $y=-% \frac{1}{3}$ a $x<-\frac{2}{3},$ to znamen\'{a}, \v{z}e $f$ nie je spojit% \'{a} na mno\v{z}ine \[ A_{1}=\left\{ z\in \mathbf{C};\,\func{Re}z<-\frac{2}{3}\wedge \func{Im}z=-% \frac{1}{3}\right\} , \]% teda $f$ je diferencovate\v{l}n\'{a} na mno\v{z}ine \[ M=\mathbf{C}\,\setminus \left[ A_{1}\cup \left\{ -\frac{2+i}{3}\right\} % \right] =\mathbf{C}\,\setminus \left\{ z\in \mathbf{C};\,\func{Re}z\leq -% \frac{2}{3}\wedge \func{Im}z=-\frac{1}{3}\right\} , \]% a jej deriv\'{a}cia je \[ f\,^{\prime }:M\longrightarrow \mathbf{C},\,f\,\,^{\prime }\left( z\right) =% \frac{3}{2+i+3z}.\,\square \] \end{solution} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{mcindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{mcindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{K2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{K2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}c}{}{}{K22.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}c}{}{}{K22.tex}% %EndExpansion %TCIMACRO{\hyperref{a str\'{a}nka}{}{}{K22.tex}}% %BeginExpansion \msihyperref{a str\'{a}nka}{}{}{K22.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{Ot2.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{Ot2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{Cv2.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{Cv2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Ind.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Ind.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za III} \section{Diferenci\'{a}lny po\v{c}et funkci\'{\i} komplexnej premennej} \end{document}