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What is the definition of a^i?

  1. Dec 11, 2004 #1
    (1)

    what is the definition of a^i ? (where a is a real number , i = (-1)^(1/2) )

    is this still a complex number ?




    (2)

    where can I find the prove of "a n degree polynomial has n roots" ??
     
  2. jcsd
  3. Dec 11, 2004 #2
    1) a^i =e^(i(log(a)))=cos(log(a)) + isin(log(a))

    2) A proof of the fundamental theorem of algebra, first formulated by Gauss is very difficult. A simpler proof can be found by the time you get into the study of Complex Functions, generally a Junior college study.
     
  4. Dec 12, 2004 #3
    thank you!

    then , is i^i = e^[(i)*(pi/2)*(i)] = e^(-pi/2) ??
     
  5. Dec 12, 2004 #4
    Yes, it is.
     
  6. Dec 12, 2004 #5

    HallsofIvy

    User Avatar
    Science Advisor

    With the proviso that ai is, like most complex functions, not single valued.
     
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