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Complex number calculation

  1. Nov 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Evaluate {(1+i)^(1-i)} and describe the set{1^x} when x is a real number, distinguish between the cases when x is rational and when x is rational. for now considering the complex number.

    2. The attempt at a solution
    i dont know how to start with,for firest part i just write it into e^((1-i)log(1+i)) then get the number with e to the power which inculding i , and the secound part , for 1=e^(i2npi) then 1^x is e^(2inxpi) then how to consider the case for rational and irrational here?????
     
  2. jcsd
  3. Nov 28, 2012 #2

    jedishrfu

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    start by breaking the factor into two factors (1+i)^1 * (1+i)^(-i)
     
  4. Nov 28, 2012 #3
    i can do the first part now, many thanks ,but i dont understand the secound part of the question.should it be if 1=exp(2ikpi) then 1^x= exp(2ikpix) then consider the rational and irrational case on this form. but whats the differences , i have no idea
     
    Last edited: Nov 28, 2012
  5. Nov 28, 2012 #4

    haruspex

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    Yes. Think about whether different values for k can produce the same value.
     
  6. Nov 29, 2012 #5
    if rational, then 1^(p/q) is the q complex roots of 1, if x is irrational then 1^x=exp(2ikpix) then k can be chosen infinitely many values then there are infinite points
     
  7. Nov 29, 2012 #6

    haruspex

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    That's it.
     
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