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E to the pi * i

  1. Apr 15, 2010 #1

    1. Compute all the values of [tex] e^ {\pi i} [/tex], indicating clearly whether there is just one or many of them.

    Trivially, exp(pi * i) = -1. However, we can also consider e to be the complex number z, and pi * i to be the complex number alpha. Then we get:

    [tex]e^{\pi i} = z^{\alpha} = e^{\alpha log(z)}
    = e^{\alpha (Log |z| + i arg(z))}
    = e^{\pi i (Log |e| + i arg(e))}
    = e^{\pi i (1 + i2k\pi)}
    = e^{\pi i}e^{-2\pi^{2}k}
    = - e^{-2\pi^{2}k}
    where k is an integer.

    So what exactly is going on here? does exp(pi*i) = -1 or -exp(-2kpi^2)??

    P.S. I hope all this tex doesn't mess up :(
  2. jcsd
  3. Apr 15, 2010 #2
    something is wrong with LaTeX... it isn't displaying my tex right...
  4. Apr 15, 2010 #3


    Staff: Mentor

    Fixed your LaTeX.
  5. Apr 15, 2010 #4
    Is the complex exponential function invertible? (What is required for a function to have an inverse?)
  6. Apr 16, 2010 #5
    The function must be 1-1, right?
  7. Apr 16, 2010 #6
    Correct. Does the complex exponential satisfy this?
  8. Apr 16, 2010 #7
    Sorry, I misread your first question. So, no, the complex exponential is not an invertible function. Where does my initial post break down then?
  9. Apr 16, 2010 #8
    When you tried to invert it.
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