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Math Help - Finding Solutions using Complex Numbers

  1. #1
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    Finding Solutions using Complex Numbers

    I have been having trouble doing practice quesions on finding the solutions using complex numbers. I feel like I understand it on a very superficial level as I can solve simple ones, but I have trouble when I need to solve questions like z^5 = 12exp(i*pi/2) which shouldn't be too hard since it is already in polar form.
    It would be greatly appreciated someone could give me a quick rundown on how to properly solve these types of questions or point me a good resource on them.

    Edit: An example of what I am having trouble with:

    z^2 = -7*sqrt(2)/2 + 7*sqrt(2)*i/2

    Using the form z = r*exp(a*i), I find r to be 7 and a to be pi/4

    So z = sqrt(7)exp(pi*n*i/8)

    This is where I get stuck. The examples we did in class only involved then plugging 0, 1, 2, 3, etc until the results looped. However, the awnsers provided for this question are sqrt(7)exp(3*pi*i/8) and sqrt(7)exp(11*pi*i/8). What should I be doing instead?
    Last edited by Vanilla; July 16th 2011 at 07:45 PM.
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  2. #2
    Grand Panjandrum
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    Re: Finding Solutions using Complex Numbers

    Quote Originally Posted by Vanilla View Post
    I have been having trouble doing practice quesions on finding the solutions using complex numbers. I feel like I understand it on a very superficial level as I can solve simple ones, but I have trouble when I need to solve questions like z^5 = 12exp(i*pi/2) which shouldn't be too hard since it is already in polar form.
    It would be greatly appreciated someone could give me a quick rundown on how to properly solve these types of questions or point me a good resource on them.

    Edit: An example of what I am having trouble with:

    z^2 = -7*sqrt(2)/2 + 7*sqrt(2)*i/2

    Using the form z = r*exp(a*i), I find r to be 7 and a to be pi/4

    So z = sqrt(7)exp(pi*n*i/8)

    This is where I get stuck. The examples we did in class only involved then plugging 0, 1, 2, 3, etc until the results looped. However, the awnsers provided for this question are sqrt(7)exp(3*pi*i/8) and sqrt(7)exp(11*pi*i/8). What should I be doing instead?
    There are exactly 2 distinct square roots, so plugging in 0 and 1 will produce both and beyond that they will just cycle through the two you already have.

    Also:

    z^2=7e^{i(3\pi/4+2n\pi)}

    (that is \arg(z)=3\pi/4 )so:

    z=\sqrt{7}e^{i(3\pi/4+2n\pi)/2}=\sqrt{7}e^{i(\pi/8)(3+8n)}

    Now try plugging in values of n

    CB
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