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: Need a quick help with a simple identity

  1. May 26, 2006 #1
    Where is the error is this 'identity'?:

    [tex](-e^i)^{\frac{1}{2}} = (-1)^{\frac{1}{2}}*e^{\frac{i}{2}}[/tex]

    My calculator says that the right side is minus one times the left but I can't see the mistake I'v made. Help me please, thanks.
  2. jcsd
  3. May 26, 2006 #2


    User Avatar
    Science Advisor

    What happened to the "i" on the left? I assume that was a typo and you mean [itex](-e^i)^{\frac{1}{2}} = (-1)^{\frac{1}{2}}*e^{\frac{i}{2}}[/itex]

    You have to be careful about apply "laws of exponents" to complex numbers- we all remember the "proof" that 1= -1:
    [tex]1= ((-1)(-1))^\frac{1}{2}= (-1)^\frac{1}{2}(-1)^\frac{1}{2}= (i)(i)= -1[/tex]
  4. May 26, 2006 #3
    - I am not sure which 'i' (and 'typo') you are reffering to, your 'indentity' is, as far as I can say, identical to mine.

    - So make this clear for me: when can I use the fact that [tex](A*B)^x = A^x*B^x[/tex] when A,B are complex and x real? (I have little knowledge of complex analysis)

    I need to get [tex](-e^i)^x = something * e^{ix}[/tex] but I am not sure what the 'something' should be, obviously it is not [tex](-1)^x[/tex], could you please help me?

    Thanks, H.
  5. May 26, 2006 #4


    User Avatar
    Science Advisor

    Okay, I didn't see that you had "i/2" rather than "1/2". Maybe I need to have my eyes checked!

    Actually it is but (-1)x, like most complex valued functions, is multi-valued. You need to state which value you are using.

    In your particular case, [itex](-1)^\frac{1}{2}[/itex] has two values: i and -i.
    Last edited by a moderator: May 26, 2006
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook