%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %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 19:12:06} %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 1\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{Komplexn\'{e} \v{c}\'{\i}sla. Z\'{a}kladn\'{e} vlastnosti, limita a spojitos\v{t} funkci\'{\i} komplexnej premennej} \subsection{Komplexn\'{e} \v{c}\'{\i}sla.} V cvi\v{c}eniach 1 - 5 n\'{a}jdite goniometrick\'{y} tvar komplexn\'{e}ho \v{c}\'{\i}sla $z,$ ak \textbf{Pr\'{\i}klad 1. }$z=3.\;% \CustomNote{Answer}{$z=3\left( \cos 0+i\sin 0\right) .$}$ \textbf{Pr\'{\i}klad 2. }$z=-7.% \CustomNote{Answer}{$\;z=7\left( \cos \pi +i\sin \pi \right) .$}$ \textbf{Pr\'{\i}klad 3. }$z=1-i\sqrt{3}.\;% \CustomNote{Answer}{$z=2\left( \cos \left( -\frac{\pi }{3}\right) +i\sin \left( -\frac{\pi }{3}\right) \right) .$}$ \textbf{Pr\'{\i}klad 4. }$z=-1+i.\;% \CustomNote{Answer}{$z=\sqrt{2}\left( \cos \left( \frac{3\pi }{4}\right) +i\sin \left( \frac{3\pi }{4}\right) \right) .$}$ \textbf{Pr\'{\i}klad 5. }$z=-2i.\;% \CustomNote{Answer}{$z=2\left( \cos \left( -\frac{\pi }{2}\right) +i\sin \left( -\frac{\pi }{2}\right) \right) .$}$ V cvi\v{c}eniach 6 - 8 n\'{a}jdite algebraick\'{y} tvar komplexn\'{e}ho \v{c}% \'{\i}sla $z$\textbf{\ } \textbf{Pr\'{\i}klad 6. \ }$z=2\left( \cos \left( \frac{2\pi }{3}\right) +i\sin \left( \frac{2\pi }{3}\right) \right) .\;% \CustomNote{Answer}{$z=-1+i\sqrt{3}.$}$ \textbf{Pr\'{\i}klad 7. \ }$z=2\left( \cos \left( \frac{\pi }{6}\right) +i\sin \left( \frac{\pi }{6}\right) \right) .% \CustomNote{Answer}{$z=\sqrt{3}+i.$}\;$ \textbf{Pr\'{\i}klad 8. \ }$z=5\left( \cos \left( \frac{\pi }{4}\right) +i\sin \left( \frac{\pi }{4}\right) \right) .\;% \CustomNote{Answer}{$z=\frac{5\sqrt{2}}{2}\left( 1+i\right) .$}$ Vypo\v{c}\'{\i}tajte: \textbf{Pr\'{\i}klad 9. }$\frac{2-3i}{3+4i}.\;% \CustomNote{Answer}{$-\frac{6}{25}-\frac{17}{25}i.$}$ \textbf{Pr\'{\i}klad 10. }$\frac{1+i}{1-i}+2i^{19}.\;% \CustomNote{Answer}{$-i.$}$ \textbf{Pr\'{\i}klad 11. }$2i-\frac{\overline{2-4\bar{\imath}}}{2}.\;% \CustomNote{Answer}{$-1.$}$ \textbf{Pr\'{\i}klad 12. \ \ a. \ }$i^{16}.\;% \CustomNote{Answer}{$1.$}$ \qquad \qquad \qquad \textbf{b. }$i^{-9}.\;% \CustomNote{Answer}{$-i.$}$ \qquad \qquad \qquad \textbf{c. }$\left( -1+i\right) ^{4}.\;% \CustomNote{Answer}{$-4.$}$ \qquad \qquad \qquad \textbf{d. }$\left( \frac{1-i}{1+\sqrt{3}i}\right) ^{24}.\;% \CustomNote{Answer}{$\frac{1}{2^{12}}.$}$ \textbf{Pr\'{\i}klad 13. }Vypo\v{c}\'{\i}tajte $z_{1}\cdot z_{2},\,\frac{% z_{1}}{z_{2}},\,$ ak: \textbf{a. }$z_{1}=1+5i,\,z_{2}=3+i.% \CustomNote{Answer}{$\;-2+16i,\,\frac{4}{5}+\frac{7}{5}i.$}$ \textbf{b. }$z_{1}=3\left( \cos \left( \frac{\pi }{3}\right) +i\sin \left( \frac{\pi }{3}\right) \right) ,\,z_{2}=\sqrt{3}+i.\;% \CustomNote{Answer}{$6i,\frac{3}{4}\left( \,\sqrt{3}+i\right) .$}$ \textbf{Pr\'{\i}klad 14. }Vypo\v{c}\'{\i}tajte $z^{5},\,$ak $z=\sqrt{3}+i.\;% \CustomNote{Answer}{$-16\sqrt{3}+16i.$}$ V pr\'{\i}kladoch 14 - 19 n\'{a}jdite v\v{s}etky rie\v{s}enia binomickej rovnice (rie\v{s}enia zobrazte aj graficky) \textbf{Pr\'{\i}klad 15. \ }$z^{4}-1=0.\;% \CustomNote{Answer}{$1,\,i,\,-1,\,-i.$}$ \textbf{Pr\'{\i}klad 16. }$z^{2}-i=0.\;% \CustomNote{Answer}{$\frac{\sqrt{2}}{2}\left( 1+\,i\right) ,\,-\frac{\sqrt{2}% }{2}\left( 1+\,i\right) .$}$ \textbf{Pr\'{\i}klad 17. }$z^{3}+1=0.\;% \CustomNote{Answer}{$\frac{1}{2}\left( 1+\sqrt{3}\,i\right) ,\,-1,\,\frac{1}{% 2}\left( 1-\sqrt{3}\,i\right) .$}$ \textbf{Pr\'{\i}klad 18. }$z^{2}+i=0.\;% \CustomNote{Answer}{$\frac{\sqrt{2}}{2}\left( 1-\,i\right) ,\,\frac{\sqrt{2}% }{2}\left( -1+\,i\right) .$}$ \textbf{Pr\'{\i}klad 19. }$z^{3}-i=0.\;% \CustomNote{Answer}{$\frac{\sqrt{3}}{2}+\frac{1}{2}\,i,\,-\frac{\sqrt{3}}{2}+% \frac{1}{2}\,i,\,-\,i.$}$ \textbf{Pr\'{\i}klad 20. }$\left( \frac{z-1}{z+1}\right) ^{2}=2i.\;% \CustomNote{Answer}{$-1+2\,i,\,-\frac{1}{5}-\frac{2}{5}\,i.$}$ V \'{u}loh\'{a}ch 21 - 31 zistite, ak\'{a} mno\v{z}ina je ur\v{c}en\'{a} dan% \'{y}m vz\v{t}ahom. Jej obraz na\v{c}rtnite v komplexnej rovine. \textbf{Pr\'{\i}klad 21. }$\left| z-z_{0}\right| =r,\,\,r>0,\,\,z_{0}$ je pevn\'{y} bod. \CustomNote{Answer}{% Kru\v{z}nica so stredom $z_{0}$ a polomerom $r.$} \textbf{Pr\'{\i}klad 22. }$\left| z-z_{1}\right| =\left| z-z_{2}\right| ,\,z_{1}\neq z_{2}$ s\'{u} pevn\'{e} body. $\mathbf{% \CustomNote{Answer}{$2x\left( x_{2}-x_{1}\right) +2y\left( y_{2}-y_{1}\right) +x_{1}^{2}-x_{2}^{2}+y_{1}^{2}-y_{2}^{2}=0,\,$% \par je to symetr\'{a}la \'{u}se\v{c}ky, s koncov\'{y}mi bodmi \par $z_{1},\,z_{2}$}}$ \textbf{Pr\'{\i}klad 23. }$\left| z+i\right| +\left| z-i\right| <4.% \CustomNote{Answer}{$\,$Vn\'{u}tro elipsy \ $\frac{x^{2}}{3}+\frac{y^{2}}{4}% =1.$}$ \textbf{Pr\'{\i}klad 24. }$\left| z+2\right| >1.$ $% \CustomNote{Answer}{% Vonkaj\v{s}ok kru\v{z}nice so stredom $S=\left( -2;0\right) $% \par a polomerom $r=1.$}$ \textbf{Pr\'{\i}klad 25. }$\left| z-2\right| <\left| z\right| .\,$ $% \CustomNote{Answer}{% Polrovina \ $\func{Re}z>1.$}$ \textbf{Pr\'{\i}klad 26. }$\left| z-1\right| \geq 2\left| z-i\right| .$ $% \CustomNote{Answer}{% Uzavret\'{y} kruh so stredom $S=\left( -\frac{1}{3},\frac{4}{3}\right) $% \par a polomerom $r=\frac{2\sqrt{2}}{3}.$}$ \textbf{Pr\'{\i}klad 27. }$\func{Im}\left( \frac{1}{z}\right) =2.$ $% \CustomNote{Answer}{% Kru\v{z}nica so stredom $S=\left( 0,-\frac{1}{4}\right) $% \par a polomerom $r=\frac{1}{4}$ a $z\neq 0.$}$ \textbf{Pr\'{\i}klad 28. }$\func{Re}\frac{1}{z+1}>2.$ $% \CustomNote{Answer}{% Vn\'{u}tro kruhu so stredom $S=\left( -\frac{3}{4},0\right) $% \par a polomerom $r=\frac{1}{4}.$}$ \textbf{Pr\'{\i}klad 29. }$1\leq \left| z-i\right| <3.$ $% \CustomNote{Answer}{% Medzikru\v{z}ie so stredom $S=\left( 0,1\right) $% \par a polomermi $r_{1}=1,\,r_{2}=3.$}$ \textbf{Pr\'{\i}klad 30. }$\left| z\right| <\left| z+i\right| .$ $% \CustomNote{Answer}{% Polrovina \ $\func{Im}z>-\frac{1}{2}.$}$ \textbf{Pr\'{\i}klad 31. }$\left| z+i\right| \leq \left| z-3\right| .$ $% \CustomNote{Answer}{% Polrovina $y\leq -3x+4.$}$ \subsection{Z\'{a}kladn\'{e} vlastnosti funkci\'{\i} komplexnej premennej.} V \'{u}loh\'{a}ch 32 -- 34 n\'{a}jdite re\'{a}lnu a imagin\'{a}rnu \v{c}as% \v{t} funkcie $f:$ \textbf{Pr\'{\i}klad 32. }$f\left( z\right) =\frac{3i}{2i-z}.% \CustomNote{Answer}{$\func{Re}f\left( z\right) =\frac{6-3y}{x^{2}+\left( 2-y\right) ^{2}},\,\func{Im}f\left( z\right) =\frac{-3x}{x^{2}+\left( 2-y\right) ^{2}}.$}$ \textbf{Pr\'{\i}klad 33. }$f\left( z\right) =z^{2}+1.$ $% \CustomNote{Answer}{$\func{Re}f\left( z\right) =x^{2}+y^{2}+1,\,\func{Im}% f\left( z\right) =2xy.$}$ \textbf{Pr\'{\i}klad 34. }$f\left( z\right) =\frac{z-\left| z\right| }{% \left| z\right| -1}.$ $% \CustomNote{Answer}{$\func{Re}f\left( z\right) =\frac{x-\sqrt{x^{2}+y^{2}}}{% \sqrt{x^{2}+y^{2}}-1},\,\func{Im}f\left( z\right) =\frac{y}{\sqrt{x^{2}+y^{2}% }-1}.$}$ \textbf{Pr\'{\i}klad 35. }N\'{a}jdite komplexne zdru\v{z}en\'{u} funkciu k funkcii $f:$ \textbf{a. }z pr\'{\i}kladu 33. \CustomNote{Answer}{% \ $\overline{f}(z)=\overline{z}^{2}+1.$} \textbf{b. }z pr\'{\i}kladu 34. $% \CustomNote{Answer}{$\overline{f}(z)=\frac{\overline{z}-\left| z\right| }{% \left| z\right| -1}.$}$ V \'{u}loh\'{a}ch 36 a 37 n\'{a}jdite defini\v{c}n\'{y} obor funkcie $f:$ \textbf{Pr\'{\i}klad 36. }$f\left( z\right) =\frac{3iz-12z+i}{iz^{2}+1-i}.$ $% \CustomNote{Answer}{$D\left( f\right) =\mathbf{C\setminus }\left\{ \sqrt[4]{2% }e^{i\frac{\pi }{8}},\,\sqrt[4]{2}e^{i\frac{9\pi }{8}}\right\} .$}$ \textbf{Pr\'{\i}klad 37. }$f\left( z\right) =\frac{\overline{z}}{\left( z^{3}-2i\right) \left( \left| z\right| -3\right) }.$ $% \CustomNote{Answer}{$D\left( f\right) =\mathbf{C\setminus }\left( \left\{ z\in \mathbf{C};\,\left| z\right| =3\right\} \cup \left\{ \frac{\sqrt[3]{2}% \sqrt{3}}{2}+i\frac{\sqrt[3]{2}}{2},\,-\frac{\sqrt[3]{2}\sqrt{3}}{2}+i\frac{% \sqrt[3]{2}}{2},\,-i\sqrt[3]{2}\right\} \right) .$}$ V \'{u}loh\'{a}ch 38 - 41 vypo\v{c}\'{\i}tajte funk\v{c}n\'{u} hodnotu funkcie $f$ v \v{c}\'{\i}sle $z_{0}:$ \textbf{Pr\'{\i}klad 38. }$f\left( z\right) =\frac{\overline{z}}{\left( z^{3}-2i\right) \left( \left| z\right| -3\right) },\,z_{0}=i.\;% \CustomNote{Answer}{$-\frac{1}{6}.$}$ \textbf{Pr\'{\i}klad 39. }$f\left( z\right) =z+\overline{z}^{2}-\func{Re}% \left( z\overline{z}\right) -\func{Im}\left( z\overline{z}\right) ,\,z_{0}=8-6i.\;% \CustomNote{Answer}{$-64+90i.$}$ \textbf{Pr\'{\i}klad 40. }$f\left( z\right) =\frac{z^{2}-\left| z\right| }{% z-8},\,z_{0}=8-6i.\;% \CustomNote{Answer}{$16+3i.$}$ \textbf{Pr\'{\i}klad 41. }$f\left( z\right) =\arg z,$ \textbf{a. }$z_{0}=8-6i.\;% \CustomNote{Answer}{$-\func{arctg}\left( \frac{3}{4}\right) .$}$ \textbf{b. }$z_{0}=-1+2i.\;% \CustomNote{Answer}{$\pi -\func{arctg}2.$}$ \textbf{c. }$z_{0}=-1-i.\;% \CustomNote{Answer}{$-\frac{3\pi }{4}.$}$ \subsection{Limita a spojitos\v{t} funkci\'{\i} komplexnej premennej.} V \'{u}loh\'{a}ch 42 - 49 vypo\v{c}\'{\i}tajte limity: \textbf{Pr\'{\i}klad 42. }$\lim_{z\longrightarrow 2i}\frac{z+3}{z^{2}+2iz}.\;% \CustomNote{Answer}{$-\frac{3+2i}{8}.$}$ \textbf{Pr\'{\i}klad 43. }$\lim_{z\longrightarrow i}\frac{z^{2}-iz+z-i}{% 3iz^{2}+3z}.\;% \CustomNote{Answer}{$-\frac{1+i}{3}.$}$ \textbf{Pr\'{\i}klad 44. }$\lim_{z\longrightarrow 2+i}\frac{3iz-6i+3}{% 2iz^{2}-4iz+2z}.\;% \CustomNote{Answer}{$\frac{6-3i}{10}.$}$ \textbf{Pr\'{\i}klad 45. }$\lim_{z\longrightarrow i}\frac{z^{2}+\left( 2-i\right) z-2i}{z^{2}+1}.\;% \CustomNote{Answer}{$\frac{1}{2}-i.$}$ \textbf{Pr\'{\i}klad 46. }$\lim_{z\longrightarrow 4}\frac{z-4}{z^{2}-\left( 4-i\right) z-4i}.\;% \CustomNote{Answer}{$\frac{4}{17}-i\frac{1}{17}.$}$ \textbf{Pr\'{\i}klad 47. }$\lim_{z\longrightarrow 0}\frac{z^{3}}{\left| z\right| ^{2}}.% \CustomNote{Answer}{$\;0.$}$ \textbf{Pr\'{\i}klad 48. }$\lim_{z\longrightarrow 0}\frac{z^{2}}{\left| z\right| ^{2}}.\;% \CustomNote{Prob_Solv_Hint}{% N\'{a}vod : vyjadrite re\'{a}lnu a imagin\'{a}rnu \v{c}as\v{t} funkcie, potom uk\'{a}\v{z}te, \v{z}e limita neexistuje.}$ \textbf{Pr\'{\i}klad 49. }$\lim_{z\longrightarrow i}\frac{z}{z^{2}+1}.\;% \CustomNote{Answer}{$\infty .$}$ \textbf{Pr\'{\i}klad 50. }Zistite, \v{c}i funkcia $f$ je spojit\'{a} v \v{c}% \'{\i}sle $2i,$ ak $f\left( z\right) =\left\{ \begin{tabular}{cc} $\frac{z^{3}-4z}{z^{2}-2iz+z-2i}$ & $z\neq 2i,\,z\neq -1$ \\ $5$ & $z=2i$% \end{tabular}% \right. .$ \CustomNote{Answer}{% V bode $z=2i$ nie je spojit\'{a}.} V \'{u}loh\'{a}ch 51 - 56 vy\v{s}etrite spojitos\v{t} funkcie $f:$ \textbf{Pr\'{\i}klad 51. }$f\left( z\right) =\frac{1}{1-z}.$ $% \CustomNote{Answer}{% Spojit\'{a} v \ $\mathbf{C\setminus }\left\{ 1\right\} .$}$ \textbf{Pr\'{\i}klad 52. }$f\left( z\right) =\frac{1}{1+z^{2}}.$ $% \CustomNote{Answer}{% Spojit\'{a} v $\mathbf{C}\setminus \left\{ -i,i\right\} .$}$ \textbf{Pr\'{\i}klad 53. }$f\left( z\right) =\left\{ \begin{tabular}{cc} $\frac{z\func{Re}z}{\left| z\right| }$ & $z\neq 0$ \\ $0$ & $z=0$% \end{tabular}% \right. .$ $% \CustomNote{Answer}{% Spojit\'{a} v$\ \mathbf{C}.$}$ \textbf{Pr\'{\i}klad 54. }$f\left( z\right) =\left\{ \begin{tabular}{cc} $\frac{\func{Re}z}{z}$ & $z\neq 0$ \\ $0$ & $z=0$% \end{tabular}% \right. .$ $% \CustomNote{Answer}{% Spojit\'{a} v$\ \mathbf{C}\setminus \left\{ 0\right\} .$}$ \textbf{Pr\'{\i}klad 55. }$f\left( z\right) =\left\{ \begin{tabular}{cc} $\frac{\func{Re}\left( z^{2}\right) }{z^{2}}$ & $z\neq 0$ \\ $0$ & $z=0$% \end{tabular}% \right. .$ $% \CustomNote{Answer}{% Spojit\'{a} v$\ \mathbf{C}\setminus \left\{ 0\right\} .$}$ \textbf{Pr\'{\i}klad 56. }$f\left( z\right) =\left\{ \begin{tabular}{cc} $\frac{\left( \func{Re}z^{2}\right) ^{2}}{z^{2}}$ & $z\neq 0$ \\ $0$ & $z=0$% \end{tabular}% \right. .$ $% \CustomNote{Answer}{% Spojit\'{a} v$\ \mathbf{C.}$}$ V \'{u}loh\'{a}ch 57 - 58 zistite, \v{c}i je mo\v{z}n\'{e} dodefinova\v{t} funkciu $f$ v bode $z_{0}$ tak, aby bola spojit\'{a} v tomto bode: \textbf{Pr\'{\i}klad 57. }$f:\mathbf{C\setminus }\left\{ 0,1+i\right\} \longrightarrow \mathbf{C},\,f\left( z\right) =\frac{% z^{3}-z^{2}-iz^{2}+iz-i+1}{z^{2}-z-iz},\,z_{0}=1+i.$% \CustomNote{Answer}{% Je mo\v{z}n\'{e}, ak $f\left( 1+i\right) =\frac{3}{2}\left( 1+i\right) .$} \textbf{Pr\'{\i}klad 58. }$f:\mathbf{C\setminus }\left\{ 4+i\right\} \longrightarrow \mathbf{C},\,f\left( z\right) =\frac{z^{2}-\left( 3+2i\right) z-6+7i}{z-4-i},\,z_{0}=4+i.$ $% \CustomNote{Answer}{% Nie je mo\v{z}n\'{e}, lebo $f\lim_{z\longrightarrow 4+i}f\left( z\right) =\infty .$}$ \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}