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Complex Numbers - Finding roots

  1. Apr 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Solve the equation z^4= -i


    2. Relevant equations
    De Moivre's Theorem


    3. The attempt at a solution
    I understand how to find the roots by equating modulus and argument but I wanted to ask how do you know which arguments to take? Because I got up to

    4*theta = -Pi/2, 3*Pi/2, 7*Pi/2

    then I thought that I should take -5*Pi/2 because I thought that the final argument should lie between -Pi and Pi. Is that wrong? Because the answers took 11*Pi/2 instead...

    I don't understand. Please help.
     
  2. jcsd
  3. Apr 15, 2009 #2

    CompuChip

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    Science Advisor
    Homework Helper

    Short answer: you are right.

    Long answer:

    First of all, note that for any integer n,
    [tex]e^{i \theta} = e^{i (\theta + 2 \pi)}.[/tex]

    However, it is convention to normalize angles by subtracting multiples of 2 pi. Unfortunately two different conventions are in use: some people prefer to take all angles between -pi and pi, some prefer to use angles between 0 and 2 pi. If you are using the first [second] convention and you get a negative angle [angle > pi] you can always go to the other convention by adding [subtracting] 2pi.

    Now, solving z4 = -i you first write [itex]z = r e^{i \theta}, i = e^{-i \pi}[/itex]. The modulus equation gives r = 1, and then you get
    [tex]e^{4 i \theta} = e^{-i \pi / 2}[/tex]
    (I am using the convention of angles between -pi and pi here, otherwise you would get 3pi/2 on the RHS).
    The solution is given by
    [tex]4 \theta = - \pi / 2 + 2 \pi n[/tex]
    so -- dividing by 4 --
    [tex]\theta = - \frac{\pi}{8} + n \frac{pi}{2}[/tex]

    Now all you have to do is plug in values of n to get all the inequivalent angles between -pi and pi. You will find
    4theta = -5*Pi/8, -Pi/2, 3*Pi/2, 7*Pi/2

    If instead, you use the convention that angles should be between 0 and pi, you have to add 2 pi to the first two (i.e. add 8pi to 4 times the angle), and you get
    4theta = 3*Pi/2, 7*Pi/2, 11*Pi/2, 15*Pi/2.

    The book is apparently mixing the two conventions. Of course the answers are right, and if you want you could have written down
    4theta = +75*Pi/2, -33*Pi/2, 3*Pi/2, -1593*Pi/2
    if you wanted.
     
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