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01MAA4:Kapitola36 - Historie editací
2024-03-29T15:57:41Z
Historie editací této stránky
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Kubuondr: důkaz neexistence derivace argumentu na P_pi.
2017-05-31T07:27:31Z
<p>důkaz neexistence derivace argumentu na P_pi.</p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 31. 5. 2017, 07:27</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L203" >Řádka 203:</td>
<td colspan="2" class="diff-lineno">Řádka 203:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{remark}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{remark}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{enumerate}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{enumerate}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\item Funkce $\arg z$ není spojitá na $P_\pi$ a nikde nemá derivaci. (<del class="diffchange diffchange-inline">K důkazu toho, že nemá derivaci</del>, <del class="diffchange diffchange-inline">lze využít vyjádření </del>z <del class="diffchange diffchange-inline">následujícího bodu</del>; pro dokázání mírně slabšího tvrzení stačí využít větu \ref{th:realnaholomorfni}.)</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\item Funkce $\arg z$ není spojitá na $P_\pi$ a nikde nemá derivaci. (<ins class="diffchange diffchange-inline">Neexistence derivace plyne z nespojitosti</ins>, <ins class="diffchange diffchange-inline">viz </ins>z <ins class="diffchange diffchange-inline">následující bod</ins>; pro dokázání mírně slabšího tvrzení stačí využít větu \ref{th:realnaholomorfni}.)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\item Pro $z = x + \im y$ lze argument vyjádřit explicitně třeba takto:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\item Pro $z = x + \im y$ lze argument vyjádřit explicitně třeba takto:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[</div></td></tr>
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Kubuondr
https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=5580&oldid=prev
Krasejak: Kompletní revize, oprava chyb, doplnění chybějících tvrzení, důkazů a poznámek
2015-09-18T23:42:51Z
<p>Kompletní revize, oprava chyb, doplnění chybějících tvrzení, důkazů a poznámek</p>
<a href="https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=5580&oldid=5428">Ukázat změny</a>
Krasejak
https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=5428&oldid=prev
Nguyebin: Odebrání přebytečných informací nad rámec výkladu.
2015-02-06T10:19:48Z
<p>Odebrání přebytečných informací nad rámec výkladu.</p>
<a href="https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=5428&oldid=5211">Ukázat změny</a>
Nguyebin
https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=5211&oldid=prev
Nguyebin: Přidání části o původu Cauchy-Riemann podmínek + celková úprava.
2014-01-24T12:48:18Z
<p>Přidání části o původu Cauchy-Riemann podmínek + celková úprava.</p>
<a href="https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=5211&oldid=5101">Ukázat změny</a>
Nguyebin
https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=5101&oldid=prev
Nguyebin: Drobná úprava.
2013-10-04T23:19:48Z
<p>Drobná úprava.</p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 4. 10. 2013, 23:19</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L2" >Řádka 2:</td>
<td colspan="2" class="diff-lineno">Řádka 2:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\section{Komplexní derivace}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\section{Komplexní derivace}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>   </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>   </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Studujeme funkce $\C\mapsto\C$. $\C$ je normovaný prostor <del class="diffchange diffchange-inline">homeomorfní</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Studujeme funkce $\C\mapsto\C$. $\C$ je normovaný prostor<ins class="diffchange diffchange-inline">, izomorfní</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>s~$\R^2$ a z~hlediska topologie nerozeznatelný. Nevyužívali jsme však</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>s~$\R^2$ <ins class="diffchange diffchange-inline">($\C \cong\R^2$) </ins>a z~hlediska topologie nerozeznatelný. Nevyužívali jsme však</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>toho, že $\C$ je těleso.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>toho, že $\C$ je těleso <ins class="diffchange diffchange-inline">(uzavřené na součin prvků)</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>   </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>   </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Jednoznačný vztah </del>mezi <del class="diffchange diffchange-inline">$\C\mapsto\C$ a </del>$\R^2\mapsto\C$:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Izomorfismus </ins>mezi $\R^2\mapsto\C$:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$f(z)=f(x+iy)=f(x,y)$.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$f(z)=f(x+iy)=f(x,y)$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>   </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>   </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="L12" >Řádka 12:</td>
<td colspan="2" class="diff-lineno">Řádka 12:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Buď $f:\C\mapsto\C$, $z_0\in\vn{(\df f)}$. Pak existuje-li limita</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Buď $f:\C\mapsto\C$, $z_0\in\vn{(\df f)}$. Pak existuje-li limita</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0},\]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0},\]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>říkáme, že funkce $f$ má v~$z_0$ derivaci.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>říkáme, že funkce $f$ má v~$z_0$ <ins class="diffchange diffchange-inline">(komplexní) </ins>derivaci.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{define}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{define}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>   </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>   </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="L144" >Řádka 144:</td>
<td colspan="2" class="diff-lineno">Řádka 144:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\frac{\pd^2 f_1}{\pd x^2}=\frac{\pd^2 f_2}{\pd x\pd y}\]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\frac{\pd^2 f_1}{\pd x^2}=\frac{\pd^2 f_2}{\pd x\pd y}\]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\frac{\pd^2 f_1}{\pd y^2}=-\frac{\pd^2 f_2}{\pd y\pd x}\]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\frac{\pd^2 f_1}{\pd y^2}=-\frac{\pd^2 f_2}{\pd y\pd x}\]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Sečtením dostaneme $\Delta f_1=0$ a analogicky $\Delta f_2=0$.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Sečtením dostaneme $\Delta f_1=0$ a analogicky $\Delta f_2=0$ <ins class="diffchange diffchange-inline">(harmonické funkce)</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{enumerate}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{enumerate}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{remark}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{remark}</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="L157" >Řádka 157:</td>
<td colspan="2" class="diff-lineno">Řádka 157:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>a definujeme {\bf logaritmus komplexního čísla}:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>a definujeme {\bf logaritmus komplexního čísla}:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$\ln z=\ln\abs{z}+\im\arg z$.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$\ln z=\ln\abs{z}+\im\arg z$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\item Má logaritmus derivaci ? $\Re\ln z=\ln\sqrt{x^2+y^2}$, $\Im\ln</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\item Má logaritmus derivaci? $\Re\ln z=\ln\sqrt{x^2+y^2}$, $\Im\ln</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>z=\arg z$.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>z=\arg z$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\left(\ln\sqrt{x^2+y^2}\right)'=\frac{x}{x^2+y^2}\d x+</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\left(\ln\sqrt{x^2+y^2}\right)'=\frac{x}{x^2+y^2}\d x+</div></td></tr>
</table>
Nguyebin
https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=4994&oldid=prev
Nguyebin v 26. 8. 2013, 13:08
2013-08-26T13:08:51Z
<p></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 26. 8. 2013, 13:08</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L1" >Řádka 1:</td>
<td colspan="2" class="diff-lineno">Řádka 1:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>%\wikiskriptum{01MAA4}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>%\wikiskriptum{01MAA4}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\section{<del class="diffchange diffchange-inline">Elementární funkce</del>}</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\section{<ins class="diffchange diffchange-inline">Komplexní derivace</ins>}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>   </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>   </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Studujeme funkce $\C\mapsto\C$. $\C$ je normovaný prostor homeomorfní</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Studujeme funkce $\C\mapsto\C$. $\C$ je normovaný prostor homeomorfní</div></td></tr>
</table>
Nguyebin
https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=4202&oldid=prev
Patakjan: doplnění chybícího x do argumentu sin
2011-05-23T17:10:55Z
<p>doplnění chybícího x do argumentu sin</p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 23. 5. 2011, 17:10</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L82" >Řádka 82:</td>
<td colspan="2" class="diff-lineno">Řádka 82:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\cos z = \cosh\im z,\quad\sin z=-\im\sinh\im z\]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\cos z = \cosh\im z,\quad\sin z=-\im\sinh\im z\]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\sin(x+\im y)=\sin x\cos\im y+\sin\im y\cos x=</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\sin(x+\im y)=\sin x\cos\im y+\sin\im y\cos x=</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\sin\cosh y+\im\sinh y\cos x\]</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\sin <ins class="diffchange diffchange-inline">x </ins>\cosh y+\im\sinh y\cos x\]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\cos(x+\im y)=\cos x\cosh y-\im\sin x\sinh y\]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\cos(x+\im y)=\cos x\cosh y-\im\sin x\sinh y\]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Nulové body:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Nulové body:</div></td></tr>
</table>
Patakjan
https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=4201&oldid=prev
Patakjan: odstranění přebytečného rovnítka
2011-05-23T17:08:13Z
<p>odstranění přebytečného rovnítka</p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 23. 5. 2011, 17:08</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L19" >Řádka 19:</td>
<td colspan="2" class="diff-lineno">Řádka 19:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{split}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{split}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}=\alpha&\iff</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}=\alpha&\iff</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\lim_{h\to 0}<del class="diffchange diffchange-inline">=</del>\frac{f(z_0+h)-f(z_0)-\alpha h}{\abs{h}}</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\lim_{h\to 0}\frac{f(z_0+h)-f(z_0)-\alpha h}{\abs{h}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\frac{\abs{h}}{h}=0\iff\\</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\frac{\abs{h}}{h}=0\iff\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&\iff\lim_{h\to 0}\frac{f(z_0+h)-f(z_0)-\alpha h}{\abs{h}}=0,</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&\iff\lim_{h\to 0}\frac{f(z_0+h)-f(z_0)-\alpha h}{\abs{h}}=0,</div></td></tr>
</table>
Patakjan
https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01MAA4:Kapitola36&diff=3252&oldid=prev
Admin: Založena nová stránka: %\wikiskriptum{01MAA4} \section{Elementární funkce} Studujeme funkce $\C\mapsto\C$. $\C$ je normovaný prostor homeomorfní s~$\R^2$ a z~hlediska topologie nerozeznatel...
2010-08-01T09:06:06Z
<p>Založena nová stránka: %\wikiskriptum{01MAA4} \section{Elementární funkce} Studujeme funkce $\C\mapsto\C$. $\C$ je normovaný prostor homeomorfní s~$\R^2$ a z~hlediska topologie nerozeznatel...</p>
<p><b>Nová stránka</b></p><div>%\wikiskriptum{01MAA4}<br />
\section{Elementární funkce}<br />
<br />
Studujeme funkce $\C\mapsto\C$. $\C$ je normovaný prostor homeomorfní<br />
s~$\R^2$ a z~hlediska topologie nerozeznatelný. Nevyužívali jsme však<br />
toho, že $\C$ je těleso.<br />
<br />
Jednoznačný vztah mezi $\C\mapsto\C$ a $\R^2\mapsto\C$:<br />
$f(z)=f(x+iy)=f(x,y)$.<br />
<br />
\begin{define}<br />
Buď $f:\C\mapsto\C$, $z_0\in\vn{(\df f)}$. Pak existuje-li limita<br />
\[\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0},\]<br />
říkáme, že funkce $f$ má v~$z_0$ derivaci.<br />
\end{define}<br />
<br />
\begin{remark}<br />
\[<br />
\begin{split}<br />
\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}=\alpha&\iff<br />
\lim_{h\to 0}=\frac{f(z_0+h)-f(z_0)-\alpha h}{\abs{h}}<br />
\frac{\abs{h}}{h}=0\iff\\<br />
&\iff\lim_{h\to 0}\frac{f(z_0+h)-f(z_0)-\alpha h}{\abs{h}}=0,<br />
\end{split}<br />
\]<br />
to je dále ekvivalentní s~nulovostí dvou reálných limit<br />
\[\lim_{(h_1,h_2)\to(0,0)}\frac{f_1(z_0+h)-f_1(z_0)-<br />
\alpha_1h_1+\alpha_2h_2}{\sqrt{h_1^2+h_2^2}}=0\]<br />
a<br />
\[\lim_{(h_1,h_2)\to(0,0)}\frac{f_2(z_0+h)-f_2(z_0)-<br />
\alpha_1h_2-\alpha_2h_1}{\sqrt{h_1^2+h_2^2}}=0\]<br />
a dále pro $h_1=0$, případně $h_2=0$ s~{\bf Cauchyho-Riemannovými podmínkami}:<br />
\[\exists f_1'(x_0,y_0)\wedge\exists f_2'(x_0,y_0)\wedge<br />
\alpha_1=\frac{\pd f_1}{\pd x}=\frac{\pd f_2}{\pd y}\wedge<br />
\alpha_2=\frac{\pd f_2}{\pd x}=-\frac{\pd f_1}{\pd y}\]<br />
\end{remark}<br />
<br />
\begin{example}<br />
$f(z)=\overline{z}$ už nemá derivaci.<br />
\end{example}<br />
<br />
\begin{theorem}<br />
Nechť $f,g$ mají derivaci v~$z_0$. Pak<br />
\begin{enumerate}[(i)]<br />
\item $(f+cg)'(z_0)=f'(z_0)+cg'(z_0)$,<br />
\item $(fg)'(z_0)=f'(z_0)g(z_0)+f(z_0)g'(z_0)$.<br />
\item Jestliže $g'(z_0)\not=0$, pak<br />
\[\left(\frac1g\right)'(z_0)=-\frac{1}{g^2(z_0)}g'(z_0).\]<br />
\end{enumerate}<br />
\end{theorem}<br />
<br />
\begin{theorem}<br />
Nechť $\exists f'(g(z_0))$, $\exists g'(z_0)$. Pak<br />
$(f\circ g)'(z_0)=f'(g(z_0))g'(z_0)$.<br />
\end{theorem}<br />
<br />
\begin{remark}<br />
\[e^z=\sum_{n=0}^\infty\frac{z^n}{n!}\]<br />
\[e^{\im z}=\cos z+\im\sin z\]<br />
\[\sin z=\frac{e^{\im z}-e^{-\im z}}{2\im},\quad<br />
\cos z=\frac{e^{\im z}+e^{-\im z}}{2}\]<br />
Platí, že $e^{z_1}e^{z_2}=e^{z_1+z_2}$:<br />
\[\sum_{n=0}^\infty\frac{z_1^n}{n!}\sum_{n=0}^\infty\frac{z_2^n}{n!}=<br />
\sum_{n=0}^\infty\frac1{n!}\sum_{n=0}^\infty n!<br />
\frac{z_1^k z_2^{n-k}}{k!(n-k)!}=<br />
\sum_{n=0}^\infty\frac{(z_1+z_2)^n}{n!}\]<br />
\[<br />
\sin z=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}z^{2n+1},\quad<br />
\cos z=\sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}z^{2n}<br />
\]<br />
\[\sin(z_1+z_2)=\sin z_1\cos z_2+\cos z_1\sin z_2\]<br />
\[\cos(z_1+z_2)=\cos z_1\cos z_2-\sin z_1\sin z_2\]<br />
\[<br />
\sin(z+2k\pi)=\sin z,\quad<br />
\cos(z+2k\pi)=\cos z,\quad<br />
\sin\left(\frac{\pi}2-z\right)=\cos z<br />
\]<br />
\[\cos^2 z+\sin^2 z= \cos z \cos -z -\sin z \sin -z = \cos(z-z)=1\]<br />
ale $\cos^2 z$ a $\sin^2 z$ už nemusí ležet v intervalu $\left<0,1\right>$<br />
\[\sinh z=\frac{e^z-e^{-z}}{2},\quad\cosh z=\frac{e^z+e^{-z}}{2}\]<br />
\[e^{x+\im y}=e^x e^{\im y}=e^x(\cos y+\im\sin y),\quad<br />
\cos z = \cosh\im z,\quad\sin z=-\im\sinh\im z\]<br />
\[\sin(x+\im y)=\sin x\cos\im y+\sin\im y\cos x=<br />
\sin\cosh y+\im\sinh y\cos x\]<br />
\[\cos(x+\im y)=\cos x\cosh y-\im\sin x\sinh y\]<br />
Nulové body:<br />
\[\sin z=\sin(x+\im y)=0\iff<br />
\sin x\cosh y=0\wedge\sin y\cos x=0\iff<br />
x=k\pi\iff y=0.\]<br />
Derivace:<br />
\[\left(e^z\right)'=e^z,\quad<br />
(\sin z)'=\cos z,\quad<br />
(\cos z)'=-\sin z\]<br />
Prostota $e^z$:<br />
\[e^{z_1}=e^{z_2}\iff e^{z_1-z_2}=1\]<br />
\[e^x(\cos y+\im\sin y)=1\]<br />
\[e^x\sin y=0\implies y=k\pi\]<br />
\[e^x\cos y=1\implies y=2k\pi\]<br />
$e^z$ není prostá, je prostá na množině<br />
\[E_\alpha=\{z\in\C|\Im z=y\in(\alpha-\pi,\alpha+\pi\ra\}\]<br />
\[z\in\C\sm\{0\},\quad z=\abs{z}(\cos\alpha+\im\sin\alpha)\]<br />
\end{remark}<br />
<br />
\begin{define}<br />
{\bf Argumentem komplexního čísla} $z$ nazýváme množinu<br />
$\{\alpha\in\R|z=\abs{z}e^{\im\alpha}\}=\Arg z$.<br />
\end{define}<br />
<br />
\begin{define}<br />
Buď $\vartheta\in\R$. Potom<br />
$\Arg z\cap(\vartheta-\pi,\vartheta+\pi\ra\ni\arg_\vartheta z$ je jednoprvková<br />
množina, tím definujeme funkci pro $z$. Zkráceně $\arg=\arg_0$.<br />
\end{define}<br />
<br />
\begin{theorem}<br />
$\arg_\vartheta z=\arg(ze^{-\im\vartheta})+\vartheta$.<br />
\end{theorem}<br />
<br />
\begin{define}<br />
Buď $\vartheta\in\R$, definujeme<br />
$P_\vartheta=\{z|z=te^{\im\vartheta},\ t>0\}$.<br />
\end{define}<br />
<br />
\begin{remark}<br />
\begin{enumerate}<br />
\item $\arg z$ nemá derivaci, není spojitá na $P_\pi$.<br />
\item<br />
\[<br />
\arg z=\begin{cases}<br />
\arccos\frac{x}{\abs z} & y\ge 0\\<br />
-\arccos\frac{x}{\abs z} & y<0.<br />
\end{cases}<br />
\]<br />
\item \[\arg z_1z_2=\arg z_1+\arg z_2+2\pi\epsilon,\]<br />
\[\arg\frac{z_1}{z_2}=\arg z_1-\arg z_2+2\pi\epsilon,\]<br />
\[\arg\frac{1}{z}=-\arg z+2\pi\epsilon,\]<br />
přičemž $\epsilon$ volím $-1$, $0$ nebo $1$ tak, abych zůstal v<br />
základním intervalu.<br />
\item Nechť platí pro funkce $f_1(x,y)$ a $f_2(x,y)$ Cauchy-Riemannovy<br />
podmínky a nechť jsou třídy $\c{2}$.<br />
\[\frac{\pd f_1}{\pd x}=\frac{\pd f_2}{\pd y},\quad<br />
\frac{\pd f_1}{\pd y}=-\frac{\pd f_2}{\pd x},\]<br />
zkoumáme<br />
\[\frac{\pd^2 f_1}{\pd x^2}=\frac{\pd^2 f_2}{\pd x\pd y}\]<br />
\[\frac{\pd^2 f_1}{\pd y^2}=-\frac{\pd^2 f_2}{\pd y\pd x}\]<br />
Sečtením dostaneme $\Delta f_1=0$ a analogicky $\Delta f_2=0$.<br />
\end{enumerate}<br />
\end{remark}<br />
<br />
\begin{remark}<br />
\begin{enumerate}<br />
\item Zavedeme množinu $\Ln z=\{w\in\C|z=e^w\}$, $w=u+\im v$,<br />
$e^w=e^u e^{\im v}$<br />
\[\ln_\vartheta z\in\Ln z\wedge<br />
\Im\Ln_\vartheta z\in(\vartheta-\pi,\vartheta+\pi\ra\]<br />
\[\ln_\vartheta z=\ln\abs{z}+\im\arg_\vartheta z\]<br />
a definujeme {\bf logaritmus komplexního čísla}:<br />
$\ln z=\ln\abs{z}+\im\arg z$.<br />
\item Má logaritmus derivaci ? $\Re\ln z=\ln\sqrt{x^2+y^2}$, $\Im\ln<br />
z=\arg z$.<br />
\[\left(\ln\sqrt{x^2+y^2}\right)'=\frac{x}{x^2+y^2}\d x+<br />
\frac{y}{x^2+y^2}\d y,\]<br />
\[\left(\arg z\right)'=-\frac{y}{x^2+y^2}\d x+<br />
\frac{x}{x^2+y^2}\d y,\]<br />
takže Cauchyho-Riemannovy podmínky platí a derivace existuje. Můžeme<br />
se proto omezit na nějakou konkrétní podmnožinu.<br />
\[(f(z_0))'=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}=<br />
\lim_{x\to x_0}\frac{f(x,y_0)-f(x_0,y_0)}{x-x_0}=<br />
\frac{\pd f_1}{\pd x}(x_0,y_0)+\im\frac{\pd f_2}{\pd x}(x_0,y_0)\]<br />
\[(\ln z)'=\frac{x}{x^2+y^2}-\im\frac{y}{x^2+y^2}=\frac{\overline<br />
z}{z\overline z}=\frac1z.\]<br />
\item Analogicky s~reálnými funkcemi definujeme<br />
\[\argsinh z=\ln\left(z+\sqrt{1+z^2}\right),\quad<br />
\argcosh z=\ln\left(z+\sqrt{z-1}\sqrt{z+1}\right),\quad<br />
\argtgh z=\frac12\ln\frac{1+z}{1-z}.\]<br />
$\sqrt{z-1}\sqrt{z+1} = \sqrt{z^2-1}$ obecně pro komplexní odmocninu neplatí. <br />
%http://en.wikipedia.org/wiki/Square_root<br />
\[<br />
\arcsin z=-\im\ln\left(\im z+\sqrt{1-z^2}\right),\quad<br />
\arccos z=-\im\ln\left(z+\sqrt{z^2-1}\right)=-\im\ln\left(z+\im\sqrt{1-z^2}\right),\]<br />
\[\arctg z=\frac{i}{2}\ln\left(\frac{1-\im z}{1+\im z}\right)\]<br />
<br />
\item Pro $z,\alpha\in\C$<br />
\[z^\alpha=e^{\alpha\ln z},\]<br />
pokud $z\not=0$, tato definice je jednoznačná. Lepší je <br />
\[<br />
z^\alpha=e^{\alpha\ln z+\alpha\,2k\pi\im} \quad k\in \Z <br />
\]<br />
exponenciála je periodická s periodou $2\pi \im$. To má za následek,<br />
že pro $\Re\alpha \in \N$ a $\Im\alpha = 0$ je $z^\alpha$ definováno jednoznačně. <br />
Pro $\Re\alpha \in \Q \Rightarrow \Re\alpha = \frac{p}{q} $ a $\Im\alpha = 0$ je možných $q$ kořenů. <br />
A pokud je $\Re\alpha$ iracionální a nebo $\Im\alpha \neq 0$, pak je kořenů dokonce nekonečně mnoho. <br />
Pro $\Im\alpha = 0$ se kořeny nachází na kružnici, pro $\Re\alpha = 0$ na polopřímce <br />
a pro $\Re\alpha \neq 0 \wedge \Im\alpha \neq 0$ jsou umístěny kořeny na spirále. <br />
<br />
Podobný problém nastává i u dalších funkcí, k jejichž definici se použil logaritmus, tedy arcsin, argsinh, $\ldots$ <br />
%http://en.wikipedia.org/wiki/Riemann_surface<br />
\begin{example}<br />
\[\im^{\im}=e^{\im\left( \frac\pi2\im +2k\pi\im \right)}=e^{-\frac\pi2 - 2k\pi} \quad k\in \Z \]<br />
\[{x}^{\frac{3}{5}} = e^{\frac{3}{5}\ln x}e^{ \frac{3}{5}\,2k\pi\im}\quad k\in \hat 5 \]<br />
\[{x}^{\sqrt{2}} = e^{\sqrt{2}\ln x}e^{ \sqrt{2}\,2k\pi\im}\quad k\in \Z \].<br />
\end{example}<br />
<br />
\end{enumerate}<br />
\end{remark}</div>
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