3.3 Übungen
Aus Online Mathematik Brückenkurs 2
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Exercise 3.3:1
Write the following number in the form \displaystyle \,a+ib\,, where \displaystyle \,a\, and \displaystyle \,b\, are real numbers:
| a) | \displaystyle (i+1)^{12} | b) | \displaystyle \displaystyle\Bigl(\frac{1+i\sqrt{3}}{2}\,\Bigr)^{12} |
| c) | \displaystyle (4\sqrt{3} -4i)^{22} | d) | \displaystyle \Bigl(\displaystyle\frac{1+i\sqrt{3}}{1+i}\,\Bigr)^{12} |
| e) | \displaystyle \displaystyle\frac{(1+i\sqrt{3}\,)(1-i)^8}{(\sqrt{3}-i)^9} |
Answer
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Solution a
Solution b
Solution c
Solution d
Solution e
Exercise 3.3:2
Solve the equations
| a) | \displaystyle z^4=1 | b) | \displaystyle z^3=-1 | c) | \displaystyle z^5=-1-i |
| d) | \displaystyle (z-1)^4+4=0 | e) | \displaystyle \displaystyle\Bigl(\frac{z+i}{z-i}\Bigr)^2 = -1 |
Answer
Solution a
Solution b
Solution c
Solution d
Solution e
Exercise 3.3:3
Complete the square of the following expressions
| a) | \displaystyle z^2 +2z+3 | b) | \displaystyle z^2 +3iz-\frac{1}{4} |
| c) | \displaystyle -z^2-2iz +4z+1 | d) | \displaystyle iz^2+(2+3i)z-1 |
Answer
Solution a
Solution b
Solution c
Solution d
Exercise 3.3:4
Solve the equations
| a) | \displaystyle z^2=i | b) | \displaystyle z^2-4z+5=0 |
| c) | \displaystyle -z^2+2z+3=0 | d) | \displaystyle \displaystyle\frac{1}{z} + z = \frac{1}{2} |
Answer
Solution a
Solution b
Solution c
Solution d
Exercise 3.3:5
Solve the equations
| a) | \displaystyle z^2-2(1+i)z+2i-1=0 | b) | \displaystyle z^2-(2-i)z+(3-i)=0 |
| c) | \displaystyle z^2-(1+3i)z-4+3i=0 | d) | \displaystyle (4+i)z^2+(1-21i)z=17 |
Answer
Solution a
Solution b
Solution c
Solution d
Exercise 3.3:6
Determine the solution to \displaystyle \,z^2=1+i\, both in polar form and in the form \displaystyle \,a+ib\,, where \displaystyle \,a\, and \displaystyle \,b\, are real numbers. Use the result to calculate \displaystyle \; \tan \frac{\pi}{8}\,.
Answer
Solution
