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Raising a complex number to the nth power

  1. Jul 16, 2008 #1
    I was looking around a little bit for an algorithm that would compute a complex number to the nth power.

    Can anyone supply me a resource that covers this? I wouldn't imagine it being different than some sort of (x+y)^n formula.

    Thanks in advance.
     
  2. jcsd
  3. Jul 16, 2008 #2
    You can convert the complex number into either polar form or exponential form and then use De Moivre's theorem.
     
  4. Jul 16, 2008 #3

    HallsofIvy

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    DeMoivre's theorem:
    If [itex]z= r e{i\theta}[/itex], then [itex]z^n= r^ne^{i n\theta}[/itex].

    r is the "modulus" or absolute value of z: if z= x+iy then [itex]|z|= \sqrt{x^2+ y^2}[/itex].

    [itex]\theta[/itex] is the "argument" or angle the line through 0 and z in the complex plane makes with the real-axis: if z= x+ iy, then [itex]\theta= arctan(y/x)[/itex].
     
  5. Jul 16, 2008 #4
    @Magda:
    Here's a question for you to ponder: if you take a complex number, z, that has length 1 (a^2 + b^2 = 1 for z = a+bi), then what happens when you keep raising it to higher and higher powers: z^1, z^2, z^3, ... z^100000, ...
     
  6. Jul 16, 2008 #5
    Thank you all very much. I was not familiar with DeMoivre's Theorem at all. I have seen Euler's identity, however.

    In response to maze, I have no idea what happens when you raise a complex number, z = length 1 to higher and higher powers. I can't even begin to make an assumption of what would happen.

    I'm not smart. :/
     
  7. Jul 16, 2008 #6
    Try some examples!

    Here are the most obvious ones:
    1 1 1 1 1 1 1 1 ...
    i -1 -i 1 i -1 ...

    Here are some for you to try:
    1/sqrt(2) + i/sqrt(2)
    -1/2 + i*sqrt(3)/2
     
  8. Jul 17, 2008 #7

    HallsofIvy

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    It's not a matter of being smart, it's a matter of having specific knowledge. It is true, generally, that |xn|= |x|n. In particular, if |z|= 1 then every power of z will also have absolute value 1. In the complex plane, the absolute value of a number is its distance from the origin. Every number with absolute value 1 lies on the unit circle. If z is on the unit circle, the so is zn for all n, although they may move around the unit circle.
     
  9. Jul 17, 2008 #8
    was hoping he would figure this out on his own....
     
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