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Complex powers/logarithmic spirals

  1. Oct 22, 2007 #1
    When looking at

    w^z = e^(z log w)

    I understand that adding any integer multiple of (2*pi*i) to log w is equivalent to a full rotation in the complex plane. I don't understand how this step is equivalent to multiplying w^z by e^(z*2*pi*i). Also, I'm missing the significance of this being represented in the complex plane as the intersections of 2 logarithmic spirals. I can see how the first spiral is given by w^z, but the other?

    If anyone has a copy handy, my questions arose from looking at pages 96-97 of The Road to Reality by Roger Penrose.
  2. jcsd
  3. Oct 22, 2007 #2
    I think it is just rules of exponents from algebra.

    (w^z)*e^(z*2*pi*i) = e^(z log w)*e^(z*2*pi*i) = e^(z log w + z*2*pi*i) =
    e^(z(log w + 2*pi*i))

    Does this help?
  4. Oct 22, 2007 #3
    click! thanks for that diffy. hopefully those spirals will start to do the same now...:smile:
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