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Math Help - Analytic in a Domain D

  1. #1
    Junior Member universalsandbox's Avatar
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    Analytic in a Domain D

    if f(z) is analytic/holomorphic in a Domain D, prove f(z) must be constant in that domain if any of these are also constant in D:

    Im(f(z)), Re(f(z)), |f(z)|

    So, f(x + iy) = u(x, y) + iv(x, y)
    u, and v have first partial derivatives and satisfy Cauchy-Riemann equations:

    \frac{du}{dx} = \frac{dv}{dy}
    \frac{du}{dy} = -\frac{dv}{dx}

    How can you go about showing this? Thanks.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by universalsandbox View Post
    if f(z) is analytic/holomorphic in a Domain D, prove f(z) must be constant in that domain if any of these are also constant in D:

    Im(f(z)), Re(f(z)), |f(z)|

    So, f(x + iy) = u(x, y) + iv(x, y)
    u, and v have first partial derivatives and satisfy Cauchy-Riemann equations:

    \frac{du}{dx} = \frac{dv}{dy}
    \frac{du}{dy} = -\frac{dv}{dx}

    How can you go about showing this? Thanks.
    Here, v is Im(f(z)) so if it is constant, the right side of each of those two equations is 0 and we have \frac{\partial u}{\partial x}= 0 and \frac{\partial u}{\partial y}= 0, the Cauchy-Riemann equations. In other words, that both u and v are constant.

    Similarly, if Re(f(z)) is constant, so is u and then you can prove that v is constant.

    Finally, if |f|= (u^2+ v^2)^{1/2} is constant, so u^2+ v^2 is constant and we have 2u\frac{\partial u}{\partial x}+ 2v\frac{\partial v}{\partial y}= 0 and 2u\frac{\partial u}{\partial y}+2v\frac{\partial v}{\partial y}= 0. Use those two equations together with the two Cauchy-Riemann equations to show that the four derivatives \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, and \frac{\partial v}{\partial y} are all 0.
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