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Need a quick help with a simple identity

  1. May 26, 2006 #1
    Hi
    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

    HallsofIvy

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    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

    HallsofIvy

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    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: May 26, 2006
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