1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Complex numbers identity

  1. May 22, 2012 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    I'm trying to follow some solution to an exercise in physics and apparently [itex]e^{-im \frac{3\pi}{2}}=i^m[/itex] where [itex]m \in \mathbb{Z}[/itex].
    I don't realize why this is true.

    2. Relevant equations
    Euler's formula.

    3. The attempt at a solution
    I applied Euler's formula but this is still a mistery.
    [itex]i^m=\cos \left ( \frac{3\pi m}{2} \right ) -i \sin \left ( \frac{3\pi m }{2} \right )[/itex].
    I've checked the formula for m=1 and 2, it works. I must be missing the obvious, but I'm very tired physically and mentally.
    Thanks for any help.

    Edit: I found it. I drew a mental sketch of [itex]e^{-i \frac{3\pi }{2}}[/itex], it's "i" in the complex plane. Then just elevate this to the power m and the job is done.
  2. jcsd
  3. May 23, 2012 #2
    What you can do is to use
    [tex]i^m = e^{\ln(i^m)} = e^{m \ln i} [/tex]
    and then calculate what is [itex] \ln i [/itex]
  4. May 23, 2012 #3


    User Avatar
    Science Advisor

    [itex]e^{im\frac{-3\pi}{2}}= \left(e^{i\frac{-3\pi}{2}}\right)^m[/itex]

    And, of course, [itex]e^{i\frac{-3\pi}{2}}= i[/itex] so that expression is just [itex]i^m[/itex].

    If you are not clear that [itex]e^{i\frac{3\pi}{2}}= i[/itex], recall that [itex]e^{i\theta}[/itex], for any real [itex]\theta[/itex], lies on the unit circle ([itex]|e^{i\theta}|= 1[/itex] at angle [itex]\theta[/itex] measured counter clockwise from the positive real axis.
    [itex]e^{i\frac{-3\pi}{2}}[/itex] lies on the unit circle, an angle [itex]3\pi/2[/itex] measured clockwise from the positive real axis.

    Another way to see this is to recall that [itex]x^{-1}= 1/x[/itex] and that [itex]1/i= -i[/itex].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook