- #1
Atena
- 1
- 0
Hello. I'm not sure whether I did this right or messed up somewhere, just need to confirm my results...thanks to anybody who bothers answering.
Find all the roots of [tex]z^{4}=1-i[/tex]
I guess I should state De Moivre's here...
[tex](r cis(\vartheta))^{n}=r^{n} cis (n\vartheta)[/tex]
Firstly I re-wrote [tex]z^{4}=1-i[/tex] as
[tex]z^{4}=\sqrt{2} cis (\frac{-\pi}{4})[/tex]
Using De Moivre's,
[tex]z=(2\frac{1}{2})^{\frac{1}{4}} cis (\frac{1}{4}(\frac{-\pi}{4}+k2\pi))[/tex]
[tex]z=2\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+k2\pi))[/tex]
I found the four roots letting k=0,1,2,3
[tex]z=2^\frac{1}{8} cis (\frac{-\pi}{16})[/tex]
[tex]z=2^\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+2\pi))=2^\frac{1}{8} cis (\frac{7\pi}{16})[/tex]
[tex]z=2^\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+4\pi))=2^\frac{1}{8} cis (\frac{15\pi}{16})[/tex]
[tex]z=2^\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+6\pi))=2^\frac{1}{8} cis (\frac{23\pi}{16})[/tex]
Homework Statement
Find all the roots of [tex]z^{4}=1-i[/tex]
Homework Equations
I guess I should state De Moivre's here...
[tex](r cis(\vartheta))^{n}=r^{n} cis (n\vartheta)[/tex]
The Attempt at a Solution
Firstly I re-wrote [tex]z^{4}=1-i[/tex] as
[tex]z^{4}=\sqrt{2} cis (\frac{-\pi}{4})[/tex]
Using De Moivre's,
[tex]z=(2\frac{1}{2})^{\frac{1}{4}} cis (\frac{1}{4}(\frac{-\pi}{4}+k2\pi))[/tex]
[tex]z=2\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+k2\pi))[/tex]
I found the four roots letting k=0,1,2,3
[tex]z=2^\frac{1}{8} cis (\frac{-\pi}{16})[/tex]
[tex]z=2^\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+2\pi))=2^\frac{1}{8} cis (\frac{7\pi}{16})[/tex]
[tex]z=2^\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+4\pi))=2^\frac{1}{8} cis (\frac{15\pi}{16})[/tex]
[tex]z=2^\frac{1}{8} cis (\frac{1}{4}(\frac{-\pi}{4}+6\pi))=2^\frac{1}{8} cis (\frac{23\pi}{16})[/tex]