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Matematika A3a 2008/12. gyakorlat - Laptörténet
2024-03-28T14:18:40Z
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Mozo: /* Egész kitevőjű hatványsorba fejtés */
2015-11-11T13:48:31Z
<p><span class="autocomment">Egész kitevőjű hatványsorba fejtés</span></p>
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<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2015. november 11., 13:48-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">9. sor:</td>
<td colspan="2" class="diff-lineno">9. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\sum\limits_{n=0}^{\infty}z^n=\frac{1}{1-z},\quad\quad (|z|<1)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\sum\limits_{n=0}^{\infty}z^n=\frac{1}{1-z},\quad\quad (|z|<1)</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Példa.''' Adjuk meg az alábbi függvény 1 körüli összes Laurent-sorát! Válasszuk ki ezek közül azt, mely <del class="diffchange diffchange-inline">előállítja </del>a 0-<del class="diffchange diffchange-inline">t</del>!</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Példa.''' Adjuk meg az alábbi függvény 1 körüli összes Laurent-sorát! Válasszuk ki ezek közül azt, mely <ins class="diffchange diffchange-inline">a függvényt </ins>a 0-<ins class="diffchange diffchange-inline">ban állítja elő</ins>!</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>f(z)=\frac{(z-1)^2}{z+i}\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>f(z)=\frac{(z-1)^2}{z+i}\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2" class="diff-lineno">20. sor:</td>
<td colspan="2" class="diff-lineno">20. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>A másik esetben a (-1)/(''z'' - 1) elsőfokú polinomjaként kell a nevezőt felírni, majd a konstans tagot kiemelni, hogy az 1 legyen. Ezt a legegyszerűbben úgy tehetjük, ha a (z-1) elyőfokú polinomjaként írjuk fel a nevezőt, majd (z-1)-et kiemelünk (amivel 1 marad ennek a tagnak a helyén).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>A másik esetben a (-1)/(''z'' - 1) elsőfokú polinomjaként kell a nevezőt felírni, majd a konstans tagot kiemelni, hogy az 1 legyen. Ezt a legegyszerűbben úgy tehetjük, ha a (z-1) elyőfokú polinomjaként írjuk fel a nevezőt, majd (z-1)-et kiemelünk (amivel 1 marad ennek a tagnak a helyén).</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Reziduum-számítás===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Reziduum-számítás===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Milyen típusú szingularitása van, mennyi a reziduuma a szingularitási helyen?</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Milyen típusú szingularitása van, mennyi a reziduuma a szingularitási helyen?</div></td></tr>
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Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A3a_2008/12._gyakorlat&diff=9254&oldid=prev
Mozo: /* Harmonikus társ keresése */
2013-10-23T05:39:59Z
<p><span class="autocomment">Harmonikus társ keresése</span></p>
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<td colspan='2' style="background-color: white; color:black;">A lap 2013. október 23., 05:39-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">133. sor:</td>
<td colspan="2" class="diff-lineno">133. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>=2\pi i\left.\frac{z^3}{z^2+1}\right|_{1}+2\pi i\left.\frac{z^3}{(z+i)(z-1)}\right|_{i}+2\pi i\left.\frac{z^3}{(z-i)(z-1)}\right|_{-i}=\frac{1}{2}+\frac{1}{4}(i+1)+\frac{1}{4}(1-i)=1</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>=2\pi i\left.\frac{z^3}{z^2+1}\right|_{1}+2\pi i\left.\frac{z^3}{(z+i)(z-1)}\right|_{i}+2\pi i\left.\frac{z^3}{(z-i)(z-1)}\right|_{-i}=\frac{1}{2}+\frac{1}{4}(i+1)+\frac{1}{4}(1-i)=1</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">===Harmonikus társ keresése===</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Azt mondjuk, hogy a kétszer differenciálható u=u(x,y) valós függvény ''harmonikus'', ha </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">:<math>u_{xx}''+u_{yy}''\equiv \Delta u\equiv 0\, </math> </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">itt &Delta; a Laplace-operátor (nem a Laplace-transzformátor!, hanem a vektoranalízisbeli vektormezőre Hesse-mátrix nyoma).</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">A C--R-egyenletek mutatják, hogy ha f=u+iv reguláris, akkor u és v harmonikus függvények. Ugyanis:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">:<math>u_x'=v_y'\,</math> és</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">:<math>v_x'=-u_y'\,</math></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">De u és v Hesse-mátrixa is szimmetrikus, ezért:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">:<math>v_{yy}''=u_{xy}''=u_{yx}''=-v_{xx}''\,</math></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">azaz </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">:<math>\Delta v\equiv 0\,</math> és fordítva.</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Általában az a feladat, hogy ha adott u, akkor keressük az ő harmonikus társát, v-t, mellyel u+iv reguláris. Ha tehát adott u, akkor van F és G, hogy </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">:<math>F=v_y'\,</math></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">:<math>G=-v_x'\,</math></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Ami az egzakt differenciálegynlet megoldásánál tanult parciális differenciálegyenlet megoldását igényli v-re, mint potenciálfüggvényre (ekkor f-et komplex pontenciálnak nevezzük, mármint a (<math>v'_x(x,y)</math>,<math>v_y'(x,y)</math>) síkbeli vektormező komplex pontenciáljának; a v valódi pontenciálja lenne. Ennek szükséges utánanézni máshol is!)</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">'''1..''' Keressünk harmonikus párt az </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">:<math>u=x^4+y^4-6x^2y^2\,</math></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">függvényhez!</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">''Mo.'' Van neki, ha &Delta;=0. Ezt ellenőrizni kell, majd az előző módszerrel megkeresi v-t, amivel u+iv reguláris.</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''2.''' Keressünk potenciált a</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''2.''' Keressünk potenciált a</div></td></tr>
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Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A3a_2008/12._gyakorlat&diff=4840&oldid=prev
Mozo: /* Vektor és skalármezők deriválása */
2008-12-11T14:38:40Z
<p><span class="autocomment">Vektor és skalármezők deriválása</span></p>
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<td colspan='2' style="background-color: white; color:black;">A lap 2008. december 11., 14:38-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">174. sor:</td>
<td colspan="2" class="diff-lineno">174. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\mathrm{rot}\,(\mathrm{div}\,(r|r|)r)=?\,\quad\quad r\in\mathbf{R}^2</math>  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\mathrm{rot}\,(\mathrm{div}\,(r|r|)r)=?\,\quad\quad r\in\mathbf{R}^2</math>  </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>u=?\quad\quad\mathrm{ha}\quad\quad\mathrm{grad}\,u=r|r|^2\,\quad\quad r\in\mathbf{R}^4\setminus \{0\}</math> </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td></tr>
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Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A3a_2008/12._gyakorlat&diff=4839&oldid=prev
Mozo: /* Harmonikus társ keresése */
2008-12-11T14:34:47Z
<p><span class="autocomment">Harmonikus társ keresése</span></p>
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<td colspan='2' style="background-color: white; color:black;">A lap 2008. december 11., 14:34-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">144. sor:</td>
<td colspan="2" class="diff-lineno">144. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>De u és v Hesse-mátrixa is szimmetrikus, ezért:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>De u és v Hesse-mátrixa is szimmetrikus, ezért:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>v_{yy}''=<del class="diffchange diffchange-inline">u_xy</del>''=u_{yx}''=-v_{xx}''\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>v_{yy}''=<ins class="diffchange diffchange-inline">u_{xy}</ins>''=u_{yx}''=-v_{xx}''\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>azaz  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>azaz  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\Delta v\equiv 0\,</math> és fordítva.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\Delta v\equiv 0\,</math> és fordítva.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"> </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Általában az a feladat, hogy ha adott u, akkor keressük az ő harmonikus társát, v-t, mellyel u+iv reguláris. Ha tehát adott u, akkor van F és G, hogy </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>F=v_y'\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>G=-v_x'\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Ami az egzakt differenciálegynlet megoldásánál tanult parciális differenciálegyenlet megoldását igényli v-re, mint potenciálfüggvényre (ekkor f-et komplex pontenciálnak nevezzük, mármint a (<math>v'_x(x,y)</math>,<math>v_y'(x,y)</math>) síkbeli vektormező komplex pontenciáljának; a v valódi pontenciálja lenne. Ennek szükséges utánanézni máshol is!)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''1..''' Keressünk harmonikus párt az </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>u=x^4+y^4-6x^2y^2\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">függvényhez!</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">''Mo.'' Van neki, ha &Delta;=0. Ezt ellenőrizni kell, majd az előző módszerrel megkeresi v-t, amivel u+iv reguláris.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''2.''' Keressünk potenciált a</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">v=(2xy^3,3x^2y^2,z^2)\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">térbeli vektormezőhöz!</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''3.''' Keressünk potenciált a</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">v=(x^3+y^3,3xy^2)\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">síkbeli vektormezőhöz!</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">===Vektor és skalármezők deriválása===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>\mathrm{grad}\,(r\cdot|r|^4)=?\,</math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>\mathrm{rot}\,(\mathrm{div}\,(r|r|)r)=?\,\quad\quad r\in\mathbf{R}^2</math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A3a_2008/12._gyakorlat&diff=4838&oldid=prev
Mozo: /* Integrálás véges sok szinguláris pontot körülhurkoló görbén */
2008-12-11T14:14:32Z
<p><span class="autocomment">Integrálás véges sok szinguláris pontot körülhurkoló görbén</span></p>
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<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2008. december 11., 14:14-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">133. sor:</td>
<td colspan="2" class="diff-lineno">133. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>=2\pi i\left.\frac{z^3}{z^2+1}\right|_{1}+2\pi i\left.\frac{z^3}{(z+i)(z-1)}\right|_{i}+2\pi i\left.\frac{z^3}{(z-i)(z-1)}\right|_{-i}=\frac{1}{2}+\frac{1}{4}(i+1)+\frac{1}{4}(1-i)=1</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>=2\pi i\left.\frac{z^3}{z^2+1}\right|_{1}+2\pi i\left.\frac{z^3}{(z+i)(z-1)}\right|_{i}+2\pi i\left.\frac{z^3}{(z-i)(z-1)}\right|_{-i}=\frac{1}{2}+\frac{1}{4}(i+1)+\frac{1}{4}(1-i)=1</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">===Harmonikus társ keresése===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Azt mondjuk, hogy a kétszer differenciálható u=u(x,y) valós függvény ''harmonikus'', ha </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>u_{xx}''+u_{yy}''\equiv \Delta u\equiv 0\, </math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">itt &Delta; a Laplace-operátor (nem a Laplace-transzformátor!, hanem a vektoranalízisbeli vektormezőre Hesse-mátrix nyoma).</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">A C--R-egyenletek mutatják, hogy ha f=u+iv reguláris, akkor u és v harmonikus függvények. Ugyanis:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>u_x'=v_y'\,</math> és</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>v_x'=-u_y'\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">De u és v Hesse-mátrixa is szimmetrikus, ezért:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>v_{yy}''=u_xy''=u_{yx}''=-v_{xx}''\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">azaz </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\Delta v\equiv 0\,</math> és fordítva.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td></tr>
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Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A3a_2008/12._gyakorlat&diff=4836&oldid=prev
Mozo: /* Típuspéldák */
2008-12-11T11:54:12Z
<p><span class="autocomment">Típuspéldák</span></p>
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<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2008. december 11., 11:54-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">105. sor:</td>
<td colspan="2" class="diff-lineno">105. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>ahol f(x) = sh(2x), a C-formulák miatt.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>ahol f(x) = sh(2x), a C-formulák miatt.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\left.\mathrm{sh}(2x)^{(c-1)}\right|_{z=0}=\left\{\begin{matrix}0, & \mathrm{ha} & c-1\;\;\mathrm{ps}\\\\2^{c-1}, & \mathrm{ha} & c-1\;\;\mathrm{pn}\end{matrix}\right.</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\left.\mathrm{sh}(2x)^{(c-1)}\right|_{z=0}=\left\{\begin{matrix}0, & \mathrm{ha} & c-1\;\;\mathrm{ps}\\\\2^{c-1}, & \mathrm{ha} & c-1\;\;\mathrm{pn}\end{matrix}\right.</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">===Integrálás véges sok szinguláris pontot körülhurkoló görbén===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Eszköz: A szinguláris helyek körül külön-külön kis körökön integrálunk, melyekre az integrált a reziduumtétellel, vagy Cauchy-formulákkal számoljuk.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''1.'''</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Legyen </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>f(z)=\frac{z^3}{(z^2+1)(z-1)}\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Mennyi a</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\oint\limits_{|z|=2}f(z)\mathrm{d}z</math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">integrál?</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">''1. Megoldás.'' Kiszámítjuk a &infin;-beli reziduumot. A &zeta; = 1/''z'' helyettesítést alkalmazva fejtjük sorba a &zeta;=0 körül:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>F(\zeta)=f(\frac{1}{\zeta})=\frac{\frac{1}{\zeta^3}}{(\frac{1}{\zeta^2}+1)(\frac{1}{\zeta}-1)}=\frac{1}{1-\zeta+\zeta^2-\zeta^3}</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Ezt a 0-ban Taylor-sorba fejthetjük:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>F(\zeta)=\sum\limits_{n=0}^\infty(\zeta-\zeta^2+\zeta^3)^n=1+\zeta-\zeta^2+\zeta^3+(...)^2+...</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Visszatranszformálva:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>f(z)=1+\frac{1}{z}+...\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">azaz a f valóban reguláris az &infin;-ben és reziduuma -1. Emiatt az integrál +2&pi;i, ugyanis</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\mathrm{Res}_1f+ \mathrm{Res}_if+ \mathrm{Res}_{-i}f+ \mathrm{Res}_\infty f=0\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">így</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\mathrm{Res}_1f+ \mathrm{Res}_if+ \mathrm{Res}_{-i}f=- \mathrm{Res}_\infty f=-(-1)\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">''2. Megoldás.'' Egyenkint, Cauchy-formulával.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\oint\limits_{|z-1|=1}\frac{\frac{z^3}{z^2+1}}{z-1}\mathrm{d}z+\oint\limits_{|z-i|=1}\frac{\frac{z^3}{(z+i)(z-1)}}{z-i}\mathrm{d}z+\oint\limits_{|z+i|=1}\frac{\frac{z^3}{(z-i)(z-1)}}{z+i}\mathrm{d}z=</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>=2\pi i\left.\frac{z^3}{z^2+1}\right|_{1}+2\pi i\left.\frac{z^3}{(z+i)(z-1)}\right|_{i}+2\pi i\left.\frac{z^3}{(z-i)(z-1)}\right|_{-i}=\frac{1}{2}+\frac{1}{4}(i+1)+\frac{1}{4}(1-i)=1</math></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A3a_2008/12._gyakorlat&diff=4835&oldid=prev
Mozo: /* Cauchy-típusú integrálok */
2008-12-11T11:49:36Z
<p><span class="autocomment">Cauchy-típusú integrálok</span></p>
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<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2008. december 11., 11:49-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">96. sor:</td>
<td colspan="2" class="diff-lineno">96. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>f^{(n)}(a)= \frac{n!}{2\pi i}\oint\limits_{G}\frac{f(z)}{(z-a)^{(n+1)}}\,\mathrm{d}z\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>f^{(n)}(a)= \frac{n!}{2\pi i}\oint\limits_{G}\frac{f(z)}{(z-a)^{(n+1)}}\,\mathrm{d}z\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''1.''' Számoljuk ki minden c &isin; '''Z'''-re<del class="diffchange diffchange-inline">:</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''1.''' Számoljuk ki minden c &isin; '''Z'''-re <ins class="diffchange diffchange-inline">az</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\oint\limits_{|z|=1}\frac{\<del class="diffchange diffchange-inline">sin</del>(2x)}{z^c}\,\mathrm{d}z\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math><ins class="diffchange diffchange-inline">I_c=</ins>\oint\limits_{|z|=1}\frac{\<ins class="diffchange diffchange-inline">mathrm{sh}</ins>(2x)}{z^c}\,\mathrm{d}z\,</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">integrált!</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">''Megoldás.''</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>I_c=0,\quad\quad\mathrm{ha}\quad\quad c\leq 0\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">mert ekkor az integrandus reguláris függvény, így a Cauchy-féle integráltétel miatt integrálja 0:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>I_c=\frac{2\pi if^{(c-1)}(0)}{(c-1)!},\quad\quad\mathrm{ha}\quad\quad c> 0\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">ahol f(x) = sh(2x), a C-formulák miatt.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>\left.\mathrm{sh}(2x)^{(c-1)}\right|_{z=0}=\left\{\begin{matrix}0, & \mathrm{ha} & c-1\;\;\mathrm{ps}\\\\2^{c-1}, & \mathrm{ha} & c-1\;\;\mathrm{pn}\end{matrix}\right.</math></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td></tr>
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Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A3a_2008/12._gyakorlat&diff=4834&oldid=prev
Mozo: /* Cauchy-típusú integrálok */
2008-12-11T11:36:01Z
<p><span class="autocomment">Cauchy-típusú integrálok</span></p>
<table class='diff diff-contentalign-left'>
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<col class='diff-marker' />
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<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2008. december 11., 11:36-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">94. sor:</td>
<td colspan="2" class="diff-lineno">94. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ha ''f'' reguláris, akkor minden az ''a''-t a belsejében tartalmazó G egyszerű zárt görbére:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Ha ''f'' reguláris, akkor minden az ''a''-t a belsejében tartalmazó G egyszerű zárt görbére:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>f^{(n)}(a)= \oint\limits_{G}\frac{f(z)}{(z-a)^{(n+1)}}\,\mathrm{d}z\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>f^{(n)}(a)= <ins class="diffchange diffchange-inline">\frac{n!}{2\pi i}</ins>\oint\limits_{G}\frac{f(z)}{(z-a)^{(n+1)}}\,\mathrm{d}z\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''1.''' Számoljuk ki minden c &isin; '''Z'''-re:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''1.''' Számoljuk ki minden c &isin; '''Z'''-re:</div></td></tr>
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Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A3a_2008/12._gyakorlat&diff=4833&oldid=prev
Mozo: /* Cauchy-integrálformulák */
2008-12-11T11:35:12Z
<p><span class="autocomment">Cauchy-integrálformulák</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
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<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2008. december 11., 11:35-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">89. sor:</td>
<td colspan="2" class="diff-lineno">89. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Res = 1/(5!)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Res = 1/(5!)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===Cauchy-<del class="diffchange diffchange-inline">integrálformulák</del>===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===Cauchy-<ins class="diffchange diffchange-inline">típusú integrálok</ins>===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Eszköz:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Eszköz:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2" class="diff-lineno">96. sor:</td>
<td colspan="2" class="diff-lineno">96. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>f^{(n)}(a)= \oint\limits_{G}\frac{f(z)}{(z-a)^{(n+1)}}\,\mathrm{d}z\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>f^{(n)}(a)= \oint\limits_{G}\frac{f(z)}{(z-a)^{(n+1)}}\,\mathrm{d}z\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''1.''' Számoljuk ki minden c &isin; '''Z'''-re:</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>\oint\limits_{|z|=1}\frac{\sin(2x)}{z^c}\,\mathrm{d}z\,</math></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Kategória:Matematika A3]]</div></td></tr>
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Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A3a_2008/12._gyakorlat&diff=4832&oldid=prev
Mozo: /* Reziduum-számítás */
2008-12-11T11:33:42Z
<p><span class="autocomment">Reziduum-számítás</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
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<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2008. december 11., 11:33-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">88. sor:</td>
<td colspan="2" class="diff-lineno">88. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>f(z)=\frac{\sin z}{z^6}=\frac{1}{z^5}-\frac{1}{3!}\frac{1}{z^3}+\frac{1}{5!}\frac{1}{z}-...\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>f(z)=\frac{\sin z}{z^6}=\frac{1}{z^5}-\frac{1}{3!}\frac{1}{z^3}+\frac{1}{5!}\frac{1}{z}-...\,</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Res = 1/(5!)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Res = 1/(5!)</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">===Cauchy-integrálformulák===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Eszköz:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Ha ''f'' reguláris, akkor minden az ''a''-t a belsejében tartalmazó G egyszerű zárt görbére:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>f^{(n)}(a)= \oint\limits_{G}\frac{f(z)}{(z-a)^{(n+1)}}\,\mathrm{d}z\,</math></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
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Mozo