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Complex exponentiation

  1. Jun 25, 2011 #1
    Suppose we have the product

    [tex][(\pm ia) (\pm ib)]^{-\alpha}[/tex]
    where[itex]a, b, \alpha >0[/itex]. For which of the combinations (+,+), (+,-), (-,+), and (-,-) is the following property satisfied?

    [tex][(\pm ia) (\pm ib)]^{-\alpha}=(\pm ia)^{-\alpha} (\pm ib)^{-\alpha}[/tex]
     
  2. jcsd
  3. Jun 25, 2011 #2

    micromass

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    Hi Bruno67! :smile:

    Using the definition of exponentiation we have that

    [tex][(\pm ia)(pm ib)]^{-\alpha}=e^{-\alpha Log((\pm ia)(\pm ib))}[/tex]

    So the question becomes when

    [tex]Log((\pm ia)(\pm ib))=Log(\pm ia)+Log(\pm ib)[/tex]

    Solve this using the definition of the logarithm.
     
  4. Jun 25, 2011 #3
    Thanks, so it holds in all cases except the (-,-) one. In that case we have

    [tex][(-ia) (-ib)]^\alpha = (-ia)^\alpha (-ib)^\alpha (-1)^{2\alpha}.[/tex]
     
  5. Jun 25, 2011 #4

    micromass

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    Indeed!:smile:
     
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