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What is x^i?

  1. Dec 8, 2011 #1
    Very simple question... What is x^i? How can you rewrite it?
    All I could figure out is that (x^i)^i = 1/x, but that doesn't help much
    Wolfram Alpha gave me this graph (real part in blue, imaginary in orange)
    http://www4c.wolframalpha.com/Calculate/MSP/MSP17119i95eid65h0gce900001e7b96h101dd87d6?MSPStoreType=image/gif&s=62&w=320&h=119&cdf=RangeControl [Broken]
    Which is a very strange graph.

    What happens?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Dec 8, 2011 #2

    Simon Bridge

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    Re: x^i

    It is probably clearer if you look at it in the complex plane.
    Apart from that, what happens is exactly what the graph says happens.

    consider:
    [itex]e^{i\theta}[/itex] is just the unit vector rotated anti-clockwise in the complex plane by [itex]\theta[/itex] radiens.

    [itex]a^b = e^{b\ln{a}}[/itex] so [itex]x^i = e^{i\ln{x}}[/itex] so [itex]x^i[/itex] is the unit vector rotated by ln(x) radiens in the complex plane.
     
    Last edited: Dec 8, 2011
  4. Dec 8, 2011 #3
    Re: x^i

    Okay, so wolfram alpha says that 3^i is about 0.455 + 0.890i
    How did it figure that out?
     
  5. Dec 8, 2011 #4

    Simon Bridge

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    Re: x^i

    Ah - you posted while I edited: that's a bad habit of mine.
    It's a rotation in the complex plane.
    The real part is the cos(ln(x)) and the imaginary part is sin(ln(x))
     
    Last edited: Dec 8, 2011
  6. Dec 8, 2011 #5
    Re: x^i

    That made so much more sense than I expected it to.
    It also explains this graph of y=Re(x^i)^2+Im(x^i)^2
    http://www4b.wolframalpha.com/Calculate/MSP/MSP237219i95h4480ahf33i00001h6c277de8811fe7?MSPStoreType=image/gif&s=34&w=307&h=136&cdf=RangeControl [Broken]
    Friggin' beautiful.
     
    Last edited by a moderator: May 5, 2017
  7. Dec 8, 2011 #6

    Simon Bridge

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    Re: x^i

    Yep - when you get used to rotating phasors lots of things get simple.
    I dredged up a link for you. It covers the whole imaginary exponent thing (like what happens when you raise a complex number to the power of another complex number) then links to a bunch of applications.

    It's also used in analyzing linear networks (electronics) and anything with waves.
     
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