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Math Help - real part of a function

  1. #1
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    real part of a function

    hi, this is a stupid question but is the real part of the function

     f(z) = e^{iz}


     cos z ?
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  2. #2
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    Quote Originally Posted by Adre5 View Post
    hi, this is a stupid question but is the real part of the function

     f(z) = e^{iz}


     cos z ?
    No, it's not.

    Remember that z is a complex number.

    In other words, z = x + iy.


    You need to write f(z) = e^{iz} as u + iv, where u, v are real functions of x and y.


    So f(z) = e^{iz}

     = e^{i(x + iy)}

     = e^{ix + i^2y}

     = e^{-y + ix}

     = e^{-y}e^{ix}

     = e^{-y}(\cos{x} + i\sin{x}) by Euler's Rule

     = e^{-y}\cos{x} + i\,e^{-y}\sin{x}.

     = u + iv.


    So the real part of e^{iz} is e^{-y}\cos{x} and the imaginary part of e^{iz} is e^{-y}\sin{x}.
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  3. #3
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    ohh thank you!! you made it so clear for me
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