3.3 Övningar
Sommarmatte 2
(Skillnad mellan versioner)
| Versionen från 28 juni 2007 kl. 13.43 (redigera) KTH.SE:u1rp004j (Diskussion | bidrag) (→Övning 3.3:7) ← Gå till föregående ändring |
Versionen från 3 juli 2007 kl. 19.17 (redigera) (ogör) KTH.SE:u1tyze7e (Diskussion | bidrag) (Korrekturläst) Gå till nästa ändring → |
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| Rad 1: | Rad 1: | ||
| ==Övning 3.3:1== | ==Övning 3.3:1== | ||
| <div class="ovning"> | <div class="ovning"> | ||
| - | Skriv följande tal i formen $a+ib$, där $a$ och $b$ är reella tal. | + | Skriv följande tal i formen $\,a+ib\,$, där $\,a\,$ och $\,b\,$ är reella tal. |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| Rad 7: | Rad 7: | ||
| <td class="ntext" width="50%">$(i-1)^{12}$</td> | <td class="ntext" width="50%">$(i-1)^{12}$</td> | ||
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$\left(\displaystyle\frac{1+i\sqrt{3}}{2}\right)^{12}$</td> | + | <td class="ntext" width="50%">$\displaystyle\Bigl(\frac{1+i\sqrt{3}}{2}\,\Bigr)^{12}$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| Rad 13: | Rad 13: | ||
| <td class="ntext" width="50%">$ (4\sqrt{3} -4i)^{22}$</td> | <td class="ntext" width="50%">$ (4\sqrt{3} -4i)^{22}$</td> | ||
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$\left(\displaystyle\frac{1+i\sqrt{3}}{1+i}\right)^{12}$</td> | + | <td class="ntext" width="50%">$\Bigl(\displaystyle\frac{1+i\sqrt{3}}{1+i}\,\Bigr)^{12}$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">e)</td> | <td class="ntext">e)</td> | ||
| - | <td class="ntext" width="50%">$\displaystyle\frac{(1+i\sqrt{3})(1-i)^8}{(\sqrt{3}-i)^9}$</td> | + | <td class="ntext" width="50%">$\displaystyle\frac{(1+i\sqrt{3}\,)(1-i)^8}{(\sqrt{3}-i)^9}$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 25: | Rad 25: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$-64$</td> | + | <td class="ntext" width="50%">$-64$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$1$</td> | + | <td class="ntext" width="50%">$1$</td> |
| - | <td class="ntext">c)</td> | + | |
| - | <td class="ntext" width="33%">$ 4\cdot 8^{21} - i \cdot 4\sqrt{3}\cdot 8^{21} $</td> | + | |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="50%">$2^{65}+2^{65}\sqrt{3}\,i$</td> | ||
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| <td class="ntext" width="50%">$-64$</td> | <td class="ntext" width="50%">$-64$</td> | ||
| + | </tr> | ||
| + | <tr align="left"> | ||
| <td class="ntext">e)</td> | <td class="ntext">e)</td> | ||
| - | <td class="ntext" width="50%">$\frac{\sqrt{3}}{32} - \frac{i}{32} $</td> | + | <td class="ntext" width="50%">$\displaystyle\frac{\sqrt{3}}{32} - \frac{i}{32} $</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 53: | Rad 55: | ||
| <td class="ntext" width="50%">$(1+i)^{1/3}$</td> | <td class="ntext" width="50%">$(1+i)^{1/3}$</td> | ||
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$i^{1/4}$</td> | + | <td class="ntext" width="50%">$i^{\,1/4}$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$((3+3i)(\sqrt{2}-\sqrt{2}i))^{1/5}$</td> | + | <td class="ntext" width="50%">$\bigl((3+3i)(\sqrt{2}-\sqrt{2}\,i\,)\bigr)^{1/5}$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 65: | Rad 67: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br/> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$ \frac{2^{2/3}(\sqrt{3}+1)}{4} + i\frac{2^{2/3}(\sqrt{3}-1)}{4} $</td> | + | <td class="ntext" width="100%">$\displaystyle 2^{1/6}\,\Bigl(\cos\Bigl(\frac{\pi}{12}+\frac{2k\pi}{3}\Bigr)+i\,\sin\Bigl(\frac{\pi}{12}+\frac{2k\pi}{3}\Bigr)\,\Bigr)\quad$ för $\ k=0,1,2$</td> |
| + | </tr> | ||
| + | <tr align="left"> | ||
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$e^{i\pi/8} = \frac{\sqrt{2+\sqrt{2}}}{2} + i\frac{\sqrt{2-\sqrt{2}}}{2} $</td> | + | <td class="ntext" width="100%">$\displaystyle \cos\Bigl(\frac{\pi}{8}+\frac{k\pi}{2}\Bigr)+i\,\sin\Bigl(\frac{\pi}{8}+\frac{k\pi}{2}\,\Bigr)\quad$ för $\ k=0,1,2,3$</td> |
| + | </tr> | ||
| + | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%">$ 3^{1/5}2^{3/10} $</td> | + | <td class="ntext" width="50%">$2^{3/10}\,3^{1/5}$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 93: | Rad 99: | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$(z-1)^4+4=0$</td> | + | <td class="ntext" width="33%">$(z-1)^4+4=0$</td> |
| <td class="ntext">e)</td> | <td class="ntext">e)</td> | ||
| - | <td class="ntext" width="50%">$\left(\displaystyle \frac{z+i}{z-i}\right)^2 = -1$</td> | + | <td class="ntext" width="33%">$\displaystyle\Bigl(\frac{z+i}{z-i}\Bigr)^2 = -1$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 103: | Rad 109: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="33%">$z= \left\{\begin{matrix} 1 \\ -1 \\ i \\ -i\\ \end{matrix}\right.$</td> | + | <td class="ntext" width="33%">$z= \left\{\begin{matrix} \phantom{-}1 \\ -1 \\ \phantom{-}i \\ -i\\ \end{matrix}\right.$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="33%">$z = \left\{\begin{matrix} \frac{1}{2}+\frac{\sqrt{3}i}{2} \\ -1 \\ \frac{1}{2}-\frac{\sqrt{3}i}{2} \\ \end{matrix}\right.$</td> | + | <td class="ntext" width="33%">$z = \left\{\begin{matrix} \frac{1}{2}+\frac{1}{2}\sqrt{3}\,i \\ -1\phantom{{}-\frac{1}{2}\sqrt{3}\,i} \\ \frac{1}{2}-\frac{1}{2}\sqrt{3}\,i \\ \end{matrix}\right.$</td> |
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="33%">$z= \left\{\begin{matrix} 2^{1/10}e^{i\pi/4} \\ 2^{1/10}e^{13i\pi/20} \\ 2^{1/10}e^{21i\pi/20} \\ 2^{1/10}e^{29i\pi/20}\\ 2^{1/10}e^{37i\pi/20}\\ \end{matrix}\right. $</td> | + | <td class="ntext" width="33%">$\displaystyle z=2^{1/10}\exp\Bigl(\frac{\pi i}{4}+\frac{2k\pi i}{5}\Bigr)$ för $\ k=0,1,2,3,4$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$z= \left\{\begin{matrix} 2+i \\ 2-i \\ i \\ -i\\ \end{matrix}\right.$</td> | + | <td class="ntext" width="33%">$z= \left\{\begin{matrix} 2+i \\ 2-i \\ \phantom{-}i \\ -i\\ \end{matrix}\right.$</td> |
| <td class="ntext">e)</td> | <td class="ntext">e)</td> | ||
| - | <td class="ntext" width="50%">$z = \left\{\begin{matrix} 1 \\ -1 \end{matrix}\right.$</td> | + | <td class="ntext" width="33%">$z = \left\{\begin{matrix} \phantom{-}1 \\ -1 \end{matrix}\right.$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 145: | Rad 151: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| Rad 151: | Rad 157: | ||
| <td class="ntext" width="50%">$(z+1)^2+2$</td> | <td class="ntext" width="50%">$(z+1)^2+2$</td> | ||
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$\left(z+\frac{3}{2}i\right)^2+2$</td> | + | <td class="ntext" width="50%">$\left(z+\frac{3}{2}i\,\right)^2+2$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$-(z+i-2)^2+4(1-i)$</td> | + | <td class="ntext" width="50%">$-(z-2+i)^2+4(1-i)$</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$i\left(z+\left(\frac{2/i+3}{2}\right)\right)^2-4-\displaystyle\frac{5}{4}i$</td> | + | <td class="ntext" width="50%">$i\bigl(z+\frac{3}{2}-i\bigl)^2-4-\frac{5}{4}\,i$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 185: | Rad 191: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="50%">$z= \left\{\begin{matrix} \frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2} \\ \end{matrix}\right.$</td> | + | <td class="ntext" width="50%">$z= \left\{\begin{matrix} \phantom{-}(1+i)/\sqrt{2}\\ -(1+i)/\sqrt{2}\\ \end{matrix}\right.$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| <td class="ntext" width="50%">$z = \left\{\begin{matrix} 2+i \\ 2-i \\ \end{matrix}\right.$</td> | <td class="ntext" width="50%">$z = \left\{\begin{matrix} 2+i \\ 2-i \\ \end{matrix}\right.$</td> | ||
| Rad 195: | Rad 201: | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$z= \left\{\begin{matrix} -1 \\ 3 \\ \end{matrix}\right. $</td> | + | <td class="ntext" width="50%">$z= \left\{\begin{matrix} -1 \\ \phantom{-}3 \\ \end{matrix}\right. $</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$z= \left\{\begin{matrix} \frac{1+i\sqrt{15}}{4}\\ \frac{1-i\sqrt{15}}{4} \\ \end{matrix}\right.$</td> | + | <td class="ntext" width="50%">$z= \left\{\begin{matrix} (1+i\sqrt{15})/4\\ (1-i\sqrt{15})/4 \end{matrix}\right.$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 209: | Rad 215: | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext" width="50%">$z^2-2(1+i)z+2i-1$</td> | + | <td class="ntext" width="50%">$z^2-2(1+i)z+2i-1=0$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| <td class="ntext" width="50%">$z^2-(2-i)z+(3-i)=0$</td> | <td class="ntext" width="50%">$z^2-(2-i)z+(3-i)=0$</td> | ||
| Rad 215: | Rad 221: | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$z^2-(1+3i)z-4+3i$</td> | + | <td class="ntext" width="50%">$z^2-(1+3i)z-4+3i=0$</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| <td class="ntext" width="50%">$(4+i)z^2+(1-21i)z=17$</td> | <td class="ntext" width="50%">$(4+i)z^2+(1-21i)z=17$</td> | ||
| Rad 225: | Rad 231: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| Rad 231: | Rad 237: | ||
| <td class="ntext" width="50%">$z= \left\{\begin{matrix} 2+1 \\ i \\ \end{matrix}\right.$</td> | <td class="ntext" width="50%">$z= \left\{\begin{matrix} 2+1 \\ i \\ \end{matrix}\right.$</td> | ||
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext" width="50%">$z = \left\{\begin{matrix} 1+i \\ 1-2i \\ \end{matrix}\right.$</td> | + | <td class="ntext" width="50%">$z = \left\{\begin{matrix} 1+i\phantom{2} \\ 1-2i \\ \end{matrix}\right.$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext" width="50%">$z= \left\{\begin{matrix} 2+i \\ -1+2i \\ \end{matrix}\right. $</td> | + | <td class="ntext" width="50%">$z= \left\{\begin{matrix} \phantom{-}2+i\phantom{2} \\ -1+2i \\ \end{matrix}\right. $</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext" width="50%">$z= \left\{\begin{matrix} -i \\ -1-4i \\ \end{matrix}\right.$</td> | + | <td class="ntext" width="50%">$z= \left\{\begin{matrix} i \\ 1+4i \\ \end{matrix}\right.$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| Rad 245: | Rad 251: | ||
| ==Övning 3.3:7== | ==Övning 3.3:7== | ||
| <div class="ovning"> | <div class="ovning"> | ||
| - | Bestäm lösningarna till $z^2=1+i$ dels i polär form, dels i formen $a+ib$, där $a$ och $b$ är reella tal. Använd resultatet för att beräkna $\tan \frac{\pi}{8}$. | + | Bestäm lösningarna till $\,z^2=1+i\,$ dels i polär form, dels i formen $\,a+ib\,$, där $\,a\,$ och $\,b\,$ är reella tal. Använd resultatet för att beräkna $\ \tan \frac{\pi}{8}\,$. |
| </div> | </div> | ||
| Rad 251: | Rad 257: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor<br \> |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">Lösningar:</td> | <td class="ntext">Lösningar:</td> | ||
| - | <td class="ntext" width="100%">$z= \left\{\begin{matrix} 2^{1/4}(\cos\frac{\pi}{8}+i\sin\frac{\pi}{8}) = \sqrt{\frac{\sqrt{2}+1 }{2}}+i \sqrt{\frac{\sqrt{2}-1 }{2}} \\ 2^{1/4}(\cos\frac{9\pi}{8}+i\sin\frac{9\pi}{8}) = -\sqrt{\frac{\sqrt{2}+1 }{2}}-i \sqrt{\frac{\sqrt{2}-1 }{2}} \\ \end{matrix}\right.$</td> | + | <td class="ntext" width="100%">$z= \left\{\eqalign{&\textstyle\sqrt[\scriptstyle 4]{2}\bigl(\cos\frac{\pi}{8}+i\,\sin\frac{\pi}{8}\bigr)\cr &\textstyle\sqrt[\scriptstyle 4]{2}\bigl(\cos\frac{9\pi}{8}+i\,\sin\frac{9\pi}{8}\bigr)}\right. = \left\{\eqalign{&\textstyle\phantom{-}{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}+2}\vphantom{2^{2^{\scriptstyle 2}}}}+i\,{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}-2}\vphantom{2^{2^{\scriptstyle 2}}}}\cr &\textstyle -{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}+2}\vphantom{2^{2^{\scriptstyle 2}}}}-i\,{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}-2}\vphantom{2^{2^{\scriptstyle 2}}}}}\right.$</td> |
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">Uttryck:</td> | <td class="ntext">Uttryck:</td> | ||
| - | <td class="ntext" width="100%">$\tan \frac{\pi}{8} = \sqrt{2} - 1$</td> | + | <td class="ntext" width="100%">$\displaystyle\tan \frac{\pi}{8} = \sqrt{2} - 1$</td> |
| </tr> | </tr> | ||
| </table> | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
Versionen från 3 juli 2007 kl. 19.17
Innehåll |
Övning 3.3:1
Skriv följande tal i formen $\,a+ib\,$, där $\,a\,$ och $\,b\,$ är reella tal.
| a) | $(i-1)^{12}$ | b) | $\displaystyle\Bigl(\frac{1+i\sqrt{3}}{2}\,\Bigr)^{12}$ |
| c) | $ (4\sqrt{3} -4i)^{22}$ | d) | $\Bigl(\displaystyle\frac{1+i\sqrt{3}}{1+i}\,\Bigr)^{12}$ |
| e) | $\displaystyle\frac{(1+i\sqrt{3}\,)(1-i)^8}{(\sqrt{3}-i)^9}$ |
Facit
Facit till alla delfrågor
| a) | $-64$ | b) | $1$ |
| c) | $2^{65}+2^{65}\sqrt{3}\,i$ | d) | $-64$ |
| e) | $\displaystyle\frac{\sqrt{3}}{32} - \frac{i}{32} $ |
Övning 3.3:2
Beräkna följande rotuttryck
| a) | $(1+i)^{1/3}$ | b) | $i^{\,1/4}$ |
| c) | $\bigl((3+3i)(\sqrt{2}-\sqrt{2}\,i\,)\bigr)^{1/5}$ |
Facit
Facit till alla delfrågor
| a) | $\displaystyle 2^{1/6}\,\Bigl(\cos\Bigl(\frac{\pi}{12}+\frac{2k\pi}{3}\Bigr)+i\,\sin\Bigl(\frac{\pi}{12}+\frac{2k\pi}{3}\Bigr)\,\Bigr)\quad$ för $\ k=0,1,2$ |
| b) | $\displaystyle \cos\Bigl(\frac{\pi}{8}+\frac{k\pi}{2}\Bigr)+i\,\sin\Bigl(\frac{\pi}{8}+\frac{k\pi}{2}\,\Bigr)\quad$ för $\ k=0,1,2,3$ |
| c) | $2^{3/10}\,3^{1/5}$ |
Övning 3.3:3
Lös ekvationerna
| a) | $z^4=1$ | b) | $z^3=-1$ | c) | $ z^5=-1-i$ |
| d) | $(z-1)^4+4=0$ | e) | $\displaystyle\Bigl(\frac{z+i}{z-i}\Bigr)^2 = -1$ |
Facit
Facit till alla delfrågor
| a) | $z= \left\{\begin{matrix} \phantom{-}1 \\ -1 \\ \phantom{-}i \\ -i\\ \end{matrix}\right.$ | b) | $z = \left\{\begin{matrix} \frac{1}{2}+\frac{1}{2}\sqrt{3}\,i \\ -1\phantom{{}-\frac{1}{2}\sqrt{3}\,i} \\ \frac{1}{2}-\frac{1}{2}\sqrt{3}\,i \\ \end{matrix}\right.$ | c) | $\displaystyle z=2^{1/10}\exp\Bigl(\frac{\pi i}{4}+\frac{2k\pi i}{5}\Bigr)$ för $\ k=0,1,2,3,4$ |
| d) | $z= \left\{\begin{matrix} 2+i \\ 2-i \\ \phantom{-}i \\ -i\\ \end{matrix}\right.$ | e) | $z = \left\{\begin{matrix} \phantom{-}1 \\ -1 \end{matrix}\right.$ |
Övning 3.3:4
Kvadratkomplettera följande uttryck
| a) | $z^2 +2z+3$ | b) | $z^2 +3iz-\frac{1}{4}$ |
| c) | $-z^2-2iz +4z+1$ | d) | $iz^2+(2+3i)z-1$ |
Facit
Facit till alla delfrågor
| a) | $(z+1)^2+2$ | b) | $\left(z+\frac{3}{2}i\,\right)^2+2$ |
| c) | $-(z-2+i)^2+4(1-i)$ | d) | $i\bigl(z+\frac{3}{2}-i\bigl)^2-4-\frac{5}{4}\,i$ |
Övning 3.3:5
Lös ekvationerna
| a) | $z^2=i$ | b) | $z^2-4z+5=0$ |
| c) | $-z^2+2z+3=0$ | d) | $\displaystyle\frac{1}{z} + z = \frac{1}{2}$ |
Facit
Facit till alla delfrågor
| a) | $z= \left\{\begin{matrix} \phantom{-}(1+i)/\sqrt{2}\\ -(1+i)/\sqrt{2}\\ \end{matrix}\right.$ | b) | $z = \left\{\begin{matrix} 2+i \\ 2-i \\ \end{matrix}\right.$ |
| c) | $z= \left\{\begin{matrix} -1 \\ \phantom{-}3 \\ \end{matrix}\right. $ | d) | $z= \left\{\begin{matrix} (1+i\sqrt{15})/4\\ (1-i\sqrt{15})/4 \end{matrix}\right.$ |
Övning 3.3:6
Lös ekvationerna
| a) | $z^2-2(1+i)z+2i-1=0$ | b) | $z^2-(2-i)z+(3-i)=0$ |
| c) | $z^2-(1+3i)z-4+3i=0$ | d) | $(4+i)z^2+(1-21i)z=17$ |
Facit
Facit till alla delfrågor
| a) | $z= \left\{\begin{matrix} 2+1 \\ i \\ \end{matrix}\right.$ | b) | $z = \left\{\begin{matrix} 1+i\phantom{2} \\ 1-2i \\ \end{matrix}\right.$ |
| c) | $z= \left\{\begin{matrix} \phantom{-}2+i\phantom{2} \\ -1+2i \\ \end{matrix}\right. $ | d) | $z= \left\{\begin{matrix} i \\ 1+4i \\ \end{matrix}\right.$ |
Övning 3.3:7
Bestäm lösningarna till $\,z^2=1+i\,$ dels i polär form, dels i formen $\,a+ib\,$, där $\,a\,$ och $\,b\,$ är reella tal. Använd resultatet för att beräkna $\ \tan \frac{\pi}{8}\,$.
Facit
Facit till alla delfrågor
| Lösningar: | $z= \left\{\eqalign{&\textstyle\sqrt[\scriptstyle 4]{2}\bigl(\cos\frac{\pi}{8}+i\,\sin\frac{\pi}{8}\bigr)\cr &\textstyle\sqrt[\scriptstyle 4]{2}\bigl(\cos\frac{9\pi}{8}+i\,\sin\frac{9\pi}{8}\bigr)}\right. = \left\{\eqalign{&\textstyle\phantom{-}{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}+2}\vphantom{2^{2^{\scriptstyle 2}}}}+i\,{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}-2}\vphantom{2^{2^{\scriptstyle 2}}}}\cr &\textstyle -{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}+2}\vphantom{2^{2^{\scriptstyle 2}}}}-i\,{\textstyle\frac{1}{2}}\sqrt{\smash{2\sqrt{2}-2}\vphantom{2^{2^{\scriptstyle 2}}}}}\right.$ |
| Uttryck: | $\displaystyle\tan \frac{\pi}{8} = \sqrt{2} - 1$ |
