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Math Help - How To Do Integral Of Complex Conjugate?

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    How To Do Integral Of Complex Conjugate?

    \int\limits_{1+i}^{2+i}s\, dx

    Where s is the complex conjugate of x

    I have the answer but I don't have the foggiest idea how it is obtained.

    answer: [(2^2 -1 )/2] -i
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  2. #2
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    Quote Originally Posted by soma View Post
    \int\limits_{1+i}^{2+i}s\, dx

    Where s is the complex conjugate of x

    I have the answer but I don't have the foggiest idea how it is obtained.

    answer: [(2^2 -1 )/2] -i
    Define x = e^{i\theta}.

    Then \overline{x} = e^{-i\theta}

    and dx = i\,e^{i\theta}.

    Also note that \theta = \frac{\log{x}}{i}.


    So the integral becomes

    \int{\overline{x}\,dx} = \int{e^{-i\theta}\,i\,e^{i\theta}\,d\theta}

     = \int{i\,d\theta}

     = i\theta + C

     = i\left(\frac{\log{x}}{i}\right) + C

     = \log{x} + C.


    Now you should be able to substitute the limits of integration.

    Remember that \log{x} = \ln{|x|} + i\,\textrm{arg}\,{x}.
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