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If n=complex number what values of n in i^n real?

  1. Nov 2, 2009 #1
    would in = an infinite amount of real possibilites if n is complex. considering that 1+0i is still a complex number, or is that wrong?
     
  2. jcsd
  3. Nov 2, 2009 #2
    For one example, ii is real. In fact,

    [tex](it)^{it}=e^{-t\pi/2}[\cos(t\ln t) + i\sin(t\ln t)][/tex]

    is real for t ln(t) = n pi, n integer.
     
  4. Nov 4, 2009 #3

    lurflurf

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    i^n=exp(n log(i))
    For the principle branch take log(i)=i*pi/2
    i^n=exp(i*n*pi/2)
    when will that be real?
     
  5. Nov 4, 2009 #4
    i just asked if it would have an infinite amount of possibilites, obviously i have already thought about this and already know some examples of how it will be real. its a yes or no + justification response.

    im using De Moirve's theorm and i can already prove that i^i is real. and hence i^ai is real even if its complex and if n=ai+c it will be real if c is an even number or o.

    the other part of my question was, can i acurately say that a+0i is a complex number???
     
  6. Nov 4, 2009 #5
    Yes. The real numbers are a subset of the complex numbers.
     
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