%% 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=Sunday, February 13, 2005 19:15:14} %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 - Zbierka 4\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{Zbierka \'{u}loh} \begin{center} \begin{tabular}{|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{mcindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{mcindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Zbierka}{}{}{KZ.tex}}% %BeginExpansion \msihyperref{Zbierka}{}{}{KZ.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \section{Deriv\'{a}cia funkcie komplexnej premennej, analytick\'{e} (holomorfn\'{e}) funkcie} \textbf{Pr\'{\i}klad 1. }Vypo\v{c}\'{\i}tajte deriv\'{a}ciu funkcie $f\left( z\right) =\frac{3i}{2i-z}.\;% \CustomNote{Answer}{$D(f)=\mathbf{C\setminus }\left\{ 2i\right\} =D\left( f\,\,^{\prime }\right) ,\,f\,\,^{\prime }\left( z\right) =\frac{3i}{\left( 2i-z\right) ^{2}}.$}$ \textbf{Pr\'{\i}klad 2. }Dan\'{a} je funkcia $f\left( z\right) =\frac{iz-1}{% iz^{2}+1+i}.$ N\'{a}jdite: \textbf{a. }defini\v{c}n\'{y} obor; $% \CustomNote{Answer}{$\mathbf{C\setminus }\left\{ \sqrt[4]{2}e^{\frac{3\pi }{8% }i},\,\sqrt[4]{2}e^{\frac{11\pi }{8}i}\right\} .$}$ \textbf{b. }$f\,\,^{\prime },\,f\,\,^{\prime }\left( i\right) $ \ $% \CustomNote{Answer}{$f\,\,^{\prime }\left( z\right) =\frac{z^{2}+2iz+1-i}{% \left( iz^{2}+1+i\right) ^{2}},\,f\,\,^{\prime }\left( i\right) =-4+i.$}$ \textbf{Pr\'{\i}klad 3. }N\'{a}jdite n-t\'{u} deriv\'{a}ciu funkcie $f\left( z\right) =\frac{1}{2z-i}.$ \ $% \CustomNote{Answer}{$f^{\left( n\right) }\left( z\right) =\frac{\left( -2\right) ^{n}n!}{\left( 2z-i\right) ^{n+1}},\,z\neq \frac{i}{2}.$}$ \textbf{Pr\'{\i}klad 4. }Dan\'{a} je funkcia $f\left( z\right) =\frac{1}{% \left( z-2i\right) ^{n}}.$ Vypo\v{c}\'{\i}tajte $f^{\left( n\right) },\,f^{\left( 3\right) }\left( 3i\right) $\ \ $% \CustomNote{Answer}{$f^{\left( n\right) }\left( z\right) =\left( -1\right) ^{n}n\left( n+1\right) \left( n+2\right) \dots \left( 2n-1\right) \left( z-2i\right) ^{-2n},$% \par $\,z\neq 2i,\,f^{\left( 3\right) }\left( 3i\right) =60.$}$ \textbf{Pr\'{\i}klad 5. }Zistite, v ktor\'{y}ch bodoch mno\v{z}iny $\mathbf{C% }$ existuje deriv\'{a}cia funkcie $f\left( z\right) =f\left( x+iy\right) =x^{2}+y^{2}+i2xy.$ Ak sa d\'{a}, vypo\v{c}\'{\i}tajte $f\,\,^{\prime }\left( 1\right) .$ $% \CustomNote{Answer}{% Existuje len v bodoch na re\'{a}lnej \v{c}\'{\i}selnej osi; \ \par $f\,\,^{\prime }\left( 1\right) =f\,\,^{\prime }\left( 1+0i\right) =2.$}$ \ V \'{u}loh\'{a}ch 6 - 15. pre funkciu $f$ \ \ \textbf{a.} zistite, kde existuje deriv\'{a}cia, \textbf{b.} n\'{a}jdite $f\,\,^{\prime }$ v bodoch, kde existuje, \textbf{c.} vy\v{s}etrite, kde je $f$ analytick\'{a} (holomorfn\'{a}) \textbf{Pr\'{\i}klad 6. }$f\left( z\right) =x^{2}+iy^{2}.$ \ $% \CustomNote{Answer}{\textbf{a. }$f\,\,^{\prime }$ existuje na \par $M=\left\{ z\in \mathbf{C}:\func{Im}z=\func{Re}z\right\} ,$% \par \textbf{b. }$f\,\,^{\prime }\left( z\right) =f\,\,^{\prime }\left( x+iy\right) =2x,$% \par \textbf{c. }nie je analytick\'{a} v \v{z}iadnom bode}$ \textbf{Pr\'{\i}klad 7. }$f\left( z\right) =\left| z\right| .$ \ $% \CustomNote{Answer}{\textbf{a. }$f\,\,^{\prime }$ neexistuje v \v{z}iadnom bode, \par \textbf{b. }$f\,\,^{\prime }\left( z\right) \nexists ,$% \par \textbf{c. }nie je analytick\'{a} v \v{z}iadnom bode.}$ \textbf{Pr\'{\i}klad 8. }$f\left( z\right) =z^{3}+z.$ \CustomNote{Answer}{% \ \textbf{a. }$f\,\,^{\prime }$ existuje na $\mathbf{C},$% \par \textbf{b. }$f\,\,^{\prime }\left( z\right) =3z^{2}+1,$% \par \textbf{c. }je analytick\'{a} na $\mathbf{C.}$} \textbf{Pr\'{\i}klad 9. }$f\left( z\right) =z\func{Re}z.$ \ $% \CustomNote{Answer}{\textbf{a. }$f\,\,^{\prime }$ existuje len v bode $z=0,$% \par \textbf{b. }$f\,\,^{\prime }\left( 0\right) =0,$% \par \textbf{c. }nie je analytick\'{a} v \v{z}iadnom bode}$ \textbf{Pr\'{\i}klad 10. }$f\left( z\right) =z\func{Im}z.$ \ $% \CustomNote{Answer}{\textbf{a. }$f\,\,^{\prime }$ existuje len v bode $z=0,$% \par \textbf{b. }$f\,\,^{\prime }\left( 0\right) =0,$% \par \textbf{c. }nie je analytick\'{a} v \v{z}iadnom bode.}$ \textbf{Pr\'{\i}klad 11. }$f\left( z\right) =f\left( x+iy\right) =\left( x^{3}-3xy^{2}\right) +i\left( 3x^{2}y-y^{3}\right) .$\ $% \CustomNote{Answer}{\textbf{a. }$f\,\,^{\prime }$ existuje na $\mathbf{C},$% \par \textbf{b.}$f\,\,^{\prime }\left( z\right) =f\,\,^{\prime }\left( x+iy\right) =\left( 3x^{2}-3y^{2}\right) +i\left( 6xy\right) ,$% \par \textbf{c. }je analytick\'{a} na $\mathbf{C}$}$ \textbf{Pr\'{\i}klad 12. }$f\left( z\right) =f\left( x+iy\right) =\left( 2xy+2x-1\right) +i\left( y^{2}-x^{2}+2y\right) .% \CustomNote{Answer}{\textbf{a. }$f\,\,^{\prime }$ existuje na $\mathbf{C},$% \par \textbf{b. }$f\,\,^{\prime }\left( z\right) =f\,\,^{\prime }\left( x+iy\right) =\left( 2y+2\right) -i\left( 2x\right) ,$% \par \textbf{c. }je analytick\'{a} na $\mathbf{C.}$}$ \textbf{Pr\'{\i}klad 13. }$f\left( z\right) =\left( e^{x}\cos y\right) -i\left( e^{x}\sin y\right) .$ $% \CustomNote{Answer}{\textbf{a. }$f\,\,^{\prime }$ neexistuje v \v{z}iadnom bode, \par \textbf{b. }$f\,\,^{\prime }\left( z\right) \nexists ,$% \par \textbf{c. }nie je analytick\'{a} v \v{z}iadnom bode.}$ \textbf{Pr\'{\i}klad 14. }$f\left( z\right) =\left( \func{Re}z\right) ^{2}+i\left( \func{Im}z\right) ^{2}.$ \ $% \CustomNote{Answer}{\textbf{a. }$f\,\,^{\prime }$ existuje na \par $M=\left\{ z\in \mathbf{C}:\func{Im}z=\func{Re}z\right\} ,$% \par \textbf{b. }$f\,\,^{\prime }\left( z\right) =f\,\,^{\prime }\left( x+iy\right) =2x,$% \par \textbf{c. }nie je analytick\'{a} v \v{z}iadnom bode}$ \textbf{Pr\'{\i}klad 15. }$f\left( z\right) =f\left( x+iy\right) =\left( x+y\right) +i\sin \left( x+y\right) .$ $% \CustomNote{Answer}{\textbf{a. }$f\,\,^{\prime }$ neexistuje v \v{z}iadnom bode, \par \textbf{b. }$f\,\,^{\prime }\left( z\right) \nexists ,$% \par \textbf{c. }nie je analytick\'{a} v \v{z}iadnom bode.}$ V \'{u}{}loh\'{a}ch 16 - 22 \ n\'{a}jdite analytick\'{u} (holomorfn\'{u}) funkciu $f(z)=f\left( x+iy\right) =u\left( x,y\right) +iv\left( x,y\right) ,$ ak je dan\'{a} jej jedna zlo\v{z}ka a pr\'{\i}padne funk\v{c}n\'{a} hodnota v jednom bode: \textbf{Pr\'{\i}klad 16. }$u\left( x,y\right) =x^{3}-3xy^{2},\,f\left( i\right) =0.$ \ $% \CustomNote{Answer}{$f\left( z\right) =f\left( x+iy\right) =\left( x^{3}-3xy^{2}\right) +i\left( 3x^{2}y-y^{3}+1\right) $}$ \textbf{Pr\'{\i}klad 17. }$u\left( x,y\right) =x^{2}-y^{2}+xy,\,f\left( 0\right) =0.$ \ $% \CustomNote{Answer}{$f\left( z\right) =f\left( x+iy\right) =\left( x^{2}-y^{2}+xy\right) +i\left( 2xy+\frac{y^{2}}{2}-\frac{x^{2}}{2}\right) .$} $ \textbf{Pr\'{\i}klad 18. }$v\left( x,y\right) =x^{2}-y^{2}-3x+2xy,\,u\left( 2,1\right) =0.$ $% \CustomNote{Answer}{$u\left( x,y\right) =x^{2}-y^{2}-2xy+3y-2.$}$ \textbf{Pr\'{\i}klad 19. }$v\left( x,y\right) =2e^{x}\sin y,\,f(0)=1.$ \ $% \CustomNote{Answer}{$f\left( z\right) =f\left( x+iy\right) =\left( 2e^{x}\cos y-1\right) +i\left( 2e^{x}\sin y\right) .$}$ \textbf{Pr\'{\i}klad 20. }$v\left( x,y\right) =\func{arctg}\left( \frac{Y}{x}% \right) ,\,x>0,\,f(1)=1.$ $% \CustomNote{Answer}{$f(z)=f\left( x+iy\right) =\left( \ln \left( \sqrt{% x^{2}+y^{2}}\right) +1\right) +i\func{arctg}\left( \frac{Y}{x}\right) .$}$ \textbf{Pr\'{\i}klad 21. }$v\left( x,y\right) =\ln \left( x^{2}+y^{2}\right) +x-2y.$ \ $% \CustomNote{Answer}{$u\left( x,y\right) =-2\func{arctg}\left( \frac{y}{x}% \right) -y-2x+k.$}$ \textbf{Pr\'{\i}klad 22. }$u\left( x,y\right) =\frac{x}{x^{2}+y^{2}}-2y.$ \ $% \CustomNote{Answer}{$v\left( x,y\right) =-\frac{y}{x^{2}+y^{2}}+2x+k.$}$ V \'{u}loh\'{a}ch 23 - 24 zistite, \v{c}i s\'{u} dan\'{e} funkcie harmonick% \'{e}: \textbf{Pr\'{\i}klad 23. }$u\left( x,y\right) =\frac{x}{x^{2}+y^{2}}-2y.$% \CustomNote{Answer}{% Je harmonick\'{a}.} \textbf{Pr\'{\i}klad 24. }$v\left( x,y\right) =$ $x^{2}$ + $y^{3}.$ \CustomNote{Answer}{% Nie je harmonick\'{a}.} V pr\'{\i}kladoch 25 - 27 n\'{a}jdite harmonicky zdru\v{z}en\'{u} funkciu k danej funkcii: \textbf{Pr\'{\i}klad 25. }$u\left( x,y\right) =xy,\,v\left( 1,2\right) =% \frac{1}{2}.$ \ $% \CustomNote{Answer}{$v\left( x,y\right) =\frac{y^{2}-x^{2}}{2}-1.$}$ \textbf{Pr\'{\i}klad 26. }$u\left( x,y\right) =x^{2}-y^{2}+xy.$ $% \CustomNote{Answer}{$v\left( x,y\right) =2xy+\frac{y^{2}}{2}-\frac{x^{2}}{2}% +k.$}$ \textbf{Pr\'{\i}klad 27. }$u\left( x,y\right) =e^{x}\left( x\cos y-y\sin y\right) .$ \ $% \CustomNote{Answer}{$v\left( x,y\right) =e^{x}\left( x\sin y+y\cos y\right) +k.$}$ \begin{center} \begin{tabular}{|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{mcindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{mcindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Zbierka}{}{}{KZ.tex}}% %BeginExpansion \msihyperref{Zbierka}{}{}{KZ.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \section{Zbierka \'{u}loh} \end{document}