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3.3 Übungen

Aus Online Mathematik Brückenkurs 2

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|width="50%"| <math>\displaystyle\frac{(1+i\sqrt{3}\,)(1-i)^8}{(\sqrt{3}-i)^9}</math>
|width="50%"| <math>\displaystyle\frac{(1+i\sqrt{3}\,)(1-i)^8}{(\sqrt{3}-i)^9}</math>
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</div>{{#NAVCONTENT:Answer|Answer 3.3:1|Solution a|Solution 3.3:1a|Solution b|Solution 3.3:1b|Solution c|Solution 3.3:1c|Solution d|Solution 3.3:1d|Solution e|Solution 3.3:1e}}
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</div>{{#NAVCONTENT:Antwort|Antwort 3.3:1|Solution a|Solution 3.3:1a|Solution b|Solution 3.3:1b|Solution c|Solution 3.3:1c|Solution d|Solution 3.3:1d|Solution e|Solution 3.3:1e}}
===Übung 3.3:2===
===Übung 3.3:2===
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|width="33%"| <math>\displaystyle\Bigl(\frac{z+i}{z-i}\Bigr)^2 = -1</math>
|width="33%"| <math>\displaystyle\Bigl(\frac{z+i}{z-i}\Bigr)^2 = -1</math>
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</div>{{#NAVCONTENT:Answer|Answer 3.3:2|Solution a|Solution 3.3:2a|Solution b|Solution 3.3:2b|Solution c|Solution 3.3:2c|Solution d|Solution 3.3:2d|Solution e|Solution 3.3:2e}}
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</div>{{#NAVCONTENT:Antwort|Antwort 3.3:2|Solution a|Solution 3.3:2a|Solution b|Solution 3.3:2b|Solution c|Solution 3.3:2c|Solution d|Solution 3.3:2d|Solution e|Solution 3.3:2e}}
===Übung 3.3:3===
===Übung 3.3:3===
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|width="50%"| <math>iz^2+(2+3i)z-1</math>
|width="50%"| <math>iz^2+(2+3i)z-1</math>
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</div>{{#NAVCONTENT:Answer|Answer 3.3:3|Solution a|Solution 3.3:3a|Solution b|Solution 3.3:3b|Solution c|Solution 3.3:3c|Solution d|Solution 3.3:3d}}
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</div>{{#NAVCONTENT:Antwort|Antwort 3.3:3|Solution a|Solution 3.3:3a|Solution b|Solution 3.3:3b|Solution c|Solution 3.3:3c|Solution d|Solution 3.3:3d}}
===Übung 3.3:4===
===Übung 3.3:4===
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|width="50%"| <math>\displaystyle\frac{1}{z} + z = \frac{1}{2}</math>
|width="50%"| <math>\displaystyle\frac{1}{z} + z = \frac{1}{2}</math>
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</div>{{#NAVCONTENT:Answer|Answer 3.3:4|Solution a|Solution 3.3:4a|Solution b|Solution 3.3:4b|Solution c|Solution 3.3:4c|Solution d|Solution 3.3:4d}}
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</div>{{#NAVCONTENT:Antwort|Antwort 3.3:4|Solution a|Solution 3.3:4a|Solution b|Solution 3.3:4b|Solution c|Solution 3.3:4c|Solution d|Solution 3.3:4d}}
===Übung 3.3:5===
===Übung 3.3:5===
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|width="50%"| <math>(4+i)z^2+(1-21i)z=17</math>
|width="50%"| <math>(4+i)z^2+(1-21i)z=17</math>
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</div>{{#NAVCONTENT:Answer|Answer 3.3:5|Solution a|Solution 3.3:5a|Solution b|Solution 3.3:5b|Solution c|Solution 3.3:5c|Solution d|Solution 3.3:5d}}
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</div>{{#NAVCONTENT:Antwort|Antwort 3.3:5|Solution a|Solution 3.3:5a|Solution b|Solution 3.3:5b|Solution c|Solution 3.3:5c|Solution d|Solution 3.3:5d}}
===Übung 3.3:6===
===Übung 3.3:6===
<div class="ovning">
<div class="ovning">
Determine the solution to <math>\,z^2=1+i\,</math> both in polar form and in the form <math>\,a+ib\,</math>, where <math>\,a\,</math> and <math>\,b\,</math> are real numbers. Use the result to calculate <math>\; \tan \frac{\pi}{8}\,</math>.
Determine the solution to <math>\,z^2=1+i\,</math> both in polar form and in the form <math>\,a+ib\,</math>, where <math>\,a\,</math> and <math>\,b\,</math> are real numbers. Use the result to calculate <math>\; \tan \frac{\pi}{8}\,</math>.
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</div>{{#NAVCONTENT:Answer|Answer 3.3:6|Solution|Solution 3.3:6}}
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</div>{{#NAVCONTENT:Antwort|Antwort 3.3:6|Solution|Solution 3.3:6}}

Version vom 13:31, 10. Mär. 2009

       Theorie          Übungen      

Übung 3.3:1

Write the following number in the form a+ib, where a and b are real numbers:

a) (i+1)12 b) 21+i312 
c) (43−4i)22  d) 1+i1+i312 
e) (3−i)9(1+i3)(1−i)8

Antwort | Solution a | Solution b | Solution c | Solution d | Solution e

Übung 3.3:2

Solve the equations

a) z4=1 b) z3=−1 c) z5=−1−i
d) (z−1)4+4=0 e) z−iz+i2=−1 

Antwort | Solution a | Solution b | Solution c | Solution d | Solution e

Übung 3.3:3

Complete the square of the following expressions

a) z2+2z+3 b) z2+3iz−41
c) −z2−2iz+4z+1 d) iz2+(2+3i)z−1

Antwort | Solution a | Solution b | Solution c | Solution d

Übung 3.3:4

Solve the equations

a) z2=i b) z2−4z+5=0
c) −z2+2z+3=0 d) z1+z=21

Antwort | Solution a | Solution b | Solution c | Solution d

Übung 3.3:5

Solve the equations

a) z2−2(1+i)z+2i−1=0 b) z2−(2−i)z+(3−i)=0
c) z2−(1+3i)z−4+3i=0 d) (4+i)z2+(1−21i)z=17

Antwort | Solution a | Solution b | Solution c | Solution d

Übung 3.3:6

Determine the solution to z2=1+i both in polar form and in the form a+ib, where a and b are real numbers. Use the result to calculate tan8.

Antwort | Solution

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/3.3_%C3%9Cbungen“