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Math Help - harmonic v(x,y)

  1. #1
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    harmonic v(x,y)

    Hi, can anyone help me prove the following.

    Suppose that u(x,y) and v(x,y) are smooth functions satisfying the Cauchy-Riemann equations. Prove that v(x,y) is harmonic.

    thank you
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  2. #2
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    Quote Originally Posted by pasleycakes View Post
    Hi, can anyone help me prove the following.

    Suppose that u(x,y) and v(x,y) are smooth functions satisfying the Cauchy-Riemann equations. Prove that v(x,y) is harmonic.

    thank you
    Yes, but the important question is, "Can you prove it?"
    The Cauchy-Riemann equations are:
    (1) \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}
    (2) \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}.

    Differentiate (1) with respect to y and (2) with respect to x.

    Remember that, as long as the second derivatives are continuous,
    \frac{\partial f}{\partial x\partial y}= \frac{\partial f}{\partial y\partial x}.
    Last edited by HallsofIvy; May 31st 2010 at 05:08 AM.
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