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Math Help - complex number calculation

  1. #1
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    complex number calculation

    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. 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?????
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  2. #2
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    Re: complex number calculation

    Quote Originally Posted by cummings123321 View Post
    Evaluate {(1+i)^(1-i)} ?

    (1+i)^{(1-i)}=\exp((1-i)\log(1+i)) where \log(1+i)=\ln \left( {\sqrt 2 } \right) + i\left( {\frac{{ - \pi }}{4} + 2k\pi } \right).

    Frankly, I have no idea what the rest of that question could mean.

    1^x=1 if x is real and 1^z=\exp(2k\pi i)z) if z is complex.
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  3. #3
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    Re: complex number calculation

    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 by cummings123321; November 28th 2012 at 02:16 PM.
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  4. #4
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    Re: complex number calculation

    Quote Originally Posted by cummings123321 View Post
    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..

    I find this a pointless question. Allow me to show you why.

    Define f(x,k)=\exp(2k\pi x i).

    f\left( {\frac{2}{3},6} \right) = 1

    f\left( {\frac{2}{3},5} \right) = -0.5+0.866i

    f\left( {\sqrt{2},5} \right) = 0.902+0.432i

    f\left( {\sqrt{2},6} \right) = -0.996+0.092i

    Given those results, can you see why I might think this is a pointless question?
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  5. #5
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    Re: complex number calculation

    i might think thats for rational 1^(p/q) just the q roots of 1 but irrational thats infinite number cause 1^x=exp(2ikpix) then i is infinite many to be chosen
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