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Math Help - show harmonic function can be the real part of an analytic one

  1. #1
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    show harmonic function can be the real part of an analytic one

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  2. #2
    Member Mentia's Avatar
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    If we can assume that the Phi(x,y) is a real valued function then this is what you do:

    Apply the Cauchy-Riemann equations and show that F is analytic. Use the chain rule:

    \frac{\partial }{\partial x}f=2\left ( \frac{\partial \phi ((z+1)/2,(z-1)/2i) }{\partial ((z+1)/2)}\left ( \frac{\partial (z+1)/2}{\partial x} \right ) \right )+
    2\left ( \frac{\partial \phi ((z+1)/2,(z-1)/2i) }{\partial ((z-1)/2i)}\left ( \frac{\partial (z-1)/2i}{\partial x} \right ) \right )+...

    and remember that z = x + iy

    You will find that F is indeed analytic.
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